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✓ 42 results · all verified · 12 also independently AI-judged

Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 30 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.

Codimension One Foliations, Secondary Classes and Characteristic Disk Foundations

1 · Prerequisites

2 · Summary

This page brings together two strands that meet on a codimension-one foliation: the secondary characteristic classes of the foliation itself, and the local planar and foliation-theoretic foundations on which the characteristic-disk arguments rest. The Bott partial connection is a derivation along leaf directions of the normal bundle of a foliation; it is well defined, flat along leaves, and an extending connection has curvature entries in the transverse differential ideal. On a transversely oriented codimension-one foliation the Godbillon-Vey form and class are built from a defining one-form and the secondary form $η$ with $d ω = η \land ω$ ; the class is independent of the choice of $η$ up to an exact form, invariant under rescaling the defining form, and invariant under smooth foliated concordance, while it vanishes for foliations defined by a closed one-form and for mapping-torus fibrations. The Bott vanishing theorem records which real Pontryagin monomials must vanish for a codimension- $q$ foliation. The second strand supplies the local analytic work: winding-number lemmas that are choice-free, a general-position statement for disk maps that makes the characteristic singularities finite, interior and nondegenerate, the index count one-more-center-than-saddle for disks with leafwise or everywhere-transverse boundary, the planar Poincare-Bendixson and saddle-graph carriers, the period-annulus product coordinates and their omega-limit realizations, leafwise cap products with exact collar data, and the limitwise-nullhomotopy subgroup of a leaf with its one-sided normal-subgroup property. The global partition-of-unity and foliation constructions carry countable choice. Bott vanishing also carries full choice through its connection-existence and real characteristic-class comparison suppliers. Local planar and winding arguments use finite or explicit constructions.

3 · Logical flowchart

4 · Definitions, theorems and proofs

LemmaStatement: Literature-sourcedProof: Literature-sourcedprecheck passOpen item page →

Divisibility by a nowhere-vanishing one-form

Statement

Assume Countable Choice ${A C}_{ω}$ . Let $M$ be a smooth manifold, with boundary allowed, and let $ω$ be a nowhere-vanishing smooth $1$ -form on $M$ . (i) If $α \in Ω^{1} (M)$ satisfies $α \land ω = 0$ , then there is a unique $f \in C^{\infty} (M)$ with $α = f ω$ . (ii) If $θ \in Ω^{2} (M)$ satisfies $θ \land ω = 0$ , then there is $β \in Ω^{1} (M)$ with $θ = β \land ω$ .

Facts & Assumptions

Given: A smooth manifold $M$ with a nowhere-vanishing smooth one-form $ω$ , a one-form $α$ with $α \land ω = 0$ , and a two-form $θ$ with $θ \land ω = 0$ .

[F1]

The graded vector space $Ω^{*} (M)$ with the wedge product is an associative graded-commutative algebra. (Differential forms form a graded commutative algebra).

[F2]

If $(f_{i})$ are smooth functions on the members of an open cover and $(ϕ_{i})$ is a smooth partition of unity subordinate to that cover, then $F (p) = \sum_{i} ϕ_{i} (p) f_{i} (p)$ is a smooth function on $M$ . (Smooth locally defined functions can be glued by a partition of unity).

[F3]

Under ${A C}_{ω}$ , every smooth-manifold open cover admits a subordinate smooth partition of unity (Smooth partitions of unity exist on manifolds, Smooth partitions of unity exist on manifolds with boundary).

Proof

technique · direct

1.1givenalgebra

Fix $p \in M$ and choose $v \in T_{p} M$ with $ω_{p} (v) \neq 0$ , which exists because $ω$ is nowhere vanishing; evaluating the two-form $α \land ω$ on a pair $(u, v)$ and using its definition for a one-form times a one-form gives $0 = (α \land ω) (u, v) = α_{p} (u) ω_{p} (v) - α_{p} (v) ω_{p} (u)$ for every $u$ , so $α_{p} (u) = (α_{p} (v) / ω_{p} (v)) ω_{p} (u)$ ; the quotient $f (p) : = α_{p} (v) / ω_{p} (v)$ is independent of the choice of $v$ because it equals the value of the unique scalar with $α_{p} = f (p) ω_{p}$ , and such a scalar is unique as $ω_{p} \neq 0$ .

2.1F1step 1.1

To see that $f$ is smooth, fix a chart domain $U$ with coordinates $x_{1}, \dots, x_{m}$ and write $ω = \sum_{i} ω_{i} d x_{i}$ , $α = \sum_{j} a_{j} d x_{j}$ with smooth coefficient functions; by [F1] the wedge product expands as $α \land ω = \sum_{j < k} (a_{j} ω_{k} - a_{k} ω_{j}) d x_{j} \land d x_{k}$ , and the $d x_{j} \land d x_{k}$ are linearly independent over the coefficient functions, so $a_{j} ω_{k} = a_{k} ω_{j}$ on $U$ ; shrinking about any point at which some $ω_{i} \neq 0$ , one gets $a_{j} = a_{i} ω_{j} / ω_{i}$ for all $j$ , hence $α = (a_{i} / ω_{i}) ω$ there with smooth quotient $a_{i} / ω_{i}$ ; comparing with step 1.1 shows $f = a_{i} / ω_{i}$ on that smaller domain, and since smoothness is local $f$ is smooth on all of $M$ , giving (i).

2.2F1step 1.1

For (ii), fix a chart domain $U$ with coordinates as above and write $θ = \sum_{i < j} c_{i j} d x_{i} \land d x_{j}$ with $c_{j i} : = - c_{i j}$ ; by [F1] the wedge $θ \land ω$ has coefficient $c_{i j} ω_{k} - c_{i k} ω_{j} + c_{j k} ω_{i}$ on $d x_{i} \land d x_{j} \land d x_{k}$ for each triple $i < j < k$ , so $θ \land ω = 0$ gives those three-term identities; shrinking to a domain on which a fixed coefficient $ω_{i_{0}}$ is nowhere zero, define $b_{i} : = c_{i i_{0}} / ω_{i_{0}}$ for $i \neq i_{0}$ and $b_{i_{0}} : = 0$ , and let $β = \sum_{i} b_{i} d x_{i}$ ; substituting these definitions into the three-term identities in each of the three index orders, and using $c_{i j} = - c_{j i}$ , gives $c_{i j} = b_{i} ω_{j} - b_{j} ω_{i}$ for every pair ${i, j}$ , hence $θ = β \land ω$ on $U$ with smooth $β$ .

3.1F2F3step 2.2

Cover $M$ by such chart domains $U$ with local solutions $β_{U}$ , and use [F3] to obtain a smooth partition of unity $(ϕ_{U})$ subordinate to the cover; for overlapping domains, $(β_{U} - β_{V}) \land ω = 0$ , so by (i) there is a smooth function $f_{U V}$ with $β_{U} - β_{V} = f_{U V} ω$ on the overlap, and the local forms $ϕ_{U} β_{U}$ , extended by zero, satisfy $\sum_{U} ϕ_{U} β_{U} \land ω = \sum_{U} ϕ_{U} θ = θ$ ; thus $β : = \sum_{U} ϕ_{U} β_{U}$ is a globally defined smooth one-form by [F2] with $θ = β \land ω$ , which is (ii).

4.1F3step 1.1step 2.1step 3.1∎

Part (i) follows from steps 1.1 and 2.1, and part (ii) from step 3.1. Countable choice is used for the subordinate partition in [F3]; the local coefficients are explicit formulas in each chart.

LemmaStatement: Literature-sourcedProof: Literature-sourcedprecheck passjudge pass (gpt-6.1-sol)Open item page →

Restriction of a foliation transverse to the boundary

Statement

Let $W$ be a smooth $(n + 1)$ -manifold with boundary and let $F$ be a codimension-one regular foliation of $W$ transverse to $\partial W$ , i.e. $T_{p} (\partial W) + T_{p} F = T_{p} W$ for every $p \in \partial W$ . Let $ω$ be a nowhere-vanishing smooth defining $1$ -form for $F$ . Then: (i) $T_{p} (\partial W) \cap T_{p} F$ has dimension $n - 1$ for every $p \in \partial W$ , and these subspaces form a codimension-one regular foliation $F ∣_{\partial W}$ of the smooth $n$ -manifold $\partial W$ ; (ii) the restriction $ω ∣_{\partial W}$ is nowhere vanishing and defines $F ∣_{\partial W}$ , with $ω ∣_{\partial W} \land d (ω ∣_{\partial W}) = 0$ ; (iii) every leaf of $F ∣_{\partial W}$ is a connected component of $L \cap \partial W$ for a leaf $L$ of $F$ , with the intersection taken in the intrinsic leaf topology; (iv) if $F$ is transversely oriented by $ω$ , then $F ∣_{\partial W}$ is transversely oriented by $ω ∣_{\partial W}$ ; (v) if $η$ is a $1$ -form on $W$ with $d ω = η \land ω$ , then the pullback of $η$ to $\partial W$ satisfies $d (ω ∣_{\partial W}) = (η ∣_{\partial W}) \land (ω ∣_{\partial W})$ .

Facts & Assumptions

Given: A smooth $(n + 1)$ -manifold $W$ with boundary, a codimension-one regular foliation $F$ of $W$ transverse to $\partial W$ , and a nowhere-vanishing smooth defining one-form $ω$ for $F$ .

[F1]

For a smooth map of manifolds, pullback sends smooth forms to smooth forms, is functorial, and satisfies $F^{*} (α \land β) = F^{*} α \land F^{*} β$ (Pullback of forms is smooth functorial and preserves wedges, and for boundary manifolds The de Rham complex and pullback extend to manifolds with boundary).

[F2]

For every smooth map $F$ and every form $ω$ on the target, $d (F^{*} ω) = F^{*} (d ω)$ (The exterior derivative commutes with pullback, and for boundary manifolds The de Rham complex and pullback extend to manifolds with boundary).

[F3]

For a nowhere-zero one-form $α$ , the hyperplane distribution $\ker α$ is integrable if and only if $α \land d α = 0$ (The codimension-one Frobenius criterion).

[F4]

For $p \in \partial M$ of a manifold with boundary and the inclusion $i : \partial M ↪ M$ , the differential $d i_{p}$ identifies $T_{p} \partial M$ with the hyperplane of boundary-tangent vectors in $T_{p} M$ (The boundary tangent space is the boundary-tangent hyperplane).

[F5]

On a manifold, an integrable rank- $k$ distribution defines a regular foliation atlas whose leaves are its maximal connected integral manifolds (Regular foliations and integrable distributions correspond).

Proof

technique · direct

1.1givenF4

Write $i : \partial W ↪ W$ for the inclusion and fix $p \in \partial W$ . By [F4] the subspace $T_{p} \partial W$ sits inside $T_{p} W$ as a hyperplane, and the transversality hypothesis reads $T_{p} \partial W + T_{p} F = T_{p} W$ ; since $ω$ defines $F$ , moreover $T_{p} F = \ker ω_{p}$ .

2.1step 1.1

If $ω_{p}$ vanished on all of $T_{p} \partial W$ , then $T_{p} \partial W \subseteq \ker ω_{p} = T_{p} F$ , so the sum $T_{p} \partial W + T_{p} F = T_{p} F$ would have dimension $n$ instead of $n + 1$ ; hence $ω ∣_{\partial W}$ is nowhere vanishing, the restricted one-form has $\ker (ω ∣_{\partial W})_{p} = T_{p} \partial W \cap T_{p} F$ as its kernel, and the dimension formula for two hyperplanes with sum $T_{p} W$ gives $\dim (T_{p} \partial W \cap T_{p} F) = n - 1$ , which is the dimension count of (i) and the kernel description of (ii).

3.1F1F2F3F5step 2.1

Pulling back $ω \land d ω = 0$ along $i$ with [F1] and [F2] gives $(ω ∣_{\partial W}) \land d (ω ∣_{\partial W}) = 0$ ; since $ω ∣_{\partial W}$ is nowhere vanishing by step 2.1, [F3] makes its kernel an integrable hyperplane distribution on the smooth $n$ -manifold $\partial W$ , and [F5] turns that distribution into a codimension-one regular foliation $F ∣_{\partial W}$ defined by $ω ∣_{\partial W}$ , completing (i) and (ii).

3.2step 2.1

A defining form that orients $F$ transversely restricts to the nowhere vanishing form $ω ∣_{\partial W}$ of step 2.1, whose kernel is the restricted distribution, so the restricted foliation is transversely oriented by $ω ∣_{\partial W}$ ; this is (iv).

4.1step 2.1step 3.1

Fix a leaf $L^{'}$ of $F ∣_{\partial W}$ and $p \in L^{'}$ . As a connected manifold tangent to $T F$ and contained in $\partial W$ , the leaf $L^{'}$ lies in a leaf $L$ of $F$ and, being connected, in the intrinsic component $C$ of $L \cap \partial W$ containing $p$ . Conversely, at a point $q$ of $C$ a boundary chart with $\partial W = {t = 0}$ and a foliation chart for $F$ present $L$ locally as a level set ${y = y_{0}}$ , and because $T_{q} L = T_{q} F$ and $T_{q} \partial W$ are transverse the functions $y$ and $t$ have independent differentials at $q$ ; hence $L \cap \partial W$ is near $q$ an integral manifold of $\ker (ω ∣_{\partial W})$ of dimension $n - 1$ , that is, a plaque of the restricted foliation, and $C$ is covered by such plaques. The set of points of $C$ lying in the leaf $L^{'}$ is then both open and closed in $C$ and nonempty, so it equals $C$ ; therefore $L^{'} = C$ , which is (iii).

5.1F1F2step 3.1step 4.1∎

Finally, pulling back the identity $d ω = η \land ω$ along $i$ and applying [F1] and [F2] gives $d (ω ∣_{\partial W}) = i^{*} (d ω) = (i^{*} η) \land (i^{*} ω) = (η ∣_{\partial W}) \land (ω ∣_{\partial W})$ , which is (v); together with steps 2.1, 3.1, 3.2 and 4.1 this proves all five assertions.

DefinitionDefinition: Literature-sourcedProof: Not applicableOpen item page →

The Bott partial connection on the normal bundle of a foliation

Definition

Assume Countable Choice ${A C}_{ω}$ . Let $F$ be a codimension- $q$ regular foliation of a smooth manifold $M$ , with tangent distribution $E = T F$ , and let $ν : = T M / E$ be the normal bundle of $F$ (Quotient vector bundles by a subbundle, A vector bundle quotient by a subbundle is a smooth vector bundle). For a leaf-tangent vector field $X \in Γ (E)$ and a section $s \in Γ (ν)$ the Bott partial connection is

$\nabla_{X}^{B} s : = π [X, \tilde{s}],$

where on each bundle chart $\tilde{s}$ is a smooth local representative of $s$ and $π : T M \to ν$ is the quotient map (The canonical map to a quotient bundle is a smooth bundle map). Local representatives exist by lifting the components of $s$ in a quotient-bundle frame; their projected brackets agree on overlaps by the next lemma and therefore define a global section. The derivation is along leaf directions only: $X$ is a section of the involutive distribution $E$ (Regular foliation atlases).

This defines a map $\nabla^{B} : Γ (E) \times Γ (ν) \to Γ (ν)$ that is $C^{\infty} (M)$ -linear in the vector-field variable $X$ , $R$ -linear in $s$ , and satisfies the Leibniz rule $\nabla_{X}^{B} (f s) = X (f) s + f \nabla_{X}^{B} s$ for $f \in C^{\infty} (M)$ . The well-definedness of the formula and its flatness along each leaf are established in the next result. In the terminology of this page $\nabla^{B}$ is a partial connection along the leaves; it is not a connection on all of $T M$ , and no splitting of $T M \to ν$ is chosen.

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

The winding number jumps by one across a regular planar arc

Statement

Let $Γ$ be an oriented closed piecewise- $C^{1}$ contour. Suppose that near $0$ it contains exactly one regular $C^{1}$ arc, traversed once with positive real tangent, and that the remaining contour is a compact set disjoint from $0$ . Then for all sufficiently small $ε > 0$ , the points $i ε$ and $- i ε$ avoid $Γ$ and $n (Γ, i ε) - n (Γ, - i ε) = 1$ .

Facts & Assumptions

Given: An oriented closed piecewise- $C^{1}$ contour $Γ$ that near $0$ contains exactly one regular $C^{1}$ arc, traversed once with positive real tangent, the remaining part of the contour being compact and disjoint from $0$ .

[F1]

For a closed complex contour $γ$ and a point $p$ off its trace, $n (γ, p) = \frac{1}{2 π i} \int_{γ} \frac{d z}{z - p}$ . (The winding number of a closed contour about a point off its trace).

[F2]

For continuous $f, g$ on the trace of a rectifiable contour $γ$ and $α, β \in C$ , $\int_{γ} (α f + β g) d z = α \int_{γ} f d z + β \int_{γ} g d z$ . (Complex line integrals are linear in the integrand).

[F3]

Complex line integrals over piecewise- $C^{1}$ paths are unchanged by an orientation-preserving piecewise- $C^{1}$ reparametrization. (Scalar line integrals are parametrization-independent; vector line integrals retain orientation and change sign when it reverses).

[F4]

If $∣ f (z) ∣ \leq M$ on the trace of a rectifiable contour $γ$ with $M \geq 0$ , then $∣ \int_{γ} f (z) d z ∣ \leq M L (γ)$ . (ML estimate: a contour integral is bounded by a supremum bound times path length).

[F5]

For a closed complex contour $γ$ and a point $p$ off its trace, $n (γ, p) \in Z$ . (The winding number of a closed contour is an integer).

[F6]

For real $a < b$ , a real-valued function continuous on $[a, b]$ and differentiable on $(a, b)$ satisfies $f (b) - f (a) = f^{'} (c) (b - a)$ for some $c \in (a, b)$ . (The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ ).

[F7]

A continuous real function on a connected space has order-convex image and attains every intermediate value. (A real-valued continuous map on a connected space has order-convex image, so it takes every value between any two of its values).

[F8]

For every real $x$ , $\frac{d}{d x} \arctan x = \frac{1}{1 + x^{2}}$ . (Principal arctangent: derivative, integral, power series, and the Gregory–Leibniz series).

Proof

technique · direct

1.1given

Write the regular arc near $0$ as $z (t) = x (t) + i y (t)$ on a parameter interval $[- δ, δ]$ , with $t = 0$ corresponding to $0$ , so $x (0) = y (0) = 0$ and after an orientation-preserving affine change of parameter the positive real tangent gives $x^{'} (0) = 1$ , $y^{'} (0) = 0$ .

2.1F6F7step 1.1

Shrink $δ$ so that $x^{'} \geq 1 / 2$ on the interval; then [F6] makes the real part strictly increasing, [F7] shows its image contains a symmetric interval $[- a, a]$ about zero; restrict the arc to the preimage of this interval, and the inverse $x \mapsto t (x)$ is $C^{1}$ with derivative $1 / x^{'} (t (x))$ by the difference quotient and the positive lower bound; hence the arc is a graph $z (x) = x + i f (x)$ for $∣ x ∣ \leq a$ with $f (0) = f^{'} (0) = 0$ , and after shrinking $a$ further one has $∣ z^{'} (x) ∣ \leq 2$ and $∣ f (x) ∣ \leq η ∣ x ∣$ for a fixed $0 < η < 1 / 2$ .

2.2F4step 1.1

The parameter pieces outside the local arc form a compact set disjoint from $0$ ; for each parameter $t$ in it continuity of the contour gives a relative interval on which $∣ z ∣$ exceeds half its positive value at $t$ , the family of all these intervals covers the compact parameter set, so finitely many cover it, and the minimum of the finitely many positive half-values is a number $d > 0$ with $∣ z ∣ \geq d$ on the remainder.

3.1step 2.1step 2.2

If $ε < d / 2$ then $∣ z ∣ \geq d > ε$ on the remainder, so $i ε$ and $- i ε$ avoid the remainder, and they avoid the arc because its real coordinate vanishes only at $x = 0$ , where $z (0) = 0$ ; hence both points lie off the trace of $Γ$ .

4.1F1F2F3step 3.1

By the winding definition, linearity and orientation-preserving reparametrization, $n (Γ, i ε) - n (Γ, - i ε) = \frac{1}{2 π i} \int_{Γ} \frac{2 i ε}{z^{2} + ε^{2}} d z$ , the integrand being continuous on the trace of $Γ$ for these values of $ε$ .

5.1F4step 4.1

On the remainder $∣ z^{2} + ε^{2} ∣ = ∣ z - i ε ∣ ∣ z + i ε ∣ \geq d^{2} / 4$ , so the ML estimate bounds the contribution of the remainder to the integral of step 4.1 by a constant times $ε$ .

5.2F4F8step 4.1

On the local graph one has $∣ z (x) - i ε ∣ ∣ z (x) + i ε ∣ \geq c (x^{2} + ε^{2})$ for a constant $c > 0$ : for $∣ x ∣ \geq ε$ each factor is at least $∣ x ∣$ , while for $∣ x ∣ < ε$ the bound $∣ f (x) ∣ \leq η ∣ x ∣ < ε / 2$ gives $∣ z (x) \pm i ε ∣ \geq ε / 2$ ; substituting $x = ε u$ turns the local contribution into $\int_{- a / ε}^{a / ε} \frac{2 i z^{'} (ε u)}{(z (ε u) / ε)^{2} + 1} d u$ , whose integrand converges uniformly on bounded $u$ -intervals to $\frac{2 i}{1 + u^{2}}$ and is dominated by $C / (1 + u^{2})$ , so the tails are uniformly of order $1 / R$ outside $[- R, R]$ and the integral tends to $\int_{R} \frac{2 i d u}{1 + u^{2}} = 4 i \lim_{R \to \infty} \arctan R = 2 π i$ by [F8]; hence the index difference tends to $1$ .

6.1F5step 5.1step 5.2∎

Each winding number is an integer by [F5], so the difference $n (Γ, i ε) - n (Γ, - i ε)$ is an integer for every sufficiently small $ε > 0$ ; since it tends to $1$ by steps 5.1 and 5.2, it equals $1$ for all sufficiently small $ε$ .

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)Open item page →

The winding number is locally constant by an integral estimate

Statement

Let $Γ$ be a closed rectifiable contour of length $L$ and let $p_{0}$ lie off its trace. If $d > 0$ satisfies $∣ z - p_{0} ∣ \geq d$ for all $z \in Γ^{*}$ and $∣ p - p_{0} ∣ < d / 2$ , then $∣ n (Γ, p) - n (Γ, p_{0}) ∣ \leq L ∣ p - p_{0} ∣ / (π d^{2})$ . In particular the winding number is locally constant on the complement of the trace.

Facts & Assumptions

Given: A closed rectifiable contour $Γ$ of length $L$ , a point $p_{0}$ off its trace, and a number $d > 0$ with $∣ z - p_{0} ∣ \geq d$ for all $z \in Γ^{*}$ .

[F1]

For a closed complex contour $γ$ and a point $p$ off its trace, $n (γ, p) = \frac{1}{2 π i} \int_{γ} \frac{d z}{z - p}$ . (The winding number of a closed contour about a point off its trace).

[F2]

[F3]

[F4]

A closed box in $R^{n}$ is a compact subset of Euclidean space. (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line).

[F5]

For a closed complex contour $γ$ and a point $p$ off its trace, $n (γ, p) \in Z$ . (The winding number of a closed contour is an integer).

Proof

technique · direct

1.1givenF1

Let $p$ satisfy $∣ p - p_{0} ∣ < d / 2$ ; then $∣ z - p ∣ \geq ∣ z - p_{0} ∣ - ∣ p - p_{0} ∣ > d - d / 2 = d / 2 > 0$ for every $z \in Γ^{*}$ , so $p$ also lies off the trace and both winding numbers are defined by the contour integral of the corresponding $1 / (z - p)$ .

2.1F1F2step 1.1

Subtracting the two integrands gives $\frac{1}{z - p} - \frac{1}{z - p_{0}} = \frac{p - p_{0}}{(z - p) (z - p_{0})}$ for $z \in Γ^{*}$ , so by linearity of complex line integrals $n (Γ, p) - n (Γ, p_{0}) = \frac{1}{2 π i} \int_{Γ} \frac{p - p_{0}}{(z - p) (z - p_{0})} d z$ .

2.2F3step 1.1

On the trace $∣ z - p ∣ \geq d / 2$ and $∣ z - p_{0} ∣ \geq d$ , so the integrand has modulus at most $2 ∣ p - p_{0} ∣ / d^{2}$ ; the ML estimate with the length $L$ of $Γ$ and the factor $(2 π i)^{- 1}$ give $∣ n (Γ, p) - n (Γ, p_{0}) ∣ \leq L ∣ p - p_{0} ∣ / (π d^{2})$ , the displayed estimate.

3.1F4step 2.2

For the local-constancy assertion fix $p_{0}$ off the trace; if $L = 0$ the estimate of step 2.2 bounds the defining integral by $0$ for every point off the trace, so the winding number vanishes near $p_{0}$ ; if $L > 0$ , then for each parameter $t$ continuity of the rectifiable contour supplies a relative interval $J$ containing $t$ on which $∣ Γ (s) - p_{0} ∣ > ∣ Γ (t) - p_{0} ∣ / 2$ , the family of all such pairs $(t, J)$ covers the compact parameter interval, so finitely many cover it by [F4], and the minimum of the finitely many positive numbers $∣ Γ (t_{i}) - p_{0} ∣ / 2$ is a $d > 0$ with $∣ z - p_{0} ∣ \geq d$ on $Γ^{*}$ .

4.1F5step 3.1∎

With that $d$ the estimate of step 2.2 gives $∣ n (Γ, p) - n (Γ, p_{0}) ∣ \leq L ∣ p - p_{0} ∣ / (π d^{2})$ , which is less than $1$ whenever $∣ p - p_{0} ∣ < \min (d / 2, π d^{2} / L)$ ; the difference of the two winding numbers is an integer by [F5], so it vanishes for every $p$ in that relative neighbourhood of $p_{0}$ , and since $p_{0}$ was an arbitrary point off the trace the winding number is locally constant on the complement of the trace; no general Jordan theorem or choice principle is used.

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)Open item page →

C² inverses and scalar return roots

Statement

Let $f$ be a $C^{2}$ map between open subsets of $R^{n}$ with invertible derivative at a point. Its local inverse is $C^{2}$ . If $g (s, t)$ is $C^{2}$ near $(s_{0}, t_{0})$ , $g (s_{0}, t_{0}) = 0$ and $g_{t} (s_{0}, t_{0}) \neq 0$ , then there is a unique local $C^{2}$ root $t = T (s)$ , with $T^{'} = - g_{s} / g_{t}$ and $T^{''} = - (g_{s s} + 2 g_{s t} T^{'} + g_{t t} (T^{'})^{2}) / g_{t}$ , evaluated at $(s, T (s))$ . No choice axiom is used.

Facts & Assumptions

Given: A $C^{2}$ map $f$ between open subsets of $R^{n}$ with invertible derivative at $x_{0}$ , and a $C^{2}$ function $g (s, t)$ near $(s_{0}, t_{0})$ with $g (s_{0}, t_{0}) = 0$ and $g_{t} (s_{0}, t_{0}) \neq 0$ .

[F1]

If $U \subseteq R^{n}$ is open, $f : U \to R^{n}$ is $C^{1}$ and $D f (a)$ is invertible, then $f$ is a local diffeomorphism at $a$ whose inverse $g$ is $C^{1}$ with $D g (y) = D f (g (y))^{- 1}$ (The Euclidean inverse function theorem).

[F2]

For composable differentiable maps the total derivative of the composite is the composite of the total derivatives (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ).

Proof

technique · direct

1.1givenF1

Let $f$ be $C^{2}$ at $x_{0}$ with $D f (x_{0})$ invertible; by [F1] there are open sets $V$ with $x_{0} \in V$ and $W$ with $f (x_{0}) \in W$ such that $f ∣_{V} : V \to W$ is a bijection with $C^{1}$ inverse $g : W \to V$ satisfying $D g (y) = D f (g (y))^{- 1}$ .

2.1step 1.1F2

The matrix $D f$ is invertible throughout a neighbourhood of $g (W)$ , and the entries of its inverse are quotients of polynomial functions of the entries of $D f$ by the determinant, hence are $C^{1}$ functions of the entries of $D f$ ; since $f$ is $C^{2}$ and $g$ is $C^{1}$ , [F2] shows that $y \mapsto D g (y) = D f (g (y))^{- 1}$ is $C^{1}$ , that is, $g$ is $C^{2}$ .

3.1step 1.1algebra

Apply step 1.1 to the $C^{2}$ map $G (s, t) : = (s, g (s, t))$ near $(s_{0}, t_{0})$ : its derivative $D G = (\begin{array}{cc} 1 & 0 \\ g_{s} & g_{t} \end{array})$ has determinant $g_{t}$ , which is nonzero at $(s_{0}, t_{0})$ by hypothesis, so $D G (s_{0}, t_{0})$ is invertible and $G$ has a local $C^{2}$ inverse by step 2.1.

4.1step 3.1

Write the second component of that local inverse as $t = T (s)$ with $T$ defined near $s_{0}$ ; then $G (s, T (s)) = (s, 0)$ , that is $g (s, T (s)) = 0$ , and the equality is unique among $t$ near $t_{0}$ because the local inverse of $G$ is a function.

5.1step 4.1F2

Differentiating the identity $g (s, T (s)) = 0$ in $s$ with [F2] gives $g_{s} + g_{t} T^{'} = 0$ at $(s, T (s))$ , hence $T^{'} = - g_{s} / g_{t}$ wherever $g_{t} \neq 0$ , which holds near $s_{0}$ .

6.1F2step 5.1∎

Differentiating the same identity twice with [F2] gives $g_{s s} + g_{s t} T^{'} + g_{t} T^{''} + (g_{t s} + g_{t t} T^{'}) T^{'} = 0$ at $(s, T (s))$ ; using $g_{s t} = g_{t s}$ and solving for $T^{''}$ because $g_{t} \neq 0$ yields $T^{''} = - (g_{s s} + 2 g_{s t} T^{'} + g_{t t} (T^{'})^{2}) / g_{t}$ , and all steps used only the stated local inverse and chain rule, so no choice axiom is invoked.

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C¹ planar fields on a closed disk extend to a neighbourhood

Statement

Let $X$ be a planar vector field $C^{1}$ up to the boundary of the closed unit disk $D$ . It has a $C^{1}$ extension to an open neighborhood of $D$ whose value and first derivative agree with $X$ on $D$ , including its boundary. This is a finite explicit extension, with no choice axiom.

Facts & Assumptions

Given: A planar vector field $X$ on the closed unit disk $D$ whose components are $C^{1}$ up to the boundary, i.e. whose value and first partial derivatives extend continuously to $D$ .

[F1]

[F2]

A function continuous on $[a, b]$ and differentiable on $(a, b)$ satisfies $f (b) - f (a) = f^{'} (c) (b - a)$ for some interior point $c$ (The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ ).

Proof

technique · direct

1.1givenconstruct

Write every nonzero point in a collar of the unit circle uniquely as $q = (1 + s) p$ with $p \in S^{1}$ and $s > - 1$ , fix once and for all a number $0 < ε < 1 / 2$ , and define $E (q) : = 3 X ((1 - s) p) - 2 X ((1 - 2 s) p)$ for $0 < s < ε$ while $E (q) : = X (q)$ for $∣ q ∣ \leq 1$ ; this is an explicit finite formula with no choice.

2.1step 1.1algebra

On the unit circle, where $s = 0$ , the outer formula gives $3 X (p) - 2 X (p) = X (p)$ , so the two definitions agree there and $E$ is a well-defined map on ${∣ q ∣ < 1 + ε}$ .

2.2step 1.1F1

The inner formula is the restriction of $X$ , which is $C^{1}$ up to the boundary; the outer formula is a composite of smooth scalar operations with the map $(s, p) \mapsto X ((1 - 2 s) p)$ , and for $0 < s < ε$ the points $(1 - 2 s) p$ lie in the interior of $D$ , where $X$ is $C^{1}$ ; hence [F1] shows that $E$ is $C^{1}$ on each of the two open regions $∣ q ∣ < 1$ and $1 < ∣ q ∣ < 1 + ε$ , with derivatives computed by the chain rule.

3.1step 2.2F1

Parametrize the circle by $p = p (θ)$ . The chain rule gives $\partial_{θ} E = 3 (1 - s) D X_{(1 - s) p} p' (θ) - 2 (1 - 2 s) D X_{(1 - 2 s) p} p' (θ)$ outside the disk. As $s ↓ 0$ this tends to $(3 - 2) D X_{p} p' (θ) = D X_{p} p' (θ)$ , the inner tangential derivative.

3.2step 2.2F1

The outer radial derivative is $\partial_{s} E = - 3 D X_{(1 - s) p} p + 4 D X_{(1 - 2 s) p} p$ . As $s ↓ 0$ it tends to $(- 3 + 4) D X_{p} p = D X_{p} p$ , the inner radial derivative. Both limiting derivatives depend continuously on $p$ .

4.1F2step 3.1step 3.2∎

The first partial derivatives of $E$ are therefore continuous across the unit circle, each side being $C^{1}$ with matching limits by step 3.1 and step 3.2; for $q$ on the circle and a small displacement $h$ , applying [F2] on the segments on either side of the circle gives $∣ E (q + h) - E (q) - D E_{q} h ∣ \leq \sup_{0 \leq t \leq 1} ∥ D E_{q + t h} - D E_{q} ∥ ∥ h ∥$ , and the supremum tends to $0$ because the partial derivatives are continuous at $q$ , so $E$ is differentiable there with total derivative $D E_{q}$ and hence $C^{1}$ on ${∣ q ∣ < 1 + ε}$ . Since $E = X$ on $D$ and the derivative identity just established gives $D E = D X$ along the circle from the inner side, the value and first derivative of the extension agree with $X$ on the closed disk; the construction uses only the explicit formula of step 1.1 and finitely many evaluations.

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A C² saddle function has C¹ Morse coordinates

Statement

Let $f$ be C² near $p \in R^{2}$ , with $d f (p) = 0$ and Hessian of signature $(1, 1)$ . There is a C¹ local diffeomorphism $(x, y)$ centered at $p$ for which $f - f (p) = x y$ . The coordinate change is only asserted to be C¹.

Facts & Assumptions

Given: A function $f$ of class $C^{2}$ near $p \in R^{2}$ with $d f (p) = 0$ and Hessian of signature $(1, 1)$ .

[F1]

If $U \subseteq R^{n}$ is open, $f : U \to R^{n}$ is $C^{1}$ and $D f (a)$ is invertible, then $f$ is a local diffeomorphism at $a$ with a $C^{1}$ inverse $g$ satisfying $D g (y) = D f (g (y))^{- 1}$ . (The Euclidean inverse function theorem).

[F2]

For composable differentiable maps the total derivative of the composite is the composite of the total derivatives. (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ).

[F3]

A function continuous on $[a, b]$ and differentiable on $(a, b)$ satisfies $f (b) - f (a) = f^{'} (c) (b - a)$ for some interior point $c$ . (The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ ).

[F4]

If every partial derivative of a map exists near a point and is continuous there, then the map is totally differentiable at that point with the Jacobian as its derivative. (If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative).

Proof

technique · direct

1.1given

Translate $p$ to the origin, so $f (0) = 0$ , $d f (0) = 0$ , and the Hessian $H$ of $f$ at the origin is a symmetric bilinear form of signature $(1, 1)$ ; fix $v$ with $H (v, v) > 0$ and a vector $e$ independent of $v$ .

2.1step 1.1algebra

Replace $e$ by $w = e - H (e, v) H (v, v)^{- 1} v$ ; then $H (v, w) = 0$ , and since the Gram determinant of the independent pair $(e, v)$ under the indefinite form $H$ equals $\det (H)$ times a nonzero square it is negative, so $H (w, w) = H (e, e) - H (e, v)^{2} / H (v, v) < 0$ ; in the linear coordinates with first axis $v$ and second axis $w$ one therefore has $f_{x x} (0) > 0$ , $f_{y y} (0) < 0$ , $f_{x y} (0) = 0$ , and after shrinking to a smaller neighbourhood also $f_{x x} > 0$ there.

3.1F1step 2.1

On that smaller neighbourhood the map $Φ (x, y) = (f_{x} (x, y), y)$ has derivative $(\begin{array}{cc} f_{x x} & f_{x y} \\ 0 & 1 \end{array})$ with determinant $f_{x x} > 0$ , so $Φ$ is a local diffeomorphism at the origin by [F1]; the preimage of the slice ${(0, y)}$ is therefore a $C^{1}$ curve that can be written $x = η (y)$ with $η (0) = 0$ , and differentiating $f_{x} (η (y), y) = 0$ gives $η^{'} (y) = - f_{x y} (η (y), y) / f_{x x} (η (y), y)$ .

4.1F2step 3.1

Define $b (y) = f (η (y), y)$ ; then $b^{'} (y) = f_{x} (η (y), y) η^{'} (y) + f_{y} (η (y), y) = f_{y} (η (y), y)$ by [F2], so $b$ is $C^{2}$ near zero with $b^{'} (0) = f_{y} (0, 0) = 0$ , and differentiating once more at the origin with $η^{'} (0) = - f_{x y} (0) / f_{x x} (0) = 0$ gives $b^{''} (0) = f_{y y} (0) < 0$ .

5.1F3F4step 4.1

For $x \neq η (y)$ set $X (x, y) = sgn (x - η (y)) \sqrt{f (x, y) - b (y)}$ and set $X = 0$ on the curve $x = η (y)$ ; positivity under the square root follows from $f (x, y) - b (y) = (x - η (y)) \int_{0}^{1} f_{x} (η (y) + t (x - η (y)), y) d t$ , a mean value formula justified by [F3], together with $f_{x x} > 0$ and $f_{x} = 0$ on the curve; off the curve $2 X X_{x} = f_{x}$ and $2 X X_{y} = f_{y} - b^{'}$ , and the limits along the curve, obtained from the second-order expansion in $x - η (y)$ and continuity of the Hessian, are $X_{x} = \sqrt{f_{x x} / 2}$ and $X_{y} = f_{x y} / \sqrt{2 f_{x x}}$ at $(η (y), y)$ ; the first of these is also the derivative of the defined $X$ on the curve by the expansion, the second by $η^{'} = - f_{x y} / f_{x x}$ , and both limits are continuous with $X_{x} (0, 0) > 0$ , so [F4] applies to the defining formula for $X$ on each side of the curve with matching limits.

5.2F3step 4.1

Define $Y (y) = sgn (y) \sqrt{b (0) - b (y)}$ for $y \neq 0$ and $Y (0) = 0$ ; since $b^{''} (0) < 0$ the one-variable form of [F3] gives $b (0) - b (y) > 0$ for small nonzero $y$ and shows that $Y$ is $C^{1}$ near zero with $Y^{'} (0) = \sqrt{- b^{''} (0) / 2} > 0$ .

6.1F1step 5.1step 5.2∎

The map $(x, y) \mapsto (X (x, y), Y (y))$ has invertible derivative $diag (X_{x} (0, 0), Y^{'} (0))$ with positive diagonal entries at the origin, so by [F1] it is a $C^{1}$ local diffeomorphism; moreover $X^{2} - Y^{2} = (f - b (y)) - (b (0) - b (y)) = f (x, y) - f (0)$ , so the new coordinates $(u, w) = (X + Y, X - Y)$ centred at the origin satisfy $f - f (p) = u w$ ; the construction used only explicit linear algebra, mean value formulas and the local inverse theorem, all with finitely many choices.

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The Bott partial connection is well defined and flat along leaves

Statement

Assume Countable Choice ${A C}_{ω}$ . In the notation of The Bott partial connection on the normal bundle of a foliation, the Bott partial connection is well defined: $π [X, \tilde{s}]$ depends only on $X \in Γ (E)$ and the section $s \in Γ (ν)$ , not on the representative $\tilde{s}$ , is $C^{\infty} (M)$ -linear in $X$ , and satisfies $\nabla_{X}^{B} (f s) = X (f) s + f \nabla_{X}^{B} s$ for $f \in C^{\infty} (M)$ . Its curvature vanishes along leaves: for all $X, Y \in Γ (E)$ and $s \in Γ (ν)$ , $R^{B} (X, Y) s : = \nabla_{X}^{B} \nabla_{Y}^{B} s - \nabla_{Y}^{B} \nabla_{X}^{B} s - \nabla_{[X, Y]}^{B} s = 0$ .

Facts & Assumptions

Given: Assume ${A C}_{ω}$ . A codimension- $q$ regular foliation $F$ of a smooth manifold $M$ with tangent distribution $E = T F$ , leaf-tangent fields $X, Y \in Γ (E)$ , a normal-bundle section $s \in Γ (ν)$ , and two smooth local representatives $\tilde{s}, {\tilde{s}}^{'}$ of $s$ on a common bundle chart.

[F1]

For $X \in Γ (E)$ and $s \in Γ (ν)$ the Bott partial connection is $\nabla_{X}^{B} s = π [X, \tilde{s}]$ with $π : T M \to ν$ the quotient map and $\tilde{s}$ any smooth representative of $s$ . (The Bott partial connection on the normal bundle of a foliation).

[F2]

An integrable distribution is involutive: the Lie bracket of two of its sections is again a section. (Integrable distributions are involutive).

[F3]

For smooth functions $f$ and vector fields $X, Y$ one has $[f X, Y] = f [X, Y] - Y (f) X$ and $[X, f Y] = f [X, Y] + X (f) Y$ . (Leibniz rules for the Lie bracket with function multiples).

[F4]

Smooth vector fields on a manifold form a Lie algebra: the bracket is bilinear, alternating and satisfies the Jacobi identity. (Smooth vector fields form a Lie algebra under the Lie bracket).

Proof

technique · direct

1.1F1F2given

In any quotient-bundle frame a local lift of $s$ is obtained by using the same smooth coefficient functions in lifted frame vectors. Two such local representatives of $s$ differ by a section $σ = {\tilde{s}}^{'} - \tilde{s} \in Γ (E)$ , and since $E$ is integrable it is involutive by [F2], so $[X, σ] \in Γ (E)$ and $[X, {\tilde{s}}^{'}] = [X, \tilde{s}] + [X, σ]$ ; applying the quotient map $π$ of [F1] kills $[X, σ]$ , so $π [X, {\tilde{s}}^{'}] = π [X, \tilde{s}]$ and $\nabla_{X}^{B} s$ is independent of the chosen local representative. Consequently these smooth local sections agree on overlaps and define a global section.

2.1F3step 1.1

For $f \in C^{\infty} (M)$ the first Leibniz rule of [F3] gives $[f X, \tilde{s}] = f [X, \tilde{s}] - (\tilde{s} f) X$ , and the correction $(\tilde{s} f) X$ is a section of $E$ , so projecting gives $\nabla_{f X}^{B} s = f \nabla_{X}^{B} s$ , that is, $C^{\infty} (M)$ -linearity in the vector-field variable.

2.2F3step 1.1

The second Leibniz rule of [F3] gives $[X, f \tilde{s}] = f [X, \tilde{s}] + X (f) \tilde{s}$ for the representative $f \tilde{s}$ of $f s$ , so projecting yields $\nabla_{X}^{B} (f s) = f \nabla_{X}^{B} s + X (f) s$ , the stated Leibniz rule.

3.1F4step 2.1step 2.2∎

For flatness, lift $\nabla_{Y}^{B} s = π [Y, \tilde{s}]$ locally by $[Y, \tilde{s}]$ and compute $\nabla_{X}^{B} \nabla_{Y}^{B} s - \nabla_{Y}^{B} \nabla_{X}^{B} s - \nabla_{[X, Y]}^{B} s = π ([X, [Y, \tilde{s}]] - [Y, [X, \tilde{s}]] - [[X, Y], \tilde{s}])$ ; the bracketed expression is the Jacobi identity of [F4] applied to $X, Y, \tilde{s}$ , hence vanishes, and well-definedness from step 1.1 makes the result independent of all lifts, so $R^{B} (X, Y) s = 0$ and the connection is flat along leaf directions; only the stated bracket and involutivity facts were used, with no additional choice principle.

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A C² leaf meets a local box transversal in at most countably many points

Statement

Assume ${A C}_{ω}$ . Let $F$ be a $C^{2}$ foliation on a second-countable smooth manifold, let $L$ be a leaf, and let $τ : J \to Q$ be a vertical transverse interval in one foliation box. Then $τ^{- 1} (L)$ is at most countable. If a countable foliation- box atlas is supplied as part of the data, the countability conclusion uses no choice principle. Dense and nonembedded leaves are allowed.

Facts & Assumptions

Given: Assume ${A C}_{ω}$ . A $C^{2}$ foliation $F$ on a second-countable smooth manifold $M$ , a leaf $L$ , and a vertical transverse interval $τ : J \to Q$ in one foliation box $Q$ .

[F1]

A second-countable space is Lindelof: every open cover has a countable subcover. (Assuming countable choice, every second countable space is Lindelöf).

[F2]

The connected components of an open subset of $R^{n}$ are open and polygonally connected. (Every connected component of an open subset of $R^{n}$ is open and polygonally connected).

[F3]

For nonnegative integers the pairing $p (a, b) = (a + b) (a + b + 1) / 2 + b$ is injective: pairs with $a + b = d$ occupy the disjoint consecutive interval from $d (d + 1) / 2$ to $(d + 1) (d + 2) / 2 - 1$ , and the offset recovers $b$ and hence $a$ . Starting with $b_{0} = 0$ , put $b_{j} = p (b_{j - 1}, a_{j})$ and encode a word $(a_{1}, \dots, a_{k})$ by $p (k, b_{k})$ . Decoding the outer pair recovers $k$ , and recursively decoding the inner pairs recovers the word. Thus finite natural-number words admit this explicit injection into $N$ , without using later computability theory.

Proof

technique · direct

1.1given

Fix the second-countable $C^{2}$ foliated manifold, the leaf $L$ , the foliation box $Q$ and the vertical transverse interval $τ$ , and if the leaf dimension is zero, every plaque and hence every leaf is a singleton, so the intersection has at most one point and the conclusion is immediate. Otherwise fix one plaque $P_{0}$ of $L$ inside the recorded atlas; a vertical interval meets each plaque of $Q$ in at most one point, because the transverse coordinate is constant on a plaque while $τ$ varies only in the transverse direction.

1.2F1given

By [F1] the second-countable manifold has a countable cover by foliation boxes; selecting one foliation chart for each member of that countable subcover uses the stated countable choice, and adjoin the specified box $Q$ and the box of the initial plaque $P_{0}$ to that countable atlas (a finite addition), and record this enlarged countable atlas as fixed data for the rest of the argument.

2.1givenstep 1.1

Say a plaque is reached when it can be joined to $P_{0}$ by a finite chain of plaques of the recorded atlas in which consecutive plaques intersect; every plaque of $L$ is reached by definition of the plaque-chain relation, and it suffices to count the reached plaques contained in $Q$ .

2.2F2step 1.2

Let $P$ be a reached plaque in a box $U$ and let $V$ be a next box of the recorded atlas; in plaque coordinates the trace of $P$ inside $V$ is an open subset of the plaque coordinate space $R^{\dim L}$ , whose connected components are open and polygonally connected by [F2]; each nonempty component lies in a single plaque of $V$ , because the plaques of $V$ partition the open set $L \cap V$ into pairwise disjoint open subsets of the leaf, so a connected subset of $L \cap V$ cannot meet two of them; code each nonempty component by the least rational-box basis index contained in it, so distinct components, being disjoint, receive distinct codes, and given the current plaque $P$ and the next box $V$ the component code therefore determines at most one successor plaque.

3.1F3step 2.2

Encode each finite chain of successor data by the natural-number coding of finite sequences from [F3], and assign to each reached plaque in $Q$ the least code of a finite chain reaching it; this is a well-defined injection of the reached plaques of $Q$ into $N$ and does not select a chain at each plaque.

4.1step 1.1step 3.1∎

The reached plaques of $L$ inside $Q$ are therefore at most countable, and by step 1.1 each of them meets $τ$ in at most one point, so $τ^{- 1} (L)$ injects into a countable set and is at most countable; a countable box atlas already supplied as data removes the only countable choice of step 1.2, and dense or nonembedded leaves are allowed since only plaque chains were used.

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A finitely cornered regular plane curve separates without choice

Statement

Let $c : S^{1} \to R^{2}$ be a piecewise- $C^{1}$ topological embedding with finitely many corner parameters. Assume each smooth edge is regular up to its endpoints and the two incident one-sided tangent rays at each corner are distinct. Then $R^{2} ∖ c (S^{1})$ has exactly two connected components, one bounded and one unbounded, and each has boundary $c (S^{1})$ . No choice axiom is assumed.

Facts & Assumptions

Given: A piecewise- $C^{1}$ topological embedding $c : S^{1} \to R^{2}$ with finitely many corner parameters, each smooth edge regular up to its endpoints and the two incident one-sided tangent rays distinct at each corner.

[F1]

If an oriented closed piecewise- $C^{1}$ contour contains exactly one regular $C^{1}$ arc near $0$ , traversed once with positive real tangent, and the remaining contour is compact and disjoint from $0$ , then for all small $ε > 0$ the points $i ε$ and $- i ε$ avoid it and the two winding numbers differ by $1$ . (The winding number jumps by one across a regular planar arc).

[F2]

If the trace stays at distance at least $d$ from $p_{0}$ and $∣ p - p_{0} ∣ < d / 2$ , then $∣ n (Γ, p) - n (Γ, p_{0}) ∣ \leq L ∣ p - p_{0} ∣ / (π d^{2})$ , and the winding number is locally constant on the complement of the trace. (The winding number is locally constant by an integral estimate).

[F3]

If $K \subseteq C$ is compact, then $C ∖ K$ has exactly one unbounded connected component and every other component is bounded. (The complement of a compact plane set has exactly one unbounded connected component).

[F4]

The connected components of a topological space are nonempty, pairwise disjoint, cover the space, and each is closed in the space. (The components of a space are its maximal connected subsets, they partition it, and each of them is closed).

[F5]

The connected components of an open subset of $R^{n}$ are open and polygonally connected. (Every connected component of an open subset of $R^{n}$ is open and polygonally connected).

[F6]

For $n \geq 1$ , $R^{n}$ is polygonally connected and connected and is locally path-connected. ( $R^{n}$ is polygonally connected, connected, locally path-connected and locally connected).

[F7]

[F8]

[F9]

Proof

technique · direct

1.1givenF7

Orient $c (S^{1})$ by the parameter; since $S^{1}$ is a closed bounded subset of the plane it is compact by [F7], any open cover of $c (S^{1})$ pulls back along the continuous bijection $c$ to an open cover of $S^{1}$ , so $c (S^{1})$ is compact, and it is closed in the plane.

2.1F8F9step 1.1

At a smooth edge point choose linear coordinates with the tangent horizontal; the first coordinate has nonzero derivative along the edge, so after shrinking it has one sign, [F8] makes it strictly monotone, [F9] shows its image is an interval, and its inverse is $C^{1}$ by the difference quotient and the derivative lower bound, exhibiting the curve as a local graph; at a corner let $u$ be the outgoing directed unit tangent and $v$ the incoming directed unit tangent. The geometric incident rays point along $u$ and $- v$ , so their distinctness excludes $v = - u$ . Thus the coordinate $ξ (w) = ⟨ w, u + v ⟩$ has rate $1 + ⟨ u, v ⟩ > 0$ along both branches, so both are $C^{1}$ graphs over $ξ$ with the corner as common endpoint and disjoint $ξ$ -ranges and their union is one local graph, and no opposite-directed tangent case remains under the hypothesis. A small disk about the corner meets the curve in two arcs meeting only at the corner and its complement in that disk has exactly two connected components; every sufficiently short parameter arc has image open in $c (S^{1})$ , so inside the corresponding ambient open set one chooses an ambient disk in which the curve portion is exactly this local model and whose complement has exactly two connected sides; the finitely many such parameter arcs cover $S^{1}$ , and compactness of $S^{1}$ yields a finite subcover. Shrink the side rectangles using positive separation of the images of compact nonadjacent parameter arcs; then every overlap near the curve concerns compatible adjacent arc charts and cannot interchange the oriented sides.

3.1step 2.1

The overlap graph of this finite cover is connected, since otherwise the unions of parameter arcs in its two vertex classes would be disjoint nonempty closed subsets covering the connected circle; whenever two parameter arcs overlap, their rectangles overlap near a common curve point and the left-side patches, respectively the right-side patches, meet there because both are the same oriented side of the same local graph, so the unions $V_{+}$ and $V_{-}$ of all left and right patches are connected subsets of the complement and every curve point is approached from each side.

4.1F1F2step 3.1

At a smooth edge point $p$ with unit tangent $τ$ use the oriented coordinate $w = \overset{ˉ}{τ} (z - p)$ : the defining integral is unchanged because $d w / (w - w_{0}) = d z / (z - z_{0})$ when $w_{0} = \overset{ˉ}{τ} (z_{0} - p)$ , so the local jump lemma [F1] gives different winding numbers on the two side unions near $p$ , while the local estimate [F2] makes the winding number constant on each connected side; hence $V_{+}$ and $V_{-}$ lie in two distinct connected components of the complement.

5.1F4F5F6step 4.1

Let $U$ be any component of the open complement; $U$ is closed in the complement by [F4], open in the plane and polygonally connected by [F5], and its boundary is contained in the curve and is nonempty, because otherwise $U$ would be a nonempty proper clopen subset of the connected plane, contradicting [F6]; at a boundary point the local graph patch has exactly two connected sides, and since $U$ meets one of them and an open connected side inside $U$ cannot meet the other, the whole side lies in $U$ , identifying $U$ with one of the two global collar-side components; hence the complement has at most two components, and at least two by step 4.1.

6.1F3step 5.1∎

The local sides of $V_{+}$ and $V_{-}$ approach every curve point and no component boundary lies off the curve, so the curve is the boundary of both components, and by [F3] exactly one of them is unbounded while the other is bounded; all choices in the collar construction are finite, and neither the Jordan-Brouwer theorem, the Jordan-Schonflies theorem, nor any choice axiom is used.

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C¹ Euclidean maximal flows, variational dependence and the finite C² upgrade

Statement

Let U⊂R^n be open and Y:U→R^n be C¹. There is a unique maximal flow Φ on an open domain D⊂R×U containing {0}×U, and Φ is jointly C¹. Its time slices are local C¹ diffeomorphisms with inverse Φ_{−t}; $\partial_{t} Φ = Y (Φ)$ and $D_{p} Φ$ is the unique solution V of $V ′ = D Y (Φ) V$ , V(0)=I. Each regular point has a C¹ flow box. If Y is C², Φ and those flow boxes are C²; the second state variation W satisfies $W ′ = D Y (Φ) W + D^{2} Y (Φ) [V e_{i}, V e_{j}]$ , W(0)=0. Individual trajectories of a C¹ field are C² in time. A trajectory remaining in a compact K⊂⊂U cannot have a finite maximal endpoint. These Euclidean conclusions use no full AC or DC.

Facts & Assumptions

Given: An open set $U \subseteq R^{n}$ and a $C^{1}$ vector field $Y : U \to R^{n}$ .

[F1]

If $F$ is continuous and locally Lipschitz in the state variable and the cylinder $[t_{0} - h, t_{0} + h] \times \overline{B} (x_{0}, r)$ lies in the open domain with $∥ F ∥_{2} \leq M$ , state-Lipschitz constant $L$ , $h M \leq r$ and $L h < 1$ , then a unique solution of the initial value problem exists on $[t_{0} - h, t_{0} + h]$ with graph in the cylinder. (Picard-Lindelöf local existence and uniqueness for first-order systems).

[F2]

Near fixed data the Picard-Lindelof solutions exist on one common compact time interval and depend jointly uniformly continuously on initial time, initial state and parameters, with the explicit exponential estimate $∥ x (t) - y (t) ∥_{2} \leq e^{L H} (∥ x_{0} - y_{0} ∥_{2} + M ∣ s - t_{0} ∣ + H ω (∥ λ - μ ∥_{2}))$ . (Continuous dependence of ODE solutions on initial data and parameters).

[F3]

If $u \geq 0$ satisfies $u (t) \leq a (t) + \int_{t_{0}}^{t} b (s) u (s) d s$ with continuous $a, b$ and $b \geq 0$ , then $u (t) \leq a (t) + \int_{t_{0}}^{t} a (s) b (s) \exp (\int_{s}^{t} b (r) d r) d s$ , and for constant $a = A$ , $b = B$ this gives $u (t) \leq A e^{B ∣ t - t_{0} ∣}$ . (Gronwall's integral inequality with variable and constant coefficients).

[F4]

$R^{n}$ with the Euclidean metric is a complete metric space. ( $R$ and $R^{n}$ for $n \geq 1$ with the Euclidean metric are complete, componentwise from the Cauchy criterion in $R$ ).

[F5]

[F6]

A $C^{2}$ map with invertible derivative at a point has a $C^{2}$ local inverse. (C² inverses and scalar return roots).

[F7]

Proof

technique · direct

1.1given

All assertions are local in the state point and the time, so it suffices to work on a cylinder $[- h, h] \times B$ with $B$ a compact convex set with $\overline{B} \subset U$ ; on such a cylinder the mean value theorem [F7] applied componentwise bounds $∥ Y ∥ \leq M$ and makes $Y$ Lipschitz in the state variable with one constant $L$ , and one chooses $h > 0$ with $h M$ below the margin and $L h < 1$ .

2.1F1F2step 1.1

By the quantitative clause of [F1] each initial point of the cylinder carries a unique local solution on $[- h, h]$ , obtained from the Picard iteration that starts at the specified constant curve and is generated by ordinary recursion, and the estimate of [F2] makes these local solutions depend uniformly continuously on the initial state.

3.1F3step 2.1

Subtracting the Volterra equations of two solutions through nearby initial points $p, p + u$ and writing $A_{u} (t) = \int_{0}^{1} D Y (Φ (t, p) + r (Φ (t, p + u) - Φ (t, p))) d r$ gives a linear integral equation for the difference, and the uniform continuity of $D Y$ on the cylinder makes $A_{u} \to D Y (Φ (\cdot, p))$ uniformly; the variational equation $V = I + \int D Y (Φ) V$ has a unique solution on each fixed compact time interval by the same local Picard argument for linear equations together with the exponential bound of [F3], and subtracting $V (t) u$ from the solution difference and applying [F3] gives an error of order $o (∣ u ∣)$ uniformly in $t$ ; hence $D_{p} Φ = V$ , the same estimate at nearby $p$ makes $V$ continuous, and $\partial_{t} Φ = Y (Φ)$ is continuous, so $Φ$ is jointly $C^{1}$ .

4.1F3step 3.1

If $Y$ is $C^{2}$ then $C (t, p) = D Y (Φ (t, p))$ is $C^{1}$ in $p$ with derivative $D^{2} Y (Φ) D_{p} Φ$ ; difference quotients of $V$ solve inhomogeneous linear integral equations, and the same uniform-continuity and Gronwall remainder argument converges to the solution $W$ of $W^{'} = C W + D^{2} Y (Φ) [V e_{i}, V e_{j}]$ , $W (0) = 0$ , continuous in $(t, p)$ , so $D_{p}^{2} Φ$ exists continuously; the mixed derivative is $D Y (Φ) D_{p} Φ$ and the second time derivative is $D Y (Φ) Y (Φ)$ , giving joint $C^{2}$ regularity, while for a $C^{1}$ field a single trajectory is $C^{2}$ in time because $x^{'} = Y (x)$ can be differentiated once.

4.2F5F6step 3.1

At a point with $Y (q) \neq 0$ choose a fixed linear transversal $σ$ to $Y (q)$ ; the derivative of $(t, z) \mapsto Φ (t, σ (z))$ at the corresponding point is invertible because its time derivative is $Y (q) \neq 0$ and its spatial part spans the transversal, so [F5] makes it a local $C^{1}$ diffeomorphism onto a $C^{1}$ flow box, and when $Y$ is $C^{2}$ the same map is $C^{2}$ and [F6] makes its inverse $C^{2}$ .

5.1step 2.1step 4.1

Uniqueness glues the local solutions into a maximal flow $Φ$ on an open domain $D$ satisfying the flow law: covering a compact solution segment by finitely many of the common local cylinders of step 2.1 and composing them proves openness of $D$ and the stated $C^{1}$ regularity, respectively $C^{2}$ regularity when $Y$ is $C^{2}$ , and the flow law inverts the time slices: $Φ_{- t}$ is the inverse of $Φ_{t}$ .

6.1F4step 2.1step 5.1∎

Finally, if a trajectory remains in a compact set $K$ with $K \subset U$ and its maximal endpoint $T$ were finite, then the bound $∥ Y ∥ \leq M$ on a compact cylinder containing $K$ gives $∥ x (t) - x (s) ∥ \leq M ∣ t - s ∣$ , so $x (t)$ has a limit in $K$ as $t \to T$ by the completeness of [F4], and the local existence clause of step 2.1 restarts the solution past $T$ , contradicting maximality; only ordinary recursion, the stated Gronwall and uniform-continuity estimates and the local inverse theorem are used, so no dependent choice or full choice principle is invoked.

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Finite cellulations of compact C² subsurfaces relative to an embedded graph

Statement

Assume ${A C}_{ω}$ . Let $N$ be a compact codimension-zero $C^{2}$ subsurface of a Hausdorff second-countable boundaryless $C^{2}$ surface $S$ , with $C^{2}$ boundary. Let $G \subset Int N$ be a finite embedded graph: its vertices are distinct points, each edge is a regular $C^{2}$ embedded arc up to its endpoints, and different edge interiors are disjoint and avoid the vertices. Closed regular edges may first be subdivided by inserting finitely many vertices. Then $N$ has a finite triangular cellulation containing $G \cup \partial N$ in its one-skeleton. Each closed triangle is an embedded topological disk, the prescribed graph and original boundary retain their regular $C^{2}$ arcs, and the cell maps induce a homeomorphism from a finite abstract simplicial complex onto $N$ after finite subdivision. Interior edges have two incident triangles and boundary edges one; vertex links are circles or intervals, respectively.

A cell map is required to be a homeomorphism; the statement does not assert a differentiable straightening of an arbitrary prescribed graph at its vertices. This permits tangencies between prescribed edge germs. The empty $N$ or empty $G$ is allowed.

Facts & Assumptions

Given: $S, N, G$ as in the statement, with countable choice.

[F1]

For a $C^{r}$ Euclidean map from dimension $m$ to dimension $n$ , critical values are null when $r > \max {m - n, 0}$ (Morse-Sard for Euclidean maps).

[F2]

A $C^{2}$ map with invertible derivative has a $C^{2}$ local inverse; a scalar equation with nonzero normal derivative has a unique local $C^{2}$ root (C² inverses and scalar return roots).

[F3]

A simple polygon has two complementary regions, one bounded and one unbounded, with that polygon as the frontier of each (Polygonal Jordan curve theorem: a polygon has exactly two complementary regions and is the frontier of each).

[F4]

A simple polygonal region has a finite face-to-face triangular subdivision (Every simple polygon admits a triangulation).

[F5]

An abstract simplicial complex has finite subsets as its simplices, with every face also a simplex (An abstract simplicial complex). A finite disk-cell structure is a CW structure if its cell maps are the stated disk attachments and its topology is the weak topology on the closed cells (CW complex with closure finiteness and weak topology).

[F6]

A $C^{1}$ map with invertible derivative has a $C^{1}$ local inverse (The Euclidean inverse function theorem).

[F7]

Under the stated countable choice, a nonempty compact connected topological one-manifold without boundary is a circle (A nonempty compact connected one-dimensional manifold without boundary is a circle).

Proof

1.1F1F7givenchoose

If $N$ is empty use the empty complex. Otherwise choose finitely many relatively compact coordinate disks in $S$ with smaller cores covering $N$ . Their closures remain inside their respective charts. The boundary $\partial N$ has finitely many components: a finite cover of this compact one-manifold by connected interval neighborhoods meets every component, so there are only finitely many. By F7 each component is a circle, covered by finitely many of its regular $C^{2}$ graph arcs. Work with these finitely many boundary curves and the finitely many edges of $G$ . Choose each coordinate-disk radius in a short open interval that leaves its smaller covering core inside it. At each stage restrict the radial function to the compact pieces of every preceding curve and graph edge in the chart annulus. Finitely many parameter intervals inside that chart cover these compact pieces. By F1 for maps of dimension one to one, a radius avoiding the critical values makes the new circle transverse to all those curves. Avoid also the finitely many distances of existing vertices and crossings. Finitely many null sets and finitely many forbidden values cannot fill the radius interval. Thus the disk boundaries, $\partial N$ and $G$ have finitely many transverse crossings, with no new triple crossing and no crossing at a prescribed vertex. Finiteness follows from compactness and the local isolation supplied by transversality. These choices are finite.

2.1F2step 1.1construct

Subdivide at all crossings and insert vertices in isolated closed edges. The union of $G$ , $\partial N$ and the portions of the disk circles inside $N$ is a finite embedded graph $H$ . Each of its nonvertex points has an arc chart by F2. Its vertex stars are tame finite stars, even when two prescribed germs are tangent: in a chart at a vertex $v$ , a regular one-sided edge $α (s)$ with $α (0) = v$ and $α^{'} (0) \neq 0$ has strictly increasing distance from $v$ for small $s > 0$ , since $(α (s) - v) \cdot α^{'} (s) = s ∣ α^{'} (0) ∣^{2} + o (s)$ . Hence every sufficiently small concentric circle meets each incident germ exactly once. Their cyclic order cannot change without an intersection. A homeomorphism on each such circle sending these finitely many ordered intersection points to fixed radial directions, extended with the same radius and sending $v$ to the center, straightens the star; continuity of it and its inverse at the center follows from preservation of radius. Thus each star has finitely many well-defined sectors, or half-sectors at $\partial N$ . Cover each remaining compact edge portion by finitely many arc charts and choose a thin strip about it; compactness and separation from the other finite edge portions make the strips disjoint away from the chosen vertex stars. Their two sides and the vertex sectors are the finite local side data of $H$ .

3.1step 1.1step 2.1

Every connected component $U$ of $N ∖ H$ is planar before we count the components: membership in any coordinate disk is constant on $U$ , because it avoids that disk's boundary. A point of $U$ belongs to a covering disk core, so all of $U$ lies in that disk and its closure lies in the corresponding compact chart disk. Its frontier lies in $H$ and has a side or vertex sector from step 2.1. An empty frontier would make it a nonempty compact boundaryless open surface contained in a planar disk, which is impossible since its chart image would be both open and compact in the plane. A frontier consisting only of finitely many vertices cannot enclose a bounded open region: a ray from an interior point avoiding their finitely many directions would exit without meeting the frontier. Each edge-side germ or vertex sector lies in just one complementary component. The finite side data therefore bounds the number of components; call them $U_{1}, \dots, U_{q}$ . This proves both planarity and finiteness without an arbitrary surface-Jordan assertion.

4.1step 2.1step 3.1construct

(Compact planar cores, including slit sides.) For each $U_{j}$ choose one point in each of its finitely many incident side sectors. Join those points to one interior point by finitely many paths in $U_{j}$ ; inside its planar chart these may be finite polygonal paths, obtained by the elementary open-and-closed argument for the set of points reachable by finite segments in small open balls. Trim the vertex stars and edge strips sufficiently thinly to miss those compact paths. On each edge use a product strip, and at each vertex trim its sectors by a small arc. The remaining portion of $U_{j}$ is a compact planar surface with boundary, with finitely many piecewise regular boundary circuits. It is connected: the selected paths join all side sectors, while any extra component would have to border one of those same trimmed edge or vertex sectors, whose connected inner boundary collar already joins the selected paths. The original face is recovered by attaching the finitely many product half-strips and vertex sectors. Distinct occurrences of a slit edge are retained as distinct sides; they are identified only when these strips are restored. Consequently repeated boundary vertices or slit sides in the closure of $U_{j}$ are not falsely regarded as a single embedded polygonal boundary.

5.1F2F3F6step 2.1step 4.1construct

(Regular planar circuits reduce to polygons.) The finitely many boundary circuits of a planar core are disjoint piecewise regular embedded circles. For a regular compact arc, F2 straightens it in finitely many charts to a coordinate line. Its compact middle portions have disjoint thin product strips. A sufficiently fine inscribed broken line meets each strip fiber once: on every chosen straightening chart the arc has nonzero derivative in one fixed direction, the chords retain this sign by uniform continuity of its tangent, and the finite subdivision is fine enough to remain in that chart. Thus the broken line is a graph over the arc there. For a closed $C^{2}$ regular circuit, its unit normal is $C^{1}$ . The normal strip map is $C^{1}$ with invertible derivative along its zero section, so F6 and compactness make a short strip injective: a hypothetical sequence of collisions in arbitrarily short strips has base points converging to one common curve point, where the local inverse excludes it. A sufficiently fine inscribed polygon projects locally increasingly to the central circle in that collar, including at its corners, by the tangent estimate just used. The projection has degree one because it is uniformly close to the identity parameterization, hence is a one-sheeted circle covering and the polygon is a single graph over the old circle. For a finitely cornered circuit first use the vertex-sector charts of step 2.1 to match the finite endpoint sectors. Graph interpolation in a strip, chosen to be the identity on its outer boundary, gives a homeomorphism carrying the arc to its broken line. At corners the radial sector interpolation agrees with the strip maps. Finite closed pasting gives an ambient homeomorphism of a neighborhood of the circuit, equal to the identity outside it. The neighborhoods of distinct core boundary circuits are disjoint, so all can be polygonalized at once. Their assigned inside/outside sides are now those of F3. This proves precisely the regular-curve adapter used here; it imports neither Jordan–Schönflies for arbitrary curves nor a general surface triangulation theorem.

6.1F3F4step 4.1step 5.1construct

(Planar subdivisions with holes.) A polygonalized compact core may have several boundary circles. Choose a direction whose projections of all its finitely many vertices are distinct. Between consecutive projections all boundary segments are ordered affine graphs. Moving vertically from outside the bounded domain, membership changes at a boundary segment and nowhere else, by the local side charts and F3. Its intersection with each open slab is therefore a finite union of bands between consecutive affine graphs. Their closures are convex triangles or quadrilaterals. Refine all vertical walls at their finitely many intersections, use the same refinement on both sides, and fan each convex cell from one interior point. This gives a finite triangulation, also in the presence of holes; for a single simple polygon this is exactly F4. Pull it back by the homeomorphisms of step 5.1. Fill each restored product half-strip by a rectangle subdivision, and fill each restored vertex sector by a finite fan. These cell maps are embeddings of closed disks in the original chart sectors. Subdivide the old edges at the union of the two incident side subdivisions, so restored pieces meet face-to-face. Prescribed edges of $G$ and the original boundary remain edges of the resulting subdivision. The additional subdivision edges are tame embedded arcs supplied by these homeomorphisms; they are not claimed to be $C^{2}$ merely because the original chart and prescribed graph are $C^{2}$ . Only their topological incidence and disk-face maps are needed below. The prescribed graph edges and original $C^{2}$ boundary arcs themselves have not been changed.

7.1F5step 2.1step 6.1∎

The finitely many embedded closed triangles cover $N$ , and their incidences give a circular link at an interior vertex and an interval link at a boundary vertex, because they fill exactly the chart sectors of step 2.1. If multiple edges have the same endpoints or a triangular incidence initially repeats a vertex, first subdivide every edge with a distinct new midpoint, fan each disk face from its distinct new center, and subdivide the resulting triangles once more. Every new small triangle then has vertices specified by its incident old vertex, edge midpoint and face center; distinct such incidence flags share exactly their common flags. Thus their vertex sets define a finite abstract simplicial complex as in F5. The face maps paste to a continuous bijection from its compact realization onto Hausdorff $N$ , hence to a homeomorphism: a compact-to-Hausdorff continuous bijection is closed. The same finite closed-cell pasting proves the weak topology and finite disk attachments required by F5. No choice beyond the stated ACω is used; all geometric selections in this proof are finite and F1 is applied only on finitely many Euclidean curve pieces.

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Finite surface normal forms, Jordan disks, and torsion control

Statement

Assume ${A C}_{ω}$ (The countable-choice principle used in the foliation pair). Let $S$ be an oriented Hausdorff $C^{2}$ surface without boundary, possibly disconnected. Then:

Every compact subset $K \subset S$ lies in the interior of a compact finitely cellulated $C^{2}$ subsurface $N$ . If $K$ is contained in one connected component of $S$ , $N$ can be chosen connected. A specified finite embedded regular $C^{2}$ graph contained in $Int N$ can be included in its one-skeleton.
Every connected closed compact subsurface has an oriented genus normal form: the sphere for $g = 0$ , or the polygon word $\prod_{i = 1}^{g} a_{i} b_{i} a_{i}^{- 1} b_{i}^{- 1}$ for $g \geq 1$ , with $χ = 2 - 2 g$ . A connected compact subsurface with $b > 0$ boundary circles has a free fundamental group; capping its boundaries defines genus $g$ and gives $χ = 2 - 2 g - b$ and free rank $2 g + b - 1$ .
The fundamental group of every connected component of $S$ is torsion-free.
Every regular embedded $C^{2}$ nullhomotopic circle $c$ bounds an embedded compact disk region in $S$ . Here a disk region is a compact subsurface homeomorphic to the closed disk, with boundary exactly $c$ ; its inherited surface structure and boundary are $C^{2}$ . It is unique unless the connected component containing $c$ is a sphere; in that component the two complementary disk regions are the two possibilities.

The normal forms in clause 2 are homeomorphism normal forms. No general differentiable smoothing theorem or arbitrary-surface triangulation-existence theorem is a premise of this item.

Facts & Assumptions

Given: $S, K$ and the countable-choice hypothesis of the statement; for clause 4 an embedded circle $c$ and an actual nullhomotopy are given.

[F1]

Under ${A C}_{ω}$ , a compact $C^{2}$ subsurface of a Hausdorff second-countable boundaryless $C^{2}$ surface has a finite triangular cellulation relative to a specified finite embedded regular $C^{2}$ graph, with its boundary included, and the cell maps induce a finite abstract simplicial model with the stated edge and vertex links (Finite cellulations of compact C² subsurfaces relative to an embedded graph).

[F2]

A $C^{2}$ map from a two-dimensional chart to the line has null critical-value set (Morse-Sard for Euclidean maps); a regular scalar equation is a $C^{2}$ coordinate by its local inverse and scalar-root construction (C² inverses and scalar return roots).

[F3]

A connected closed surface with a supplied finite simplicial model having two triangles at each edge and cyclic vertex links has a one-polygon schema, by finite choices only (A finite triangulated surface has a one-polygon schema).

[F4]

Finite polygonal side subdivision, inverse-pair cancellation outside the terminal sphere digon, splits and merges, and interlaced-handle extraction preserve the surface quotient. The extraction sends $a U b V a^{- 1} X b^{- 1} Y$ to $c d c^{- 1} d^{- 1} Y X V U$ (Homeomorphism-preserving polygonal schema moves).

[F5]

A finite wedge of $r$ circles has free fundamental group on its $r$ circle loops (The fundamental group of a finite wedge of circles is free of that rank); reduced words give the free group and nonempty reduced words are nonidentity (Reduced words form the free group on an alphabet). A nullhomotopy lifts to a covering after its initial lift is prescribed (Existence and uniqueness of homotopy lifts through a covering map).

[F6]

Every free group is torsion-free (Free groups are torsion-free).

[F7]

With fixed coset-transversal data, the factor actions on normal words are consistent permutations (Factor elements act consistently by permutations on amalgamated normal words); every element of an amalgam has a unique normal form and a positive-length normal word is nonidentity (Normal form theorem for free products with amalgamation). In this item those data are constructed canonically in finite-rank free groups, not chosen by the general full-AC transversal-existence argument.

[F8]

For a two-set open cover with path-connected sets and overlap, fundamental groups give a pushout; injectivity of the overlap maps must be checked separately (Seifert–van Kampen identifies the fundamental group with a group pushout).

[F9]

For a finite CW complex, its Euler characteristic is the alternating sum of integral homology ranks (Euler–Poincare formula for finite CW complexes); its cellular homology equals singular homology (Cellular homology computes singular homology), and homeomorphisms induce homology isomorphisms by functoriality (Singular chains and singular homology are covariantly functorial).

[F10]

The sphere is simply connected ( $S^{n}$ is simply connected for every $n \geq 2$ ), and $π_{1} (T^{2}) = Z^{2}$ ( $π_{1} (T^{2}) ≅ Z \times Z$ ).

[F11]

The standing choice principle is choice for a sequence of nonempty sets, rather than arbitrary-index choice (The countable-choice principle used in the foliation pair).

[F12]

A $C^{1}$ Euclidean field has a $C^{1}$ local flow (C¹ Euclidean maximal flows, variational dependence and the finite C² upgrade), and a $C^{1}$ map with invertible derivative has a local $C^{1}$ inverse (The Euclidean inverse function theorem).

Proof

1.1F1F2construct

(Compact subsurfaces.) For a nonempty compact $K$ choose finitely many chart disks with relatively compact larger chart disks. A $C^{\infty}$ Euclidean bump supported in a larger disk, pulled back by its $C^{2}$ chart and extended by zero, is a $C^{2}$ function on $S$ . Finitely many such bumps, positive on the smaller disks covering $K$ , have a compactly supported sum $f$ with $f > 0$ on $K$ . Choose $0 < a < \min_{K} f$ outside the critical values of $f$ on its support. This is possible by F2 applied in finitely many charts covering that compact support, since a finite union of null subsets of the line cannot contain an interval. At every point of $f^{- 1} (a)$ , F2 makes $f - a$ a $C^{2}$ coordinate. Hence $N_{0} = {f \geq a}$ is a compact $C^{2}$ subsurface and $K \subset Int N_{0}$ . A finite cover of $N_{0}$ by connected disk or half-disk neighborhoods shows it has finitely many components; their union gives the required $N$ . If $K$ lies in one component of $S$ , first connect the finitely many covering disk centers by finitely many paths in that component, and enlarge $K$ by those compact paths and the finitely many closed coordinate-core disks. The resulting compact connected set is inside ${f > a}$ after repeating the bump construction, so it lies in one component of $N_{0}$ ; take that component. For disconnected $S$ , compactness meets only finitely many open components and the construction is performed in each of them. For $K = \emptyset$ , $N = \emptyset$ suffices. For every compact subsurface used here, finitely many open chart domains cover it. Their union $W \subset S$ is an open Hausdorff boundaryless $C^{2}$ surface with a countable basis: take the union of the finitely many countable chart bases. Regard the subsurface as a subsurface of $W$ and apply F1 there. This supplies the finite cellulation, including any prescribed finite regular $C^{2}$ graph once $N$ contains it in its interior, without assuming that all of $S$ is second-countable.

1.2F1construct

(A boundary surface has a finite graph spine.) Let $N$ be connected and $\partial N \neq \emptyset$ , with the triangulation from F1. Its triangle dual graph, with an additional exterior vertex joined to triangles along their boundary edges, is connected: the cyclic or interval vertex links join all triangles incident to a vertex, and connectedness then joins all triangles. Choose a finite spanning tree rooted at the exterior vertex. Remove triangles in order from the root outward, always removing a triangle together with the edge connecting it to its already removed parent. That edge is free at its removal: its only other incident triangle has already been removed, or it was a boundary edge. A triangle with a free side strongly deformation retracts onto its other two sides; in coordinates on a planar reference triangle this is the elementary linear edge-collapse retraction. Performing these finitely many collapses leaves a connected finite embedded graph $B$ , and $N$ deformation retracts onto $B$ . More is true geometrically: $N$ is homeomorphic to a thickening of $B$ . To check it, take small vertex disks and edge rectangles in the triangle charts. Reversing one free-side collapse attaches the missing triangular bulge along the two retained side strips. The union of those two strips and the bulge is a disk with the same two attaching arcs; parametrize its boundary arcs in the same order and extend their circle homeomorphism radially across the reference disk. This replaces the bulged neighborhood by the unbulged one relative to its attaching arcs. Induction reverses every collapse and identifies a neighborhood of the final graph with all of $N$ , carrying boundary to boundary. Thus the thickening is made of finitely many vertex disks and edge bands with the orientation-induced cyclic orders. This argument proves the needed thickening assertion, rather than inferring it from a deformation retraction alone.

1.3F1F3F4F9construct

(Closed oriented polygon normal forms.) For a connected closed $N$ , F1 verifies every edge and vertex hypothesis of F3, giving a one-face polygon schema. Opposite face-side orientations pair at every edge because $N$ is oriented. Reduce its vertex graph to one vertex by contracting a finite embedded spanning tree, unless the terminal sphere digon is reached first. Each edge contraction preserves the surface: a finite disk neighborhood of an edge with distinct endpoints is straightened to a segment $J$ strictly inside a convex disk; for $x \notin J$ let $p (x)$ be its nearest point on $J$ and $b (x)$ the boundary point on the ray from $p (x)$ through $x$ . The map $x \mapsto z_{0} + \frac{∣ x - p (x) ∣}{∣ b (x) - p (x) ∣} (b (x) - z_{0})$ , sending $J$ to its midpoint $z_{0}$ , induces a homeomorphism from the disk modulo $J$ to the disk, fixing the outer boundary. Its inverse follows the normal ray indexed by the radial boundary point. Extend it by the identity outside that neighborhood. In the one-face polygon, collapsing the two occurrences of a tree side separately on the boundary still leaves a disk when other sides remain: a nondecreasing circle parameter constant on those intervals and strictly increasing elsewhere extends by $(r, t) \mapsto (r, (1 - r) t + r q (t))$ , which is strictly increasing for $r < 1$ . Thus a genuine polygon remains; when only $a a^{- 1}$ remains, retain that actual sphere digon. For the remaining one-vertex opposite-pair word, there is no adjacent inverse pair, since its intermediate corner would be a separate vertex. An unprocessed pair must interlace another: otherwise $a X a^{- 1} Y$ separates the corners of $X$ and $Y$ into distinct classes. F4 extracts the interlaced pair as a commutator block and leaves the residual word in order $Y X V U$ . Previously extracted contiguous blocks stay intact because the endpoints of the selected new letters cannot cut their interiors. Each extraction processes two new pairs; finite repetition yields $\prod_{i = 1}^{g} [a_{i}, b_{i}]$ . The terminal digon is a sphere, as follows by splitting it into two disks with their whole boundary circles identified. The one-handle square is the usual opposite-side torus. The cell counts are $(1, 2 g, 1)$ for $g \geq 1$ and $(2, 1, 1)$ for the sphere, giving $χ = 2 - 2 g$ ; F9 makes this invariant and hence makes $g$ unique. No general triangulation-existence or full-AC classification theorem has been used.

2.1F5step 1.2construct

(Free groups and essential boundary words.) Collapse a finite spanning tree of $B$ . Each remaining edge becomes one circle, giving a homotopy equivalence with a finite wedge; the finite tree contraction and its homotopy extend over the adjacent edge intervals by their endpoint parameters. F5 makes $π_{1} (N)$ free of rank $r = ∣ E (B) ∣ - ∣ V (B) ∣ + 1$ . If $B$ is a tree, its thickening is a disk: remove a leaf disk and its incident band, which is a disk attached along one arc, and induct to one vertex disk. Conversely, if $B$ has a cycle, repeatedly delete leaf edges and their end disks; this leaves a nonempty core whose vertex degrees are at least two, and does not change the boundary-loop classes except for deletion of immediate edge-and-inverse excursions. A boundary circle of the thickening follows an edge band and, at its next vertex disk, takes the next germ in cyclic order. In the core this is never the germ that would immediately reverse the incoming edge, because there are at least two germs. Every boundary circuit therefore gives a nonempty cyclically nonbacktracking closed edge path. Such a path has no null positive power: the covering graph whose vertices are reduced edge paths from a fixed vertex and whose edges append an edge and cancel an immediate backtrack is a tree (every nonroot path has its unique shorter prefix as parent). Its local edge stars map bijectively onto those of $B$ , so it is a covering. A nonbacktracking path of positive length lifts from the root to its distinct path vertex; its repeated cyclically nonbacktracking powers have the same property. By F5, a nullhomotopy would lift and make their endpoints agree, a contradiction. Thus every boundary circle of a connected compact boundary surface other than a disk generates an injected infinite cyclic subgroup. This verifies the boundary injections that will be used below.

2.2F1F9F12step 1.1construct

(A null simple curve separates the finite subsurface.) Include $c$ and a compact nullhomotopy in a connected $N$ as in step 1.1, and use F1 with $c$ as a prescribed graph. A regular embedded compact $C^{2}$ circle in an oriented surface has a two-sided $C^{1}$ collar: the tangent is $C^{1}$ , and the induced normal orientation selects the positive transverse cone. Patch finitely many local transverse fields with chart bumps to a $C^{1}$ field near $c$ ; its short flow in F12 gives a map $c \times (- δ, δ) \to N$ . Its differential on the zero section is invertible, and compactness plus the local inverse in F12 excludes collisions after a common shrink. This proves the collar, with no $C^{2}$ collar-flow assertion needed. If $N ∖ c$ were connected, choose a simple path there joining opposite sides of a small crossing segment. Close it across that segment to obtain a dual circle $d$ meeting $c$ once. Choose $d$ as a finite normal path through triangles: the triangle dual graph of the connected cut surface is connected by its vertex links, so it joins the two triangles beside a selected interior point of a $c$ -edge without crossing another $c$ -edge. The joining path inside those triangles closes across that one edge, misses vertices, and meets the other edges transversely. Define an integer cochain on oriented triangulation edges by their signed crossings with $d$ . On every triangle the entering and exiting crossings cancel, so this cochain annihilates its boundary. Its evaluation on the edge cycle $c$ is $\pm 1$ . It therefore detects a nonzero cellular homology class of $c$ , hence a nonzero singular class by F9. The supplied nullhomotopy makes that class zero: triangulate the parameter disk and push its finite singular two-chain into $N$ ; its boundary is the subdivided curve cycle. This contradiction shows that $c$ separates $N$ . The collar has two connected sides, and every component of $N ∖ c$ has frontier on one of them (otherwise it would be open and closed in connected $N$ ); hence there are exactly two components. Their closures $N_{1}, N_{2}$ are compact connected boundary surfaces, each with the distinguished boundary $c$ .

3.1F9step 1.2step 2.1step 1.3construct

(Boundary genus and the separating handle curve.) For a connected boundary surface with $b$ circles, cap them by $b$ abstract disks, triangulated by finite fans along the existing boundary subdivisions. The capped surface is an oriented closed topological surface with a supplied finite triangulation, so the finite reduction of step 1.3 applies, with no need for a differentiable smoothing of the capped charts. Its genus $g$ defines the genus of $N$ . Each cap adds one face in the disk-cell count and no new boundary cells, so F9 gives $χ (N) = 2 - 2 g - b$ . Since the spine of step 1.2 is a connected graph, its free rank is $1 - χ (N) = 2 g + b - 1$ . The closed normal word is also the connected sum of $g$ tori: in two normal polygons remove small interior disks and identify their boundary circles with reversed orientations. Cut the resulting polygonal annuli along a bridge between their outer marked vertices. The resulting disk word is $W t V t^{- 1}$ ; the bridge is an embedded edge with distinct endpoints, and the contraction in step 1.3 gives precisely $W V$ . Iterating proves this assertion from the actual finite disk and annulus gluings. For $g \geq 2$ , the seam splitting the first torus from the other $g - 1$ tori is therefore an embedded separating circle. Its two sides are compact boundary surfaces that are not disks: their spines have ranks $2$ and $2 g - 2$ . Step 2.1 supplies their injective infinite cyclic boundary subgroups.

3.2F6F7step 2.1construct

(The choice cost of amalgam normal form.) When two finite-rank free groups are amalgamated over such injected boundary circles, order each finite free alphabet and list all reduced words by length and then lexicographically. For every left coset of the cyclic boundary subgroup, take its least word in this well-order. This defines all representatives at once by a formula, with the identity representing the subgroup; it chooses no element from an arbitrary-index family. The coefficient in the boundary subgroup is unique because its generator has infinite order by step 2.1. Thus the fixed transversal data needed by F7 actually exist without full AC. Use the factor actions on these data and the uniqueness/nonidentity conclusion of F7; the general supplier's preliminary appeal to AC to find unspecified transversals is not a premise here. Every amalgam element is conjugate either into a factor or to a cyclically reduced alternating word of syllable length at least two: if the first and last syllables are in the same factor, conjugate by the first syllable and merge the new terminal pair, shortening the finite word; repeat. When the two ends are in different factors, concatenating any positive number of copies has no merging seam, so F7 makes it nonidentity. Consequently finite-order elements are conjugate into a factor. If both factors are torsion-free by F6, so is this amalgam. F7 also makes the common cyclic subgroup inject into the amalgam, including its nonidentity length-zero coefficients.

4.1F6F8F10step 1.1step 2.1step 1.3step 3.1step 3.2

(Torsion-freeness for compact and arbitrary surfaces.) A compact boundary surface has a free group by step 2.1, hence is torsion-free by F6. A closed genus-zero surface has trivial group, and a genus-one surface has $Z^{2}$ , by F10 and step 1.3. For genus at least two, enlarge the two sides of the seam in step 3.1 by open annular collars; these are open path-connected sets with connected overlap retracting onto that circle. F8 identifies the group with the amalgam of the two free groups over the injected cyclic boundary group; step 3.2 proves torsion-freeness. Finally, if a loop in an arbitrary component of $S$ has a positive power nullhomotopic, the loop and an actual nullhomotopy have compact connected image. Step 1.1 places that image inside a connected compact subsurface $N$ . The power is null in $N$ , whose group has just been proved torsion-free; hence the original loop is null in $N$ and therefore in $S$ . No injectivity of $π_{1} (N) \to π_{1} (S)$ has been assumed.

4.2F8step 2.1step 3.2step 2.2

(One side is an embedded disk.) If neither $N_{i}$ were a disk, step 2.1 would make the distinguished circle inject as an infinite cyclic subgroup in both free fundamental groups. Enlarge the two sides by open collars of $c$ ; their overlap is a connected annulus. F8 and step 3.2 then give an amalgam in which the common cyclic subgroup is injective, so the loop $c$ is nontrivial in $π_{1} (N)$ . This contradicts the actual nullhomotopy placed in $N$ . At least one $N_{i}$ is therefore a disk, and its inclusion in $S$ is the required embedded compact disk region. Its boundary is the original regular $C^{2}$ circle, so the region's half-space charts are $C^{2}$ by the local graph inverse coordinates; its interior carries the given $C^{2}$ surface structure. The proof produces an embedded region, rather than an immersed null cap or a claim that a nullhomotopy is already embedded.

5.1F9F11step 1.1step 2.1step 1.3step 3.1step 4.1step 4.2∎

(Uniqueness and components.) If $D \subset S$ is any such disk region, $D ∖ c$ is connected and open in $S ∖ c$ , and is closed there because compact $D$ is closed in Hausdorff $S$ . Hence its interior is an entire component of $S ∖ c$ . There are at most two components in the connected component of $S$ containing $c$ , by the same two-sided collar and frontier argument as in step 2.2. Two disk regions on the same side must consequently have the same interior and closure. If both sides are disks, their union fills a neighborhood also at every point of $c$ , so it is an open and closed compact surface in that ambient connected component, hence equals the whole component. Gluing two disks by their boundary-circle homeomorphism gives a sphere: extend that homeomorphism radially across one disk and identify the resulting pair with the two hemispheres. Thus two distinct disk regions are possible only when that connected component is a sphere; conversely, if the ambient component is a sphere, take the disk already supplied by step 4.2. The closure of its other side is a compact connected boundary surface. Additivity of the finite cell count along their common circle gives $2 = 1 + χ (N_{2})$ , since a disk has Euler characteristic one and a circle zero. Thus $χ (N_{2}) = 1$ , its graph spine has rank zero by step 3.1, and step 2.1 makes it a disk too. These constructions prove all clauses. Every geometric choice and word reduction was finite, and the only stated background choice is F11; the canonical coset formula of step 3.2 adds no full AC.

Remarks

The argument applies to an oriented leaf universal cover (Universal covering spaces) by pulling back its local surface charts. It gives the embedded Jordan disk and, in the nonspherical component, its uniqueness without identifying that universal cover globally with the plane. The construction uses finite charts around each compact loop and cap image; no global triangulation or smoothing assertion for the whole noncompact cover is required.

LemmaStatement: Literature-sourcedProof: Literature-sourcedprecheck passOpen item page →

Frobenius divisibility: d omega equals eta wedge omega

Statement

Assume Countable Choice ${A C}_{ω}$ . Let $F$ be a transversely oriented codimension-one foliation of a smooth manifold $M$ with nowhere-vanishing defining $1$ -form $ω$ , so $T F = \ker ω$ . Then $ω \land d ω = 0$ and there exists a smooth $1$ -form $η$ on $M$ with $d ω = η \land ω$ . If $η^{'}$ is another such form, then $η^{'} - η = f ω$ for a unique $f \in C^{\infty} (M)$ .

Facts & Assumptions

Given: A transversely oriented codimension-one foliation $F$ of a smooth manifold $M$ with nowhere-vanishing defining one-form $ω$ , so $T F = \ker ω$ , and the standing countable choice assumption.

[F1]

For a nowhere-zero one-form $α$ , the hyperplane distribution $\ker α$ is integrable if and only if $α \land d α = 0$ . (The codimension-one Frobenius criterion).

[F2]

If $ω$ is nowhere vanishing, then $α \land ω = 0$ for a one-form $α$ forces $α = f ω$ for a unique smooth $f$ , and $θ \land ω = 0$ for a two-form $θ$ forces $θ = β \land ω$ for a smooth one-form $β$ . (Divisibility by a nowhere-vanishing one-form).

[F3]

The wedge product of alternating forms is associative and graded-commutative, so for forms of odd degree $α \land β = - β \land α$ . (The wedge product is associative and graded commutative).

Proof

technique · direct

1.1F1F3given

Since $F$ is integrable with $T F = \ker ω$ , the Frobenius criterion [F1] gives $ω \land d ω = 0$ , and by the graded commutativity of [F3] with degrees one and two (and hence sign $(- 1)^{1 \cdot 2} = 1$ ) this is equivalent to $d ω \land ω = 0$ .

2.1F2step 1.1

Applying the divisibility lemma [F2] to the two-form $θ = d ω$ with $θ \land ω = 0$ produces a smooth one-form $η$ with $d ω = η \land ω$ .

3.1F2step 2.1∎

If $η^{'}$ is another one-form with $d ω = η^{'} \land ω$ , then $(η^{'} - η) \land ω = 0$ , so by part (i) of [F2] there is a unique smooth $f$ with $η^{'} - η = f ω$ ; evaluating at any vector field $X$ with $ω (X) = 1$ gives $f = η^{'} (X) - η (X)$ , which both exhibits $f$ and proves its uniqueness, and no choice principle beyond the standing vocabulary is used.

DefinitionDefinition: Literature-sourcedProof: Not applicableOpen item page →

Smooth foliated concordance of codimension-one foliations

Definition

Assume Countable Choice ${A C}_{ω}$ . Let $M$ be a closed smooth manifold and let $F_{0}, F_{1}$ be transversely oriented codimension-one foliations of $M$ with defining forms $ω_{0}, ω_{1}$ . A smooth foliated concordance from $F_{0}$ to $F_{1}$ is a codimension-one regular foliation $G$ of $W : = M \times [0, 1]$ , transverse to the two boundary slices $M \times {0}$ and $M \times {1}$ , together with a nowhere-vanishing defining $1$ -form $ω$ for $G$ such that for $j = 0, 1$ the pullback $i_{j}^{*} ω$ along $i_{j} (x) : = (x, j)$ is a positive smooth multiple of $ω_{j}$ , and hence a defining form for $F_{j}$ with its prescribed coorientation. By Restriction of a foliation transverse to the boundary each slice $M \times {j}$ carries the codimension-one foliation $G ∣_{M \times {j}}$ with defining form $i_{j}^{*} ω$ , so the requirement is exactly that these boundary foliations be $F_{j}$ with the prescribed co- orientations.

The smooth structure on $M \times [0, 1]$ is given explicitly by product charts: near $s = 0$ use $(x, s)$ and near $s = 1$ use $(x, 1 - s)$ , with x a smooth chart of M. These are half-space charts with smooth product transitions, as required by Smooth charts, atlases, and structures with boundary.

LemmaStatement: Literature-sourcedProof: Literature-sourcedprecheck passOpen item page →

Curvature of an extending Bott connection lies in the transverse differential ideal

Statement

Assume Countable Choice ${A C}_{ω}$ . Let $F$ be a codimension- $q$ regular foliation of a smooth manifold $M$ with normal bundle $ν = T M / T F$ , let $\nabla^{B}$ be the Bott partial connection, and let $\nabla$ be any connection on $ν$ extending it: $\nabla_{X} = \nabla_{X}^{B}$ for every $X \in Γ (T F)$ . Let $I \subseteq Ω^{*} (M)$ be the differential ideal generated by the $1$ -forms that vanish on $T F$ , i.e. the ideal locally generated by a coframe of the annihilator bundle of $T F$ . Then, in a local frame of $ν$ that is parallel along the leaves for $\nabla^{B}$ , the curvature matrix entries of $\nabla$ lie in $I$ ; consequently every coefficient of the curvature two-form $R^{\nabla} \in Ω^{2} (M; E n d ν)$ lies in $I$ , and $I^{q + 1} = 0$ .

Facts & Assumptions

Given: Assume ${A C}_{ω}$ . A codimension- $q$ regular foliation $F$ with tangent distribution $T F$ and normal bundle $ν = T M / T F$ , the Bott partial connection $\nabla^{B}$ on $ν$ , and a connection $\nabla$ on $ν$ with $\nabla_{X} = \nabla_{X}^{B}$ for every $X \in Γ (T F)$ .

[F1]

The Bott partial connection is well defined, is $C^{\infty}$ -linear in the vector-field variable, satisfies the Leibniz rule, and has vanishing curvature $R^{B} (X, Y) s = 0$ for leaf-tangent fields $X, Y$ . (The Bott partial connection is well defined and flat along leaves).

[F3]

In a local frame with connection matrix $ω$ and curvature matrix $Ω$ , the structure equation $Ω = d ω + ω \land ω$ holds. (Curvature two-form structure equation).

[F4]

For homogeneous smooth forms $α, β$ of degrees $p, q$ one has $d (α \land β) = d α \land β + (- 1)^{p} α \land d β$ . (The exterior derivative is a graded derivation).

Proof

technique · direct

1.1givenF1F4

In a foliation chart $(x_{1}, \dots, x_{n - q}, y_{1}, \dots, y_{q})$ , the annihilator of $T F$ is spanned by $d y_{1}, \dots, d y_{q}$ . Thus the ideal $I$ consists locally of sums $\sum_{j} d y_{j} \land α_{j}$ , and is closed under $d$ , because $d (d y_{j}) = 0$ and $d (d y_{j} \land α_{j}) = - d y_{j} \land d α_{j}$ . The classes $e_{j} = π (\partial_{y_{j}})$ form a local frame of $ν$ parallel for the Bott connection: if $X = \sum_{i} X^{i} \partial_{x_{i}}$ then $[X, \partial_{y_{j}}] = - \sum_{i} (\partial_{y_{j}} X^{i}) \partial_{x_{i}}$ is leaf-tangent.

2.1givenstep 1.1

Use the extending connection supplied in the statement. Its connection entries $θ^{a}_{b}$ in the frame of step 1.1 vanish on every leaf direction, since $\nabla_{X} e_{b} = \nabla_{X}^{B} e_{b} = 0$ there. Hence $θ^{a}_{b} \in I$ .

3.1F3step 1.1step 2.1

The structure equation gives $Ω^{a}_{b} = d θ^{a}_{b} + \sum_{c} θ^{a}_{c} \land θ^{c}_{b}$ . Both summands lie in $I$ , by its differential-ideal property and step 2.1. A frame change conjugates the curvature matrix by smooth function matrices, so every curvature entry in every frame lies in $I$ .

4.1step 1.1step 3.1∎

Every product of $q + 1$ local elements of $I$ contains $q + 1$ factors drawn from the $q$ forms $d y_{j}$ ; alternating multiplication forces a repeated factor and gives zero. Thus $I^{q + 1} = 0$ , including $q = 0$ , where $I = 0$ . This proves the assertions without asserting existence of an extension or choosing a global cover.

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

Relative generic position for characteristic disk maps

Statement

Assume Countable Choice ${A C}_{ω}$ (The countable-choice principle used in the foliation pair). Let $M$ be a smooth $3$ -manifold, let $F$ be a cooriented codimension-one foliation of $M$ given by a $C^{2}$ foliated atlas (C¹ codimension-one regular foliations and transverse orientation, read with two continuous derivatives), and let $ω$ be a nowhere-vanishing $C^{2}$ defining $1$ -form with $\ker ω = T F$ . Let $h : D^{2} \to M$ be a $C^{2}$ map and put $β : = h^{*} ω$ , the characteristic covector of $h$ ; its zero set $Sing (β) : = {x \in D^{2} : β_{x} = 0}$ is the set of characteristic singularities of $h$ . Write $C_{+}$ for any fixed closed collar of $\partial D^{2}$ in $D^{2}$ on which $β$ is already nowhere vanishing.

(a) If $h ∣_{\partial D^{2}}$ is a closed transversal to $F$ , that is, $β (τ) \neq 0$ at every point of $\partial D^{2}$ for the unit tangent $τ$ , then $β$ is nowhere vanishing on a collar of $\partial D^{2}$ .

(b) If $h (\partial D^{2})$ lies in a single leaf, that is, $β (τ) = 0$ at every point of $\partial D^{2}$ , then $h$ is homotopic relative to $\partial D^{2}$ to a $C^{2}$ map $h_{1}$ whose characteristic covector $h_{1}^{*} ω$ is nowhere vanishing on a collar of $\partial D^{2}$ ; the homotopy may be chosen with tracks supported in an arbitrarily small collar of $\partial D^{2}$ , and $h_{1}$ may be chosen arbitrarily $C^{0}$ -close to $h$ . Arbitrary $C^{1}$ or $C^{2}$ closeness is not asserted in (b).

(c) In either case, let $h$ now be a map whose characteristic covector is nowhere vanishing on the fixed collar $C_{+}$ . Then for every $C^{2}$ neighbourhood $U$ of $h$ there is a $C^{2}$ map $g \in U$ , equal to $h$ on an open neighbourhood of $C_{+}$ and homotopic to $h$ by a homotopy fixed there, such that $Sing (g^{*} ω)$ is finite, contained in the interior of $D^{2}$ , and consists of nondegenerate points: at each singular point $p$ there is a foliation chart in which the local transverse function $u$ of $g$ has $\nabla u (p) = 0$ and $D^{2} u (p)$ invertible. Each singular point is a center (if $D^{2} u (p)$ is definite, the characteristic line field near $p$ has a family of small closed orbits around $p$ ) or a saddle (if $D^{2} u (p)$ is indefinite, the characteristic line field near $p$ has the usual four-sector hyperbolic picture).

The statement does not assert that distinct singular points map into distinct ambient leaves.

Facts & Assumptions

Given: A cooriented codimension-one $C^{2}$ foliation $F$ of a smooth $3$ -manifold $M$ with nowhere-vanishing $C^{2}$ defining form $ω$ , and a $C^{2}$ map $h : D^{2} \to M$ .

[F1]

In a foliation chart $χ = (x, z)$ of the given $C^{2}$ atlas the leaves are the level sets of the transverse coordinate $z$ , one has $d z \neq 0$ and $T F = \ker d z$ on the chart, and on an overlap the transverse coordinates satisfy $z^{'} = φ (z)$ with $φ$ a $C^{2}$ diffeomorphism of intervals (C¹ codimension-one regular foliations and transverse orientation, Regular foliation atlases). Pulling back $ω = f d z$ with $f \neq 0$ gives $h^{*} ω = (f \circ h) d (z \circ h)$ on the chart, so the singularities of $h^{*} ω$ are exactly the critical points of the local transverse function $u : = z \circ h$ ; and $D^{2} (φ \circ u) = φ^{'} (u) D^{2} u$ at a critical point, so nondegeneracy and the type (definite or indefinite) do not depend on the chart. [F1]

[F2]

A $C^{1}$ map that is nonzero at a point is bounded away from zero on a neighbourhood of it; a continuous function on a compact set attains a positive minimum when it is everywhere positive. (Direct compactness argument, using Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line.)

[F3]

Compactly supported smooth bumps: for $K \subseteq W \subseteq D^{2}$ with $K$ compact and $W$ open there is a smooth $ρ : D^{2} \to [0, 1]$ equal to $1$ near $K$ and supported in $W$ (A manifold bump for a compact set inside an open set).

[F4]

Morse-Sard in Euclidean space: for open $U \subseteq R^{2}$ and a $C^{1}$ map $G : U \to R^{2}$ , the set of critical values of $G$ is a null subset of $R^{2}$ (Morse-Sard for Euclidean maps with $m = n = 2$ , $r = 1$ ); nullity is the cover notion of Measure zero and content zero in $R^{m}$ by countable and finite cube covers.

[F5]

A closed square $Q \subseteq R^{2}$ of side $L$ is compact (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line), and a finite cover of $Q$ by axis-parallel rectangles of total area $V$ admits, for every $η > 0$ , a grid of $Q$ whose cells meeting the covered set have total area below $V + η$ (A finite rectangle cover admits grid control with arbitrarily small volume excess).

[F6]

A $C^{1}$ map between open subsets of $R^{2}$ with invertible derivative at a point has a local $C^{1}$ inverse (The Euclidean inverse function theorem).

[F7]

A $C^{2}$ function whose gradient vanishes and whose Hessian $H$ is invertible satisfies $u (p + w) = u (p) + \frac{1}{2} ⟨ H w, w ⟩ + o (∣ w ∣^{2})$ ; along each ray $w = t ξ$ , $∣ ξ ∣ = 1$ , the radial function $t \mapsto u (p + t ξ)$ is $C^{1}$ with derivative $⟨ H ξ, ξ ⟩ t + o (t)$ . (Taylor expansion of a $C^{2}$ function; The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ applied componentwise to $\nabla u (p + t ξ) - H (t ξ)$ .)

[F8]

Every continuous real function on an interval has a primitive there (Every continuous function on an interval has a primitive; two primitives differ by a constant; and $\int_{a}^{b} f = G (b) - G (a)$ for any primitive $G$ ).

[F9]

A smooth field has a jointly $C^{2}$ local flow; the Euclidean flow formulas glue in finitely many manifold charts by uniqueness (C¹ Euclidean maximal flows, variational dependence and the finite C² upgrade).

Proof

technique · direct

1.1givenF1

Fix a unit tangent $τ$ along $\partial D^{2}$ and let $β = h^{*} ω$ ; by [F1] the singularities of $β$ are exactly the critical points of the transverse functions, so it suffices to manipulate $β$ . In a foliation chart with transverse coordinate $z$ one has $ω = f d z$ with $f \neq 0$ , and a modification of $β$ that does not change $ω$ is the same as a modification of $d (z \circ h)$ ; the condition " $β \neq 0$ " is chart-independent because $β$ is a globally defined $C^{1}$ $1$ -form on $D^{2}$ .

1.2F3F9given

Leafwise boundary: a transverse field. Write $(θ, r)$ for a collar with inward coordinate $r \geq 0$ . The compact image $h (\partial D^{2})$ has a neighborhood carrying a smooth field $V$ with $ω (V) > 0$ : at each image point choose a constant field in a smooth ambient chart with positive evaluation; shrink its domain to retain positivity, take finitely many smaller compact cores covering the image, and patch those finitely many fields with nonnegative smooth bumps from [F3]. Their sum is smooth and has positive evaluation near the image. Its flow $Φ$ is jointly $C^{2}$ on a uniform short time interval there, by [F9] and compactness.

2.1step 1.1F2

The transversal boundary case (a). If $h ∣_{\partial D^{2}}$ is a closed transversal, then by definition $β (τ) \neq 0$ at every boundary point; since $β$ is continuous, [F2] gives $\min_{\partial D^{2}} ∣ β (τ) ∣ = m > 0$ , and by continuity of $β$ and compactness of $\partial D^{2}$ there is a collar $C$ of $\partial D^{2}$ with $β \neq 0$ on $C$ , which is (a).

2.2F2F9step 1.2

Leafwise boundary: normal derivative. Set $a (θ) = ω (d h (\partial_{r}))$ and $b (θ) = ω (V) > 0$ at $h (θ, 0)$ . Choose one constant $c > 0$ with $a + c b > 0$ on the compact boundary. With a smooth cutoff $χ (r)$ equal to one near zero and supported in $[0, ε)$ , define $h_{1} (θ, r) = Φ_{c r χ (r)} (h (θ, r))$ , and keep $h$ outside this collar. This is $C^{2}$ , fixed on the boundary, and there its characteristic covector has zero tangential component and normal component $a + c b > 0$ . Continuity and compactness therefore give a zero-free collar. Scaling the flow time gives a homotopy fixed on the boundary and supported in the chosen collar. Taking $ε$ small makes the value displacement arbitrarily small, while the normal derivative change $c V$ need not be small. This proves (b) with $C^{0}$ closeness.

3.1step 2.1step 2.2F2

Preparation for (c). Enlarge the fixed collar $C_{+}$ slightly to an open collar $C \supset \overline{C_{+}}$ with compact closure on which $β \neq 0$ , and let $D_{1} : = D^{2} ∖ C$ , a compact disk contained in the interior of $D^{2}$ ; all singularities lie in $D_{1}$ . Choose finitely many open disks $V_{1}, \dots, V_{N} \subseteq int D^{2}$ , each with closure disjoint from $C_{+}$ and a compact core $K_{i} \subset V_{i}$ and with $h (\overline{V_{i}})$ contained in a single foliation chart $Q_{i}$ of $F$ with positive margin from its boundary, such that the interiors of the $K_{i}$ cover $D_{1}$ ; this is possible because $D_{1}$ is compact and $h$ is continuous. Since the conditions $h (\overline{V_{i}}) \subseteq Q_{i}$ are open in the $C^{2}$ topology, there is a $C^{2}$ neighbourhood $V \subseteq U$ of $h$ such that every map in $V$ still sends each $\overline{V_{i}}$ into $Q_{i}$ and is still regular on $\overline{C}$ .

4.1step 3.1F3

The local perturbation of (c). Fix $i$ and write the current map on $V_{i}$ as $x \mapsto χ_{i}^{- 1} (Y_{i} (x), u_{i} (x))$ , where $u_{i}$ is the local transverse function. Choose a bump $ρ_{i}$ equal to $1$ near $K_{i}$ and supported in $V_{i}$ [F3], and for a parameter $a \in R^{2}$ define the modified map on $V_{i}$ by replacing $u_{i} (x)$ with $u_{i} (x) + ρ_{i} (x) (a \cdot x)$ , leaving the foliation coordinates $Y_{i}$ and the map outside $V_{i}$ unchanged; since $ρ_{i}$ is compactly supported in the interior of $V_{i}$ , the result is a $C^{2}$ map on $D^{2}$ agreeing with the previous map near $\partial V_{i}$ with all derivatives. On the open set where $ρ_{i} = 1$ the new transverse function is $u_{i} + a \cdot x$ , whose critical points are the solutions of $\nabla u_{i} (x) = - a$ ; for small $a$ the map stays in $V$ .

5.1step 4.1F4F5

Sard makes the core nondegenerate. The gradient map $\nabla u_{i} : V_{i} \to R^{2}$ is a $C^{1}$ map, so by [F4] its set of critical values is null in $R^{2}$ . A null set has empty interior: if a null set contained a closed square $Q$ of side $L \leq 1$ , nullity would give a sequence of closed cubes covering $Q$ with total area at most $L^{2} / 4$ ; thickening the $n$ -th cube by $δ_{n} = \min (1, L^{2} / (1024 \cdot 2^{n} (ℓ_{n} + 1)))$ on each side makes a cover by open cubes whose total area exceeds $L^{2} / 4$ by at most $\sum_{n} (4 ℓ_{n} δ_{n} + 4 δ_{n}^{2}) < L^{2} / 64$ , hence has total area below $17 L^{2} / 64$ ; by compactness of $Q$ [F5] finitely many of them cover $Q$ with total area $V_{f} < L^{2} / 2$ , and [F5] turns this finite cover into a grid of $Q$ whose cells meeting $Q$ (that is, all cells) have total area below $V_{f} + L^{2} / 2 < L^{2}$ , contradicting that the cells of a grid of $Q$ have total area $L^{2}$ . Hence the critical values of $\nabla u_{i}$ have empty interior and arbitrarily small vectors $- a$ are regular values, so all solutions of $\nabla u_{i} (x) = - a$ in $V_{i}$ have invertible Hessian $D^{2} u_{i} (x)$ .

6.1step 3.1step 4.1step 5.1

Preservation of the earlier cores and the fixed charts. At the moment core $i$ has been treated, its singularities are the finite set $\nabla u_{i}^{- 1} (- a) \cap K_{i}$ (finite because $D^{2} u_{i}$ is invertible at each solution, so the solutions are isolated, and $K_{i}$ is compact): they are nondegenerate, and on the compact complement of small isolating disks the gradient of the new transverse function is bounded away from zero. This property, "all singularities in $K_{i}$ are nondegenerate and isolated", is open in the $C^{2}$ topology: near each singularity the Hessian determinant stays nonzero, and on the compact remainder the gradient norm stays positive. Since there are only finitely many earlier cores and finitely many chart conditions, the regular value $- a$ in step 5.1 may be chosen arbitrarily small, and the perturbation in chart $i$ then preserves every earlier core and every fixed chart inclusion; moreover each earlier core property in turn is preserved by all later perturbations for the same reason.

7.1step 3.1step 6.1

Finiteness, interiority and the homotopy. After the finitely many steps, every point of $D_{1}$ lies in the interior of some core $K_{i}$ ; at the end all singularities in each $K_{i}$ remain nondegenerate (their positions may move), because all later perturbations preserve this property, and there are none in the collar $C$ . Hence $Sing (g^{*} ω)$ is a finite set of interior nondegenerate points. Scaling the finitely many parameters $a_{i}$ linearly from $0$ to their chosen values and concatenating the resulting homotopies gives a homotopy from $h$ to $g$ that fixes an open neighbourhood of the original collar $C_{+}$ and keeps every intermediate map $C^{2}$ .

8.1step 7.1F6F7F8

A nondegenerate singularity is a center or a saddle. Let $p$ be a nondegenerate singularity with local transverse function $u$ and Hessian $H = D^{2} u (p)$ , and translate so that $p = 0$ and $u (0) = 0$ . If $H$ is definite, then by [F7] each ray $t \mapsto u (t ξ)$ is strictly monotone in $t$ near $0$ ; for a small positive level $c$ (or negative, according to the sign of $H$ ) every ray meets ${u = c}$ in exactly one point near $0$ , by the intermediate value theorem, and the resulting radius is continuous in $ξ$ ; the levels are therefore small closed curves around $0$ , so the singularity is a center. If $H$ is indefinite, diagonalize $H$ linearly to assume $u_{x x} (0) > 0 > u_{y y} (0)$ ; the map $(x, y) \mapsto u_{x} (x, y)$ has invertible $x$ -derivative $u_{x x} (0)$ at the origin, so [F6] solves $u_{x} = 0$ locally as a $C^{1}$ curve $x = η (y)$ with $η (0) = η^{'} (0) = 0$ . Put $b (y) : = u (η (y), y)$ ; then $b$ is $C^{2}$ with $b^{'} (y) = u_{y} (η (y), y)$ and $b^{''} (0) = u_{y y} (0) < 0$ , and Taylor's theorem with [F8] applied twice in the $x$ variable gives $u (x, y) - b (y) = (x - η (y))^{2} A (x, y)$ and $b (0) - b (y) = y^{2} B (y)$ with $A, B$ continuous and $A (0, 0) > 0$ , $B (0, 0) > 0$ . The changes $X = sign (x - η (y)) \sqrt{u (x, y) - b (y)}$ and $Y = sign (y) \sqrt{b (0) - b (y)}$ are continuous and strictly monotone in $x$ and $y$ respectively near the origin, hence define local coordinates there, and in them $u - u (0) = X^{2} - Y^{2}$ ; the level sets of $u$ therefore have the four-sector saddle picture.

9.1step 2.1step 2.2step 7.1step 8.1∎

By steps 2.1, 2.2, 7.1 and 8.1 assertions (a), (b) and (c) hold. The construction selects only finitely many objects at each stage (finitely many charts, finitely many bumps, finitely many arbitrarily small regular values), so the proof's own choices are finite and need no choice principle; the stated hypothesis ${A C}_{ω}$ is inherited from the cooriented smooth-distribution interface used to speak of the foliation, its defining form and its flat charts, exactly as recorded in Transversely oriented codimension-one foliations. Nothing here separates distinct singularities into distinct leaves, since in a nonproper foliation two different transverse coordinates may lie in the same leaf; this is why no such separation is asserted.

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A C² product coordinate on a planar period annulus

Statement

Let $X$ be a $C^{2}$ vector field on an open subset of $R^{2}$ , and let $A$ be an open annulus saturated by $X$ on which $X$ is nowhere zero and every orbit is a simple periodic curve. Assume these periodic curves are strictly nested Jordan curves with a consistent orientation. Then there are an interval $I = (0, 1)$ and a $C^{2}$ diffeomorphism $Ψ : S^{1} \times I \to A$ taking each circle $S^{1} \times {s}$ onto one orbit and a positive $C^{2}$ function $a : I \to (0, \infty)$ such that $X = a (s) \partial_{θ}$ in these coordinates. The coordinate may be chosen to increase from the inner end to the outer end of the annulus.

Facts & Assumptions

Given: A $C^{2}$ vector field $X$ on an open set containing the open annulus $A$ , on which $X$ is nowhere zero, every orbit is a simple periodic curve, the periodic curves are strictly nested Jordan curves, and their boundary orientations agree.

[F1]

If $Y$ is $C^{1}$ on an open $U \subseteq R^{n}$ , its maximal flow is jointly $C^{1}$ with a $C^{1}$ flow box at each regular point; if $Y$ is $C^{2}$ the flow and those boxes are $C^{2}$ ; individual trajectories of a $C^{1}$ field are $C^{2}$ in time; and a trajectory remaining in a compact subset of $U$ has no finite maximal endpoint (C¹ Euclidean maximal flows, variational dependence and the finite C² upgrade).

[F2]

A $C^{2}$ map between open subsets of $R^{n}$ with invertible derivative at a point has a $C^{2}$ local inverse; and if $g (s, t)$ is $C^{2}$ near $(s_{0}, t_{0})$ with $g (s_{0}, t_{0}) = 0$ and $g_{t} (s_{0}, t_{0}) \neq 0$ , then there is a unique local $C^{2}$ root $t = T (s)$ , with $T^{'} = - g_{s} / g_{t}$ and $T^{''} = - (g_{s s} + 2 g_{s t} T^{'} + g_{t t} (T^{'})^{2}) / g_{t}$ (C² inverses and scalar return roots).

[F3]

Closed and bounded subsets of $R^{2}$ are compact; a decreasing nested family of nonempty compact subsets has nonempty intersection; a continuous real function on a nonempty compact set attains its maximum and minimum (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line).

[F6]

A topological embedding $c : S^{1} \to R^{2}$ that is piecewise $C^{2}$ with finitely many corners, each with two distinct one-sided tangent rays and regular edges, has a complement with exactly two connected components, one bounded and one unbounded (A finitely cornered regular plane curve separates without choice).

Proof

technique · direct

1.1givenF1F6

Let $J$ be the quarter-turn $J (u_{1}, u_{2}) = (- u_{2}, u_{1})$ and set $Z = \pm J X$ with the sign chosen so that $Z$ crosses each orbit from its bounded Jordan domain to the exterior; at a fixed orbit the crossing sense of $Z$ is a continuous nowhere-zero directional datum along the compact orbit and the consistent orientation hypothesis keeps its sign fixed, while the sense depends locally constantly on the orbit and the orbit family is connected, so one global sign makes every crossing of every orbit by $Z$ outward; hence $Z$ is a nowhere-zero $C^{2}$ field on $A$ transverse to $X$ .

2.1step 1.1F6

Fix $x_{0} \in A$ and let $σ : (t_{-}, t_{+}) \to A$ be the maximal $Z$ -trajectory with $σ (0) = x_{0}$ ; then $σ$ crosses each orbit at most once: if $t_{1} < t_{2}$ were consecutive crossing times of one orbit $C$ and $σ ([t_{1}, t_{2}])$ avoided $C$ , that connected arc would lie in one component of $R^{2} ∖ C$ by [F6], yet outward crossings at $t_{1}$ and $t_{2}$ put the points just after $t_{1}$ and just before $t_{2}$ on opposite sides of $C$ , a contradiction.

3.1step 2.1F6

All orbits lying strictly between two orbits crossed by $σ$ are crossed: if $C_{1}$ is inside $C_{2}$ and $σ (t_{1}) \in C_{1}$ , $σ (t_{2}) \in C_{2}$ , then $σ (t_{1})$ lies in the bounded component of $R^{2} ∖ C$ and $σ (t_{2})$ in the unbounded one for every orbit $C$ between them, so the connected arc $σ ([t_{1}, t_{2}])$ cannot avoid $C$ and some intermediate time lies on $C$ .

4.1F1F2F3step 1.1step 2.1step 3.1

The crossed orbits exhaust $A$ . First each orbit $C$ has a local period tube: a short local $Z$ -trajectory through a point of $C$ , which meets each orbit at most once by the argument of step 2.1, and the $C^{2}$ flow give a first-return map near its least period by [F2]; compactness of one traversal excludes returns away from the endpoints. The returned point lies on the same periodic orbit and on that local section, which meets every orbit at most once. Thus the return point is the initial point. Thus the nearby return time gives a $C^{2}$ circle product, with a transverse leaf coordinate $r$ . On a smaller closed tube the outward transverse field satisfies $Z r \geq c > 0$ by compactness. By step 3.1 the crossed family is order-convex; if it stopped at an orbit $C$ inside $A$ , the section would eventually lie in such a tube on the inner side of $C$ . It cannot leave through that side because $Z r > 0$ , and the bound $Z r \geq c$ forces it to reach $C$ in finite time. Compact flow continuation from [F1] excludes an earlier maximal endpoint. The reversed argument treats an inner stopping orbit. Hence every orbit is crossed once.

5.1step 4.1F1

The maximal trajectory $σ$ is $C^{2}$ , and after composing its parameter with one explicit increasing $C^{2}$ diffeomorphism of its open time interval onto $(0, 1)$ (affine when both ends are finite, and an arctan-type explicit map when an end is infinite) the section may be written $σ : (0, 1) \to A$ , is still $C^{2}$ , and meets every orbit exactly once with the parameter increasing from the inner to the outer end.

6.1step 5.1F1F2F3

For each $s$ the orbit of $σ (s)$ is a simple periodic curve of a nowhere-zero field, so its period set is a closed additive subgroup of $R$ whose discreteness gives a least positive period $T (s)$ ; fixing $s_{0}$ and a $C^{2}$ flow box at $σ (s_{0})$ with $X = \partial_{y}$ and the section near $σ (s_{0})$ a graph $y = η (x)$ , the function $F (s, t) : = y (s, t) - η (x (s, t))$ , built from the jointly $C^{2}$ flow of the $C^{2}$ field, is $C^{2}$ with $F (s_{0}, T (s_{0})) = 0$ and $F_{t} (s_{0}, T (s_{0})) = 1$ , so [F2] gives a unique local $C^{2}$ return time $T (s)$ near $s_{0}$ ; for $s$ near $s_{0}$ no smaller positive return occurs, because the trajectory of $σ (s)$ stays uniformly close to the reference orbit on the compact time interval $[δ, T (s_{0}) - δ]$ and avoids the section there by [F1] and [F3], while inside the flow box the section is met only at $t = 0$ ; hence $T$ is $C^{2}$ on all of $(0, 1)$ .

7.1step 6.1F1F2

Define $Ψ (θ, s) : = Φ_{θ T (s) / (2 π)}^{X} (σ (s))$ on $S^{1} \times (0, 1)$ ; it is $C^{2}$ and $2 π$ -periodic in $θ$ , and it is bijective because every orbit meets $σ$ exactly once and $θ$ modulo $2 π$ parametrizes that orbit once; its columns $\partial_{θ} Ψ = T (s) X (Ψ) / (2 π)$ and $\partial_{s} Ψ$ , the latter being a scalar multiple of $X$ plus the pushforward $D Φ_{θ T (s) / (2 π)}^{X} [Z (σ (s))]$ of the transverse vector $Z (σ (s))$ , are everywhere independent because a time slice of the flow is a linear isomorphism carrying the line spanned by $X$ onto the line spanned by $X$ ; so $D Ψ$ is invertible everywhere, [F2] gives $C^{2}$ local inverses, and they agree globally by bijectivity, making $Ψ$ a $C^{2}$ diffeomorphism onto $A$ .

8.1step 7.1F2∎

Since $\partial_{θ} Ψ = T (s) X (Ψ (θ, s)) / (2 π)$ , the pushforward satisfies $Ψ_{*}^{- 1} X = a (s) \partial_{θ}$ with $a (s) = 2 π / T (s) > 0$ of class $C^{2}$ on $(0, 1)$ ; the section parameter increases from the inner to the outer end by construction, and the argument used one specified initial point, finitely many flow boxes and compactness arguments and the explicit reparametrization, hence no choice principle, so $A$ , $X$ , $Ψ$ and $a$ have all the asserted properties.

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Local generalized Poincare-Bendixson theorem for a precompact planar orbit

Statement

Let $U \subseteq R^{2}$ be open and let $Y : U \to R^{2}$ be a $C^{1}$ vector field with flow $Φ$ . Let $O^{+} (y) = {Φ_{t} (y) : t \geq 0}$ be a positive orbit whose closure $K_{0} = \overline{O^{+} (y)}$ is compact and satisfies $K_{0} ⋐ U$ . Put $K = ω^{+} (y) = ⋂_{T \geq 0} \overline{{Φ_{t} (y) : t \geq T}}$ , and assume $K$ contains only finitely many equilibria (zeros of $Y$ ). Then exactly one of the following forms holds:

(i) $K$ is a singleton equilibrium;

(ii) $K$ is one regular periodic orbit;

(iii) $K$ is a nonempty finite set $E$ of equilibria together with at least one regular trajectory, and every regular point of $K$ lies on such a trajectory whose alpha- and omega-limit sets are points of $E$ .

The family of connecting trajectories in (iii) need not be finite. The conclusion uses no choice principle beyond the ambient Euclidean completeness.

Facts & Assumptions

Given: A $C^{1}$ field $Y$ on an open set $U \subseteq R^{2}$ , a positive orbit $O^{+} (y)$ with compact closure $K_{0} ⋐ U$ , and the limit set $K = ω^{+} (y) = ⋂_{T \geq 0} \overline{{Φ_{t} (y) : t \geq T}}$ , which contains only finitely many equilibria.

[F1]

The field $Y$ has a unique maximal flow $Φ$ , jointly $C^{1}$ and satisfying $\partial_{t} Φ = Y (Φ)$ ; two trajectories through one point agree on the common part of their time intervals; each time slice is injective; a trajectory remaining in a compact subset of $U$ has no finite maximal endpoint; each trajectory of a $C^{1}$ field is $C^{2}$ in time; and at a regular point there is a $C^{1}$ flow box whose plaques are carried by the flow (C¹ Euclidean maximal flows, variational dependence and the finite C² upgrade).

[F2]

A piecewise- $C^{2}$ topological embedding $c : S^{1} \to R^{2}$ with finitely many corners, each having two distinct one-sided tangent rays and regular edges, has a complement with exactly two connected components, one bounded and one unbounded (A finitely cornered regular plane curve separates without choice).

[F3]

Closed and bounded subsets of $R^{2}$ are compact, so a nested decreasing family of nonempty compact subsets of $R^{2}$ has nonempty intersection, and a continuous function on a compact set attains its bounds (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line).

Proof

technique · direct

1.1givenF1F3

The sets $K_{T} : = \overline{{Φ_{t} (y) : t \geq T}}$ are nested nonempty compact connected subsets of $K_{0} ⋐ U$ ; by the finite-intersection property for nested compacta [F3], $K = ⋂_{T} K_{T}$ is nonempty and compact, and it is connected because the intersection of a decreasing family of continua is a continuum. Flow continuity and the flow law make $K$ invariant: $Φ_{s} (K) \subseteq K$ for every real $s$ for which it is defined near $K$ , since $Φ_{s} (K_{T}) \subseteq K_{T + s}$ and $K$ is closed.

1.2givenF1F2

A transverse section and monotone crossings. Fix a regular point $w \in K_{0}$ and a compact embedded segment $Σ$ through $w$ , contained in one flow box of [F1] and transverse to $Y$ at every point; parameterize $Σ$ by an interval coordinate and write $p_{0} < p_{1} < \dots$ for the intersection points of an orbit with $Σ$ in the order of visiting times $t_{0} < t_{1} < \dots$ . Such times are isolated, because the flow box straightens the field and the section is transverse to its direction. Let $p_{0}, p_{1}$ be two consecutive intersections of some orbit with $Σ$ and consider the closed curve $C$ formed by the orbit arc $γ ∣_{[t_{0}, t_{1}]}$ together with the subarc of $Σ$ from $p_{0}$ to $p_{1}$ . It is a simple closed piecewise- $C^{2}$ curve with two corners at $p_{0}, p_{1}$ , each having distinct one-sided tangents, because the orbit is transverse to $Σ$ and, by uniqueness [F1], the arc has no self-intersection and, by consecutiveness, meets $Σ$ only at its endpoints; both facts use that the orbit is not periodic on $[t_{0}, t_{1}]$ . By [F2] the complement of $C$ has exactly two components. The forward orbit leaves $p_{1}$ on the side of the crossing opposite to the incoming arc and hence enters the component of $R^{2} ∖ C$ whose closure meets $Σ$ in the ray beyond $p_{1}$ : it cannot cross the orbit arc by uniqueness and cannot meet $Σ$ until its next visit, so the next intersection satisfies $p_{2} > p_{1}$ when $p_{1} > p_{0}$ , and symmetrically $p_{2} < p_{1}$ when $p_{1} < p_{0}$ . Repeating the same argument for each consecutive triple makes the sequence of crossing coordinates strictly monotone in one direction; a repeated intersection, instead, makes the orbit periodic by uniqueness.

2.1step 1.2F1

At most one point of $K$ on a compact section. Let $Σ$ be compact and transverse as in step 1.2, extend it slightly within its flow box so its endpoints are interior to the extended section, and let $q \in K \cap Σ$ . Then $q$ is a limit of crossing points of the original orbit with $Σ$ : late orbit points $Φ_{t} (y)$ with $t \to \infty$ approach $q$ , and in a small flow box around $q$ the section is crossed within a uniformly bounded signed time, so some crossing point lies arbitrarily close to $q$ . Since a transverse section meets each time-parametrised orbit in isolated times, all these crossing points avoid neighbours of $q$ only finitely often; more precisely, the monotone sequence of crossing coordinates of the orbit converges to the coordinate of the unique limit point. By the strict monotonicity of step 1.2 (for the orbit's crossings, or for the crossings of any invariant orbit inside $K$ ) two distinct points of $K \cap Σ$ would give two different limits of the same monotone sequence, which is impossible; hence $K \cap Σ$ has at most one point, and any orbit contained in $K$ has at most one distinct intersection point with $Σ$ , although a periodic orbit returns to that point repeatedly.

3.1step 1.1step 2.1F1

The dichotomy for a regular point of $K$ . Fix a regular point $z \in K$ ; since $K$ is invariant [step 1.1], the whole trajectory of $z$ lies in $K$ . Its forward limit set $ω^{+} (z) = ⋂_{T \geq 0} \overline{{Φ_{t} (z) : t \geq T}}$ is nonempty, compact, connected and contained in $K$ by the same nested-tail argument as in step 1.1, and it is invariant. If $ω^{+} (z)$ contains a regular point $w$ , choose a compact transverse section $Σ$ through $w$ ; the forward orbit of $z$ crosses $Σ$ infinitely often at points accumulating at $w$ , and all these crossing points lie in $K$ by invariance and closedness, hence in the at-most-single-point set $K \cap Σ$ of step 2.1; thus two such crossings coincide, and by uniqueness the orbit of $z$ is periodic, with $ω^{+} (z)$ equal to that periodic orbit. If $ω^{+} (z)$ contains no regular point, then every point of it is an equilibrium, so $ω^{+} (z)$ is a nonempty connected subset of the finite equilibrium set and hence a singleton equilibrium.

4.1step 3.1F1

Assembling the three alternatives. If $K$ has no regular points, then $K$ is a connected nonempty subset of the finite equilibrium set, hence a singleton equilibrium, which is (i). If $K$ has no equilibrium, take any regular $z \in K$ ; by step 3.1 either its orbit is periodic, in which case the periodic orbit $P \subseteq K$ is compact, or $ω^{+} (z)$ is a singleton equilibrium, contrary to the absence of equilibria; so $P$ exists. The periodic orbit is open in $K$ : a finite flow-box tube around $P$ meets $K$ only in $P$ , because any point of $K$ in such a tube is carried by the flow to a transverse section that already meets $P$ in at most one point, and would either produce a second point of $K$ on that section or lie on $P$ ; formally, apply step 2.1 to a short section through a point of $P$ and to the crossings forced by the tube. Being also closed in the compact $K$ and nonempty, $P = K$ by connectedness of $K$ , which is (ii). Finally suppose $K$ has both a regular point $z$ and an equilibrium. Then $ω^{+} (z)$ is a singleton equilibrium by step 3.1, and the same section argument applies to the alpha-limit of $z$ : a regular alpha-limit point would force two negative-time crossings in the singleton $K \cap Σ$ , and hence periodicity; thus its alpha-limit is a singleton equilibrium; for an arbitrary regular point $z^{'}$ of $K$ the same dichotomy gives that $ω^{+} (z^{'})$ is a singleton equilibrium as well, since if it were the periodic orbit $P$ of step 3.1 then the tube argument of the second case above would make $P$ open and closed in the connected $K$ , so $K = P$ would carry no equilibrium, contrary to the present case, and the same negative-time section argument makes $α (z^{'})$ a singleton equilibrium; writing $E$ for the finite set of equilibria of $K$ , every regular point of $K$ lies on its own trajectory and has both one-sided limit sets in $E$ , so $K = E \cup {the regular trajectories in K}$ , which is (iii).

5.1step 4.1∎

Steps 1.1–4.1 cover the three cases exhaustively and each alternative holds exactly when the corresponding case does, so exactly one of (i), (ii), (iii) occurs; the arguments used only the flow box and uniqueness clauses of the $C^{1}$ flow [F1], the finite-corner Jordan separation [F2] and compactness [F3], all of which are choice-free, so no choice principle is invoked.

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A C1 planar gradient at a nondegenerate saddle has local stable and unstable curves

Statement

Let $u$ be of class $C^{2}$ near a point $p$ of the Euclidean plane $R^{2}$ , and suppose $p$ is a nondegenerate saddle of $u$ : $\nabla u (p) = 0$ and $D^{2} u (p)$ is invertible with one positive and one negative eigenvalue. Then the $C^{1}$ field $X : = - \nabla u$ has a local stable curve $S$ and a local unstable curve $U$ through $p$ : both are $C^{1}$ embedded curves containing $p$ , the tangent line $T_{p} S$ is the positive eigenline of $D^{2} u (p)$ and $T_{p} U$ is its negative eigenline, and each of $S ∖ {p}$ and $U ∖ {p}$ consists of exactly two half-trajectories of $X$ . There is a neighbourhood $V$ of $p$ such that every trajectory of $X$ that is defined and stays in $V$ for all $t \geq 0$ lies on $S$ , and every trajectory that is defined and stays in $V$ for all $t \leq 0$ lies on $U$ . The convergence to $p$ along $S$ is exponentially fast as $t \to + \infty$ and the convergence along $U$ is exponentially fast as $t \to - \infty$ : there are constants $δ, C, β > 0$ with $∣ Φ_{t} (x) - p ∣ \leq C e^{- β t} ∣ x - p ∣$ for all $x \in S$ with $∣ x - p ∣ < δ$ and all $t \geq 0$ , and $∣ Φ_{t} (x) - p ∣ \leq C e^{β t} ∣ x - p ∣$ for all $x \in U$ with $∣ x - p ∣ < δ$ and all $t \leq 0$ . No choice principle is used.

Facts & Assumptions

Given: A $C^{2}$ function $u$ near a point $p \in R^{2}$ with $\nabla u (p) = 0$ and $D^{2} u (p)$ invertible with one positive and one negative eigenvalue.

[F1]

For a $C^{1}$ field $Y$ on an open set of $R^{n}$ there is a unique maximal jointly $C^{1}$ flow $Φ$ with $\partial_{t} Φ = Y (Φ)$ , trajectories of $Y$ are $C^{1}$ in time, and two trajectories through the same point agree on the common part of their time intervals (C¹ Euclidean maximal flows, variational dependence and the finite C² upgrade).

[F2]

A contraction of a nonempty complete metric space has a unique fixed point (A contraction of a nonempty complete metric space into itself has exactly one fixed point, the limit of the iterates from any starting point).

[F3]

If $T$ is a bounded linear operator on a Banach space with $∥ T ∥ < 1$ , then $I - T$ is invertible with $(I - T)^{- 1} = \sum_{n \geq 0} T^{n}$ and $∥ (I - T)^{- 1} ∥ \leq (1 - ∥ T ∥)^{- 1}$ , and $S \mapsto (I - S)^{- 1}$ is continuous at every such $S$ (Neumann series and small perturbations of bounded inverses).

[F4]

A pointwise limit of continuous real functions that is uniform on the domain is continuous (The uniform limit of continuous real-valued functions on a metric space is continuous).

[F5]

$R^{n}$ with the Euclidean metric is complete ( $R$ and $R^{n}$ for $n \geq 1$ with the Euclidean metric are complete, componentwise from the Cauchy criterion in $R$ ).

[F6]

For real $a < b$ , a real function continuous on $[a, b]$ and differentiable on $(a, b)$ satisfies $f (b) - f (a) = f^{'} (c) (b - a)$ for some $c \in (a, b)$ (The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ ).

[F7]

A continuous map from a nonempty compact space to a metric space is uniformly continuous (Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous).

[F8]

A real symmetric endomorphism of $R^{n}$ has an orthonormal basis of eigenvectors with real eigenvalues (Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis).

[F9]

Every continuous real function on an order-convex interval with at least two elements has a primitive there, and primitives differ by constants (Every continuous function on an interval has a primitive; two primitives differ by a constant; and $\int_{a}^{b} f = G (b) - G (a)$ for any primitive $G$ ).

Proof

technique · direct

1.1givenF8

Translating the source point to the origin and subtracting the constant $u (p)$ , assume $p = 0$ , $u (0) = 0$ and $\nabla u (0) = 0$ . Put $H : = D^{2} u (0)$ and $A : = - H$ . By [F8] the symmetric matrix $A$ has an orthonormal basis of eigenvectors; its eigenvalues are $- a$ and $b$ for some $a, b > 0$ , because $H$ has one negative and one positive eigenvalue. Let $E^{s}$ be the eigenline of $- a$ and $E^{u}$ the eigenline of $b$ , and let $P_{s}, P_{u}$ be the orthogonal projections onto them, so $P_{s} + P_{u} = I$ , $e^{t A} = e^{- a t} P_{s} + e^{b t} P_{u}$ , and for $s \geq t \geq 0$ one has $∣ e^{(t - s) A} P_{u} ∣ \leq e^{- b (s - t)}$ while $∣ e^{t A} P_{s} ∣ \leq e^{- a t}$ . Fix $β$ with $0 < β < \min (a, b)$ . Since $\nabla u$ is $C^{1}$ with derivative $H$ at the origin, the field $X : = - \nabla u$ has the form $X (x) = A x + R (x)$ with $R$ of class $C^{1}$ , $R (0) = 0$ and $D R (0) = 0$ .

2.1step 1.1F6F7

A compactly supported perturbation of $R$ with small derivative. Let $ε > 0$ (to be fixed below). Choose $ρ > 0$ with $∣ D R (x) ∣ \leq ε$ for $∣ x ∣ \leq 2 ρ$ ; this is possible because $D R (0) = 0$ and $D R$ is continuous. Choose a smooth cutoff $χ : R^{2} \to [0, 1]$ with $χ = 1$ on the ball $B_{ρ}$ and $χ = 0$ outside $B_{2 ρ}$ , and put $\tilde{R} : = χ R$ . On $B_{ρ}$ one has $\tilde{R} = R$ , and $\tilde{R} (0) = 0$ and $D \tilde{R} = 0$ there at the origin. On the support of $D χ$ , which is a compact subset of $B_{2 ρ} ∖ B_{ρ}$ , the mean value theorem [F6] gives $∣ R (x) ∣ \leq ε ∣ x ∣ \leq 2 ε ρ$ , while $∣ D χ ∣ \leq C / ρ$ for a constant $C$ depending only on the fixed cutoff profile; hence $∣ D \tilde{R} (x) ∣ \leq ε + (2 C) ε = (1 + 2 C) ε$ for every $x$ , using $D \tilde{R} = χ D R + R D χ$ . Since $ε$ is arbitrary, $ε^{'} : = \sup_{x} ∣ D \tilde{R} (x) ∣$ can be made as small as we please. Moreover $D \tilde{R}$ is continuous with compact support, hence uniformly continuous on $R^{2}$ by [F7]. Set $\tilde{X} (x) : = A x + \tilde{R} (x)$ ; then $\tilde{X} = X$ on $B_{ρ}$ .

2.2step 1.1F4F5

The weighted space of paths. Let $B$ be the set of continuous $z : [0, \infty) \to R^{2}$ with $∥ z ∥_{β} : = \sup_{t \geq 0} e^{β t} ∣ z (t) ∣ < \infty$ . This is a normed vector space, and it is complete: if $(z_{n})$ is $∥ \cdot ∥_{β}$ -Cauchy then each sequence $(z_{n} (t))$ is Cauchy in $R^{2}$ , which is complete by [F5], so $z (t) : = \lim_{n} z_{n} (t)$ exists; the Cauchy bound $e^{β t} ∣ z_{n} (t) ∣ \leq M$ passes to the limit, so $z \in B$ , and on each $[0, T]$ the convergence is uniform, so every component of $z$ is continuous by [F4]; finally $∥ z_{n} - z ∥_{β} \to 0$ . Also $∣ z (t) ∣ \leq e^{- β t} ∥ z ∥_{β}$ for every $z \in B$ and $t \geq 0$ .

3.1step 1.12.12.2F6F9

The integral operator and its contraction constant. For $z \in B$ define $N (z) (t) : = \int_{0}^{t} e^{(t - s) A} P_{s} \tilde{R} (z (s)) d s - \int_{t}^{\infty} e^{(t - s) A} P_{u} \tilde{R} (z (s)) d s .$ Both integrals converge absolutely, because $∣ \tilde{R} (p) ∣ \leq ε^{'} ∣ p ∣$ by [F6] and the kernel bounds of step 1.1 give $∣ e^{(t - s) A} P_{u} \tilde{R} (z (s)) ∣ \leq ε^{'} e^{- b (s - t)} e^{- β s} ∥ z ∥_{β}$ whose $s$ -integral over $[t, \infty)$ is $ε^{'} ∥ z ∥_{β} e^{- β t} / (b + β)$ . The same kernel bounds give, after multiplying by $e^{β t}$ , $∥ N (z) ∥_{β} \leq ε^{'} (\frac{1}{a - β} + \frac{1}{b + β}) ∥ z ∥_{β}, ∥ N (z) - N (w) ∥_{β} \leq ε^{'} (\frac{1}{a - β} + \frac{1}{b + β}) ∥ z - w ∥_{β},$ because $∣ \tilde{R} (p) - \tilde{R} (q) ∣ \leq ε^{'} ∣ p - q ∣$ by the componentwise mean value theorem [F6]; recall $a > β$ . Also $N (z)$ is continuous, being the difference of two continuous functions of $t$ . Fix $ε$ in step 2.1 so small that $k : = ε^{'} (1 / (a - β) + 1 / (b + β)) \leq \frac{1}{2}$ and $k_{0} : = ε^{'} (1 / a + 1 / b) \leq \frac{1}{2}$ .

4.1step 2.13.1F6F7

$B$ -Fréchet differentiability of $N$ . For $z, w \in B$ let $D N (z) w$ be given by the same formula with $\tilde{R} (z (s))$ replaced by $D \tilde{R} (z (s)) w (s)$ . The estimates of step 3.1 show that $D N (z)$ is linear and bounded with $∥ D N (z) w ∥_{β} \leq k ∥ w ∥_{β}$ . By the componentwise mean value theorem [F6] applied on the segment from $z (s)$ to $z (s) + w (s)$ , $∣ \tilde{R} (z (s) + w (s)) - \tilde{R} (z (s)) - D \tilde{R} (z (s)) w (s) ∣ \leq ω (∣ w (s) ∣) ∣ w (s) ∣,$ where $ω (δ) : = \sup {∣ D \tilde{R} (p^{'}) - D \tilde{R} (p) ∣ : ∣ p^{'} - p ∣ \leq δ}$ satisfies $ω (δ) \to 0$ as $δ \to 0$ by the uniform continuity of $D \tilde{R}$ established in step 2.1. Since $∣ w (s) ∣ \leq ∥ w ∥_{β}$ , the weighted kernel estimates give $∥ N (z + w) - N (z) - D N (z) w ∥_{β} \leq C_{β} ω (∥ w ∥_{β}) ∥ w ∥_{β}$ , where $C_{β} = 1 / (a - β) + 1 / (b + β)$ ; this is $o (∥ w ∥_{β})$ : thus $D N (z)$ is the Fréchet derivative of $N$ at $z$ , and $z \mapsto D N (z)$ is continuous in operator norm because $ω$ is a modulus of continuity.

4.2step 1.13.1F2

Fixed points of the contractions $T_{ξ}$ . For $ξ \in E^{s}$ put $y_{ξ} (t) : = e^{t A} ξ$ , so $y_{ξ} \in B$ and $∥ y_{ξ} ∥_{β} \leq ∣ ξ ∣$ by step 1.1, and define $T_{ξ} (z) : = y_{ξ} + N (z)$ . By step 3.1, $T_{ξ}$ is a $k$ -contraction of the nonempty complete space $B$ with $k \leq \frac{1}{2}$ ; by the Banach fixed point theorem [F2] it has a unique fixed point $z_{ξ} \in B$ . Since $N (0) = 0$ , the estimates of step 3.1 give $∥ z_{ξ} ∥_{β} \leq ∥ y_{ξ} ∥_{β} + k ∥ z_{ξ} ∥_{β}$ , hence $∥ z_{ξ} ∥_{β} \leq 2 ∣ ξ ∣$ and $∣ z_{ξ} (t) ∣ \leq 2 ∣ ξ ∣ e^{- β t} (t \geq 0) .$

4.3step 2.13.14.2F1F9

$z_{ξ}$ is a trajectory of $X$ with exponential decay. Write the first integral of $N (z_{ξ}) (t)$ as $e^{t A} P_{s} \int_{0}^{t} e^{- s A} P_{s} \tilde{R} (z_{ξ} (s)) d s$ and the second as $e^{t A} P_{u} (I_{\infty} - \int_{0}^{t} e^{- s A} P_{u} \tilde{R} (z_{ξ} (s)) d s)$ with $I_{\infty} : = \int_{0}^{\infty} e^{- s A} P_{u} \tilde{R} (z_{ξ} (s)) d s$ ; the integrands are continuous and the integrals converge absolutely as in step 3.1. Differentiating with the product rule and the primitive of a continuous function [F9] gives $z_{ξ}^{'} (t) = A z_{ξ} (t) + P_{s} \tilde{R} (z_{ξ} (t)) + P_{u} \tilde{R} (z_{ξ} (t)) = A z_{ξ} (t) + \tilde{R} (z_{ξ} (t)) = \tilde{X} (z_{ξ} (t))$ for every $t \geq 0$ , so $z_{ξ}$ solves the ODE of $\tilde{X}$ ; by [F1] it is the trajectory $Φ_{t} (z_{ξ} (0))$ . If $∣ ξ ∣ \leq ρ / 2$ then $∣ z_{ξ} (t) ∣ \leq ρ$ for all $t \geq 0$ by the exponential bound, so the trajectory stays in the ball where $\tilde{X} = X$ ; hence it is a trajectory of $X$ and $z_{ξ} (0) = ξ + h (ξ), h (ξ) : = - \int_{0}^{\infty} e^{- s A} P_{u} \tilde{R} (z_{ξ} (s)) d s \in E^{u},$ is the initial point of a trajectory of $X$ converging exponentially to the origin with rate $β$ .

5.1step 4.14.24.3F3

The initial points depend $C^{1}$ on $ξ$ . Let $Ψ (y)$ be the unique fixed point of $z \mapsto y + N (z)$ ; it exists by the same contraction argument as in step 4.2 and $z_{ξ} = Ψ (y_{ξ})$ , where $ξ \mapsto y_{ξ} = e^{\cdot A} ξ$ is linear and bounded into $B$ . We show that $Ψ$ is Fréchet differentiable with $D Ψ (y) = (I - D N (Ψ (y)))^{- 1}$ . Let $u : = Ψ (y)$ , $v : = Ψ (y + η)$ and $r : = N (v) - N (u) - D N (u) (v - u)$ ; step 4.1 gives $∥ r ∥ \leq C_{β} ω (∥ v - u ∥) ∥ v - u ∥$ , and $(I - D N (u)) (v - u) = η + r$ . Since $∥ D N (u) ∥ \leq k < 1$ , the operator $I - D N (u)$ is invertible with inverse of norm at most $(1 - k)^{- 1}$ by the Neumann series [F3], so $∥ v - u ∥ \leq (1 - k)^{- 1} (∥ η ∥ + ∥ r ∥)$ , while the contraction inequality directly gives $∥ v - u ∥ \leq (1 - k)^{- 1} ∥ η ∥$ . Therefore $v - u - (I - D N (u))^{- 1} η = (I - D N (u))^{- 1} r = o (∥ η ∥)$ , so $D Ψ (y)$ exists and has the stated form; it is a bounded linear map. Continuity of $y \mapsto D Ψ (y)$ follows from the Lipschitz continuity of $Ψ$ , the continuity of $D N$ (step 4.1) and the continuity of the inversion map $S \mapsto (I - S)^{- 1}$ [F3]. Consequently $ξ \mapsto z_{ξ} = Ψ (e^{\cdot A} ξ)$ is $C^{1}$ , and since the evaluation $z \mapsto z (0)$ is linear and bounded and $D N (0) = 0$ , differentiating $z_{ξ}$ at $ξ = 0$ in a direction $δ \in E^{s}$ gives $e^{\cdot A} δ$ , so $D G (0) = i d$ for $G (ξ) : = z_{ξ} (0) = ξ + h (ξ), D h (0) = P_{u} D G (0) ∣_{E^{s}} = 0.$

5.2F2F4F5step 2.2step 3.1step 4.2step 4.3

Bounded trajectories. A bounded trajectory $y$ in $B_{ρ}$ satisfies variation of constants. In its unstable component, $e^{- b t} P_{u} y (t) \to 0$ and the integral of $e^{- b s} P_{u} \tilde{R} (y (s))$ converges, since its integrand is bounded by $ε' ρ e^{- b s}$ . Therefore $y = T_{ξ} y$ with $ξ = P_{s} y (0)$ . This does not yet put $y$ in the weighted space. Instead use the complete space $C_{b} ([0, \infty), R^{2})$ with the supremum norm; its completeness follows by the pointwise-limit and uniform-continuity argument of step 2.2. The same operator has contraction constant $k_{0} = ε' (1 / a + 1 / b) \leq \frac{1}{2} < 1$ there. The weighted fixed point $z_{ξ}$ is bounded and solves the same equation, so uniqueness in this larger space gives $y = z_{ξ}$ , and hence exponential decay.

6.1step 4.35.15.2∎

The stable curve, the unstable curve and the half-trajectories. Choose $δ_{0} \in (0, ρ / 2)$ with $∣ h (ξ) ∣ \leq ∣ ξ ∣$ for $∣ ξ ∣ < δ_{0}$ ; this holds by step 5.1 because $D h (0) = 0$ . Then $G = (i d, h)$ maps $(- δ_{0}, δ_{0}) \subset E^{s}$ $C^{1}$ -injectively onto a $C^{1}$ embedded curve $S$ , because the linear projection $P_{s}$ restricts to the inverse of $G$ on $S$ and $S$ is the graph of the $C^{1}$ function $h$ ; and $T_{0} S = E^{s}$ because $D G (0) = i d$ . For $x = G (ξ)$ , uniqueness of the fixed point after shifting time gives $z_{ξ} (t) = G (P_{s} z_{ξ} (t))$ . On this graph, $∣ h (ξ) ∣ \leq ∣ ξ ∣$ and the small derivative bound give $\dot{ξ} = - a ξ + P_{s} \tilde{R} (G (ξ))$ , with $∣ P_{s} \tilde{R} (G (ξ)) ∣ \leq 2 ε^{'} ∣ ξ ∣ < a ∣ ξ ∣ / 2$ after decreasing $ε$ initially. Thus $∣ ξ ∣$ strictly decreases toward zero and each local half-graph is invariant. By step 4.3 every point of $S$ has its forward trajectory in $S$ , converging to $0$ at rate $β$ ; by step 5.2 every trajectory of $\tilde{X}$ , hence of $X$ , that stays in a sufficiently small ball $V \subset B_{ρ}$ for all $t \geq 0$ equals some $z_{ξ}$ , with $∣ ξ ∣ < δ_{0}$ , and therefore starts on $S$ . If $x = G (ξ) \neq 0$ , the forward trajectory of $x$ is a connected subset of $S ∖ {0}$ whose parameter values $P_{s} z_{ξ} (t)$ tend to $0$ and contain $ξ$ , so by the intermediate value property it meets both components of $S ∖ {0}$ only according to the sign of $ξ$ : the two components ${G (ξ) : ξ > 0}$ and ${G (ξ) : ξ < 0}$ are each a single half-trajectory. Applying the same construction to the function $\tilde{u} : = - u$ , whose Hessian $- H$ is again a nondegenerate saddle and whose gradient field is $- X$ , produces the unstable curve $U$ of $X$ , tangent to the negative eigenline of $H$ and swept out by the two backward half-trajectories with exponential backward convergence; the neighbourhood $V$ is the intersection of the two trapping balls. This proves all the assertions; every step used only the displayed estimates, the Banach fixed point theorem, the Neumann series and the primitive of continuous functions, none of which needs a choice principle.

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The Godbillon-Vey form eta wedge d eta is closed

Statement

Assume Countable Choice ${A C}_{ω}$ . Let $F$ be a transversely oriented codimension-one foliation with defining form $ω$ and $d ω = η \land ω$ . Then $d η \land ω = 0$ ; the $2$ -form $d η$ is divisible by $ω$ , i.e. $d η = ζ \land ω$ for a smooth $1$ -form $ζ$ ; $d η \land d η = 0$ ; and $d (η \land d η) = 0$ . Consequently $η \land d η$ is a closed $3$ -form on $M$ .

Facts & Assumptions

Given: A transversely oriented codimension-one foliation $F$ of a smooth manifold $M$ with defining one-form $ω$ and a one-form $η$ satisfying $d ω = η \land ω$ , and the standing countable choice assumption.

[F1]

Under ${A C}_{ω}$ , if $ω$ is nowhere vanishing and $θ \land ω = 0$ for a smooth two-form $θ$ , then $θ = β \land ω$ for a smooth one-form $β$ . (Divisibility by a nowhere-vanishing one-form).

[F2]

For homogeneous smooth forms $α, β$ one has $d (α \land β) = d α \land β + (- 1)^{\deg α} α \land d β$ . (The exterior derivative is a graded derivation).

[F3]

For every differential form $ω$ , $d (d ω) = 0$ . (The exterior derivative squares to zero).

[F4]

The wedge product is associative and graded-commutative, so $ζ \land ζ = 0$ for a one-form $ζ$ and $ω \land ω = 0$ . (The wedge product is associative and graded commutative).

Proof

technique · direct

1.1F2F3F4given

Differentiating $d ω = η \land ω$ with the Leibniz rule [F2] and $d^{2} = 0$ [F3] gives $0 = d η \land ω - η \land d ω = d η \land ω - η \land η \land ω$ , and $η \land η = 0$ by graded commutativity [F4], so $d η \land ω = 0$ .

2.1F1step 1.1

Since $ω$ is nowhere vanishing and the two-form $d η$ satisfies $d η \land ω = 0$ , the divisibility lemma [F1] gives a smooth one-form $ζ$ with $d η = ζ \land ω$ .

3.1F4step 2.1

Then $d η \land d η = (ζ \land ω) \land (ζ \land ω) = - ζ \land ζ \land ω \land ω = 0$ by associativity and graded commutativity with $ζ \land ζ = 0$ and $ω \land ω = 0$ [F4].

4.1F2F3step 3.1∎

Finally $d (η \land d η) = d η \land d η - η \land d (d η) = 0$ by the graded Leibniz rule [F2] and $d^{2} = 0$ [F3], so $η \land d η$ is a closed three-form. The standing ${A C}_{ω}$ assumption licenses the global divisibility result in step 2.1; the remaining calculations are formal exterior-algebra identities.

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Bott vanishing for real Pontryagin monomials of a foliation

Statement

Assume the full Axiom of Choice (The Axiom of Choice). Let $F$ be a codimension- $q$ regular foliation of a smooth manifold $M$ with normal bundle $ν = T M / T F$ . Then every real Pontryagin monomial of total cohomological degree greater than $2 q$ in the normal bundle vanishes: for every homogeneous ${GL}_{q} (R)$ -invariant polynomial $φ$ representing a Pontryagin monomial and of polynomial degree $k > q$ , the characteristic class $φ (ν) \in H_{d R}^{2 k} (M; R)$ obtained from the Pontryagin classes of $ν$ by Pontryagin classes by complexification is zero. No integral statement is made: the vanishing is of the real Chern-Weil form, hence of the real class.

Facts & Assumptions

Given: A codimension- $q$ regular foliation $F$ of a smooth manifold $M$ with normal bundle $ν = T M / T F$ and an invariant polynomial $φ$ of degree $k$ on the structure group of $ν$ .

[F1]

For a connection $\nabla$ on $ν$ extending the Bott partial connection, in a leaf-parallel frame the curvature matrix entries lie in the differential ideal $I$ generated by the one-forms vanishing on $T F$ , and $I^{q + 1} = 0$ . (Curvature of an extending Bott connection lies in the transverse differential ideal).

[F2]

The Chern-Weil construction is independent of the choice of connection and is natural under pullback. (Connection independence and naturality of Chern–Weil classes).

[F3]

The de Rham class of the Chern-Weil form of an invariant polynomial represents the corresponding real characteristic class of the bundle. (Characteristic forms represent topological characteristic classes over the reals).

[F4]

Under AC a smooth real bundle admits a connection (Every smooth vector bundle admits a connection), and a smooth manifold admits a Riemannian metric under the implied countable choice (Every smooth manifold admits a riemannian metric).

Proof

technique · direct

1.1F1F4givenconstruct

By [F4] choose a metric on $T M$ , the orthogonal projection $Π : T M \to T F$ , and a connection $\nabla^{0}$ on $ν$ . Define $\nabla_{X} s = \nabla_{Π X}^{B} s + \nabla_{X - Π X}^{0} s$ . Its direction-linearity and Leibniz rule follow from those of the two summands, since $(Π X) (f) + (X - Π X) (f) = X (f)$ ; it extends the Bott partial connection. Now [F1] puts all its curvature entries in $I$ with $I^{q + 1} = 0$ .

2.1step 1.1algebra

Evaluating the invariant polynomial $φ$ of degree $k$ on the curvature gives a sum of products of $k$ matrix entries $Ω^{a_{i}}_{b_{i}}$ , possibly with constant coefficients and traces; each such product is an element of $I^{k}$ , so $φ (R^{\nabla}) \in I^{k}$ .

3.1F2F3step 2.1∎

If $k > q$ then $I^{k} \subseteq I^{q + 1} = 0$ , so the Chern-Weil form $φ (R^{\nabla})$ is identically zero and represents the zero class; by [F2] the Chern-Weil class is independent of the connection and by [F3] it represents the real characteristic class of $ν$ , so the real Pontryagin monomial $φ (ν)$ vanishes in $H_{d R}^{2 k} (M; R)$ ; no integral refinement is claimed with AC used for [F4] and the topological-to-real comparison [F3].

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C² plaque transport and finite transverse fences preserve C² regularity

Statement

Let $F$ be a codimension-one foliation of a smooth $n$ -manifold $M$ given by a $C^{2}$ foliation atlas: charts $φ = (x, t) : U \to R^{n - 1} \times R$ whose components and inverses are of class $C^{2}$ and whose transitions have the form $(x^{'}, t^{'}) = (g (x, t), h (t))$ with $g$ of class $C^{2}$ and $h$ a one-dimensional $C^{2}$ local diffeomorphism of intervals.

(a) Every finite plaque transport between $C^{2}$ local transversals is a $C^{2}$ local diffeomorphism germ.

(b) A finite family of $C^{2}$ traces agreeing on open overlap collars glues to a $C^{2}$ trace, and if the parameter derivative of every piece has nonzero transverse component then the glued trace is transverse at every parameter, including its one-sided derivatives at parameter endpoints.

(c) Well-definedness of holonomy along leafwise loops and its invariance under leafwise homotopies relative to endpoints are supplied by the underlying $C^{1}$ atlas.

These assertions concern the regularity of specified compatible pieces; they do not assert the existence of a polycycle fence or of an extremal cycle.

Facts & Assumptions

Given: A $C^{2}$ foliation atlas for $F$ , a finite plaque transport between $C^{2}$ local transversals, and finitely many $C^{2}$ traces on open intervals that agree on open overlap collars.

[F1]

A $C^{2}$ foliation atlas as in the statement is a $C^{1}$ foliation atlas in the sense of C¹ codimension-one regular foliations and transverse orientation: every chart and its inverse is of class $C^{1}$ , and on every overlap the transition has the form $(x^{'}, t^{'}) = (g (x, t), h (t))$ with $h$ a one-dimensional $C^{1}$ local diffeomorphism.

[F2]

For a transversely oriented $C^{1}$ codimension-one foliation, plaque transport along leafwise loops defines a homomorphism into $C^{1}$ transverse germs that is independent of the foliation chart chain and invariant under leafwise homotopies relative to endpoints (Holonomy of a C¹ foliation is a representation into C¹ transverse germs).

[F3]

A $C^{2}$ map between open subsets of $R^{m}$ with invertible derivative at a point has a $C^{2}$ local inverse there (C² inverses and scalar return roots).

Proof

technique · direct

1.1F1F2given

By [F1] the given atlas is a $C^{1}$ foliation atlas. The chart-chain and homotopy argument of [F2] does not require transverse orientation: compose the transverse coordinate changes along a finite subdivision; a common refinement preserves the composite, and a finite rectangle subdivision of a leafwise homotopy changes paths only inside plaques, where transverse transport is unchanged. These statements use finite compact covers; they apply to germs of either orientation. Under the library concatenation convention, transport on the reversed loop gives the homomorphism. Thus clause (c) holds for the given atlas.

1.2F1given

Let $φ = (x, t) : U \to R^{n - 1} \times R$ be a chart of the given $C^{2}$ atlas, let $T, T^{'}$ be $C^{2}$ local transversals through points $p, p^{'}$ of one common plaque of $U$ , and parametrize $T$ near $p$ and $T^{'}$ near $p^{'}$ by $C^{2}$ curves $γ : J \to U$ and $γ^{'} : J^{'} \to U$ with $γ (0) = p$ , $γ^{'} (0) = p^{'}$ , $(t \circ γ)^{'} (0) \neq 0$ and $(t \circ γ^{'})^{'} (0) \neq 0$ . The plaques of $U$ are the level sets of $t$ , so the plaque transport between $T$ and $T^{'}$ matches points with equal $t$ -coordinate.

1.3given

Let $I_{1}, \dots, I_{m} \subseteq R$ be open intervals covering a compact parameter interval $[a, b]$ , and let $γ_{i} : I_{i} \to M$ be $C^{2}$ traces that agree on $I_{i} \cap I_{j}$ for all $i, j$ (in particular on a collar neighbourhood of every seam), so that $γ (θ) : = γ_{i} (θ)$ for $θ \in I_{i}$ is a well-defined map on $⋃_{i} I_{i} \supseteq [a, b]$ . At a parameter interior to some $I_{i}$ the glued map coincides on an open neighbourhood with the $C^{2}$ map $γ_{i}$ , hence is $C^{2}$ there; at a parameter endpoint of $[a, b]$ , restriction of any $γ_{i}$ whose interval contains that endpoint gives continuous one-sided derivatives of orders one and two, so $γ$ is $C^{2}$ on $[a, b]$ in the one-sided sense.

2.1F3step 1.2

The function $s \mapsto t (γ^{'} (s))$ is $C^{2}$ with nonzero derivative at $0$ , so by [F3] it has a $C^{2}$ local inverse $s = σ (z)$ near $z = t (p^{'})$ ; likewise $θ \mapsto t (γ (θ))$ has nonzero derivative at $0$ and is a $C^{2}$ local diffeomorphism. The single-chart transport written in the parameters of $T$ and $T^{'}$ is therefore $Θ : = σ \circ t \circ γ$ near $θ = 0$ , it is $C^{2}$ as a composite of $C^{2}$ maps, and $Θ^{'} (0) = (t \circ γ)^{'} (0) / (t \circ γ^{'})^{'} (0) \neq 0$ . Hence the piece is a $C^{2}$ local diffeomorphism germ.

2.2step 1.3given

At every parameter the derivative of the glued trace equals the derivative of a piece defined on a neighbourhood of that parameter, and by hypothesis that derivative has nonzero transverse component in a foliation chart; consequently the glued trace is transverse to $F$ at every parameter, and at the endpoints its one-sided derivative equals the one-sided derivative of any piece containing that endpoint, so transversality persists there as well. This is clause (b).

3.1F1step 2.1

Suppose a plaque transport meets the transversals $T_{0}, \dots, T_{m}$ successively and the piece from $T_{i - 1}$ to $T_{i}$ lies in the chart $U_{i}$ . Inside $U_{i}$ step 2.1 exhibits that piece as a $C^{2}$ local diffeomorphism germ with nonzero derivative. If two consecutive pieces are computed in different charts, then on their common domain the transverse coordinates are related by the transition function $h$ , which is a $C^{2}$ diffeomorphism by the atlas hypothesis, and composition with $h$ and with its inverse preserves both $C^{2}$ regularity and the nonvanishing of the derivative. A finite composition of $C^{2}$ local diffeomorphism germs with nonzero derivative is again such a germ, so every finite plaque transport between $C^{2}$ local transversals is a $C^{2}$ local diffeomorphism germ, which is clause (a).

4.1step 1.1step 3.1step 2.2∎

Clause (a) is step 3.1, clause (b) is step 2.2, and clause (c) is step 1.1; the argument used finitely many charts, finitely many pieces and local $C^{2}$ inverses only, so no choice principle is invoked.

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

The characteristic disk has one more center than saddle

Statement

Assume Countable Choice ${A C}_{ω}$ (The countable-choice principle used in the foliation pair). Let $F$ be a cooriented codimension-one foliation of a smooth $3$ -manifold $M$ with nowhere-vanishing defining form $ω$ (Transversely oriented codimension-one foliations), and let $h : D^{2} \to M$ be a $C^{2}$ disk map whose characteristic covector $β = h^{*} ω$ is nowhere vanishing on $\partial D^{2}$ and has finitely many interior zeros, all nondegenerate, each a center or a saddle (Relative generic position for characteristic disk maps). Assume moreover that the boundary loop is either leafwise ( $β (τ) = 0$ everywhere) or a closed transversal ( $β (τ) \neq 0$ everywhere), where $τ$ is its tangent. Then the characteristic line field of $h$ has finitely many nondegenerate centers and saddles, and their numbers satisfy $c - s = 1.$ In particular there is at least one center.

Facts & Assumptions

Given: A cooriented codimension-one foliation $F$ with defining form $ω$ , and a $C^{2}$ map $h : D^{2} \to M$ whose characteristic covector $β = h^{*} ω$ is nowhere vanishing on $\partial D^{2}$ and whose interior zeros are finitely many nondegenerate center/saddle points, with the boundary leafwise or a closed transversal as stated.

[F1]

Relative generic position supplies exactly the stated boundary and interior behaviour, and it also identifies each singularity with a nondegenerate critical point of the local transverse function, definite Hessian for a center and indefinite Hessian for a saddle (Relative generic position for characteristic disk maps).

[F2]

In a foliation chart with transverse coordinate $z$ one has $ω = a d z$ with $a \neq 0$ , and $β = (a \circ h) d (z \circ h)$ ; writing $β = P d x + Q d y$ in oriented source coordinates, $\nabla u$ with $u = z \circ h$ satisfies $(P, Q) = (a \circ h) \nabla u$ , so at a zero $p$ the chain rule gives $D (P, Q) (p) = (a \circ h) (p) D^{2} u (p)$ (Transversely oriented codimension-one foliations, The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ).

[F3]

Closed bounded subsets of $R^{2}$ are compact, and a continuous positive function on a compact set has a positive minimum (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line).

[F4]

Degree of circle loops: for a based loop $α$ in $S^{1}$ the degree $\deg (α) = \tilde{α} (1)$ is computed by the lift from $0$ , descends to $π_{1}$ , equals $0$ exactly for nullhomotopic loops, and adds under concatenation and changes sign under reversal; moreover there is a continuous argument along any continuous path in $S^{1}$ by path lifting, and the increment of an argument along a path is invariant under homotopies of paths with fixed endpoints (The degree of a based circle loop, Degree defines a function $Deg : π_{1} (S^{1}, [0]) \to Z$ , A based circle loop is nullhomotopic exactly when its degree is zero, Lifts of circle-loop concatenations and reversals, Existence and uniqueness of path lifts through a covering map, Existence and uniqueness of homotopy lifts through a covering map).

[F5]

The identity map of $S^{1}$ has degree $1$ and a single coordinate reflection has degree $- 1$ (Degree of identity constant reflection and antipodal sphere maps).

Proof

technique · direct

1.1givenF1F2

Write $β = h^{*} ω = P d x + Q d y$ in the oriented source coordinates of the disk and define the characteristic field $X : = (Q, - P)$ , so that $ι_{X} (d x \land d y) = β$ ; then $X$ vanishes exactly at the zeros of $β$ , which are the finitely many nondegenerate interior points $p_{1}, \dots, p_{N}$ of the hypothesis, and $X$ is nowhere zero on $\partial D^{2}$ and on a collar of it.

2.1step 1.1F4F5

Boundary degree. The normalized field $x \mapsto X (x) / ∣ X (x) ∣$ is a continuous loop in $S^{1}$ on the counterclockwise circle $\partial D^{2}$ , and its degree is $1$ . If $h (\partial D^{2})$ lies in one leaf, then $β (τ) = 0$ for the unit tangent $τ$ of $\partial D^{2}$ , while $β \neq 0$ on the boundary; from $0 = β (τ) = (d x \land d y) (X, τ)$ the vector $X$ is tangent to the boundary circle and nonzero, hence $X (x) = ε (x) τ (x)$ with $ε \neq 0$ , and $ε$ is continuous on the connected circle, so it has a constant sign: the normalized field is $\pm τ$ and has the same degree as the unit tangent $τ$ , which is the rotation of the identity map and so has degree $1$ [F5]. If $h ∣_{\partial D^{2}}$ is a closed transversal, write $X = a n + b τ$ with $n$ the outward unit normal; then $β (τ) = (d x \land d y) (X, τ) = a (d x \land d y) (n, τ)$ with $(d x \land d y) (n, τ) > 0$ for the counterclockwise orientation, so $a$ has a fixed nonzero sign, and the straight homotopy $X_{s} = (1 - s) X + s σ n$ with $σ = sign a$ has normal component $σ ((1 - s) ∣ a ∣ + s) \neq 0$ , hence is nonzero for all $s \in [0, 1]$ ; the normalized fields are therefore homotopic loops, and the normalized outward normal $n$ has degree $1$ [F5], so the boundary degree is $1$ in both cases.

3.1step 2.1F3F4

Outer polygon and holes. Since $\partial D^{2}$ has a zero-free collar and the zeros are interior, by [F3] we may choose a regular polygon $Q_{0}$ , star-shaped about the origin with positive radial function $r_{0} (θ)$ , whose boundary lies in the zero-free collar and whose closed convex hull contains all $p_{i}$ . Around each $p_{i}$ choose pairwise disjoint disks $D_{i}$ with closures in the interior of $Q_{0}$ and containing no zero of $X$ other than $p_{i}$ , and inside $D_{i}$ a centered closed square $Q_{i}$ with positive radial function $ρ_{i} (θ)$ about $p_{i}$ . The radial homotopies $θ \mapsto ((1 - s) r_{0} (θ) + s) e^{i θ}$ and $θ \mapsto p_{i} + ((1 - s) ρ_{i} (θ) + s R_{i}) e^{i θ}$ , with $R_{i}$ the radius of $D_{i}$ , move $\partial Q_{0}$ to the boundary circle of $D^{2}$ and $\partial Q_{i}$ to $\partial D_{i}$ through loops on which $X$ never vanishes; by the homotopy invariance of the argument increment [F4] the degree of the normalized field on $\partial Q_{0}$ equals the boundary degree $1$ of step 2.1, and the degree on $\partial Q_{i}$ equals the degree on $\partial D_{i}$ .

4.1step 3.1F1F2F4

The local degree at a zero is the sign of the Hessian. Fix $p_{i}$ and work in a foliation chart around $h (p_{i})$ with transverse coordinate $z$ and $u = z \circ h$ . By [F2] the derivative of $(P, Q)$ at $p_{i}$ is $(a \circ h) (p_{i}) D^{2} u (p_{i})$ , and $X = (Q, - P)$ , so $D X (p_{i}) = (a \circ h) (p_{i}) J D^{2} u (p_{i})$ with $J$ the quarter-turn matrix of determinant $1$ . The normalized field near $p_{i}$ is homotopic through nonzero fields on a small circle to the normalized linear field of $D X (p_{i})$ . If $D^{2} u (p_{i})$ is definite, write $H = D^{2} u (p_{i})$ and let $λ$ be any eigenvalue; the family $H_{s} = (1 - s) H + s λ I$ is invertible for every $s \in [0, 1]$ , so the normalized fields of $J H_{s}$ give a homotopy, and for $H = λ I$ the field $J H x$ is, in the complex notation $x_{1} + i x_{2}$ , the map $z \mapsto - λ i z$ , of degree $1$ . If $D^{2} u (p_{i})$ is indefinite, choose coordinates diagonalizing it with eigenvalues $λ_{1} > 0 > λ_{2}$ ; the family $(1 - s) H + s diag (λ_{1}, - λ_{1})$ stays invertible, and for $H = diag (1, - 1)$ one computes $J H x = (- \sin θ, - \cos θ) = - i e^{- i θ}$ on the unit circle, of degree $- 1$ . Hence a center contributes local degree $+ 1$ and a saddle contributes $- 1$ .

5.1step 3.1step 4.1F4

The index sum. Cover the closed polygon $Q_{0}$ by a finite grid of closed axis-parallel rectangles chosen so that every grid line through a side of some square $Q_{i}$ is a grid line; then each grid cell either lies inside one of the squares $Q_{i}$ or has interior disjoint from all of them. Discard the cells lying inside a square $Q_{i}$ and the cells disjoint from $Q_{0}$ ; for every remaining cell $C$ , the set $C \cap Q_{0}$ is convex, hence contractible, and is contained in the closed zero-free region $A = Q_{0} ∖ ⋃_{i} int (Q_{i})$ , so the normalized field $g = X / ∣ X ∣$ is defined on $C \cap Q_{0}$ and the loop $g ∣_{\partial (C \cap Q_{0})}$ extends to a map of the convex set $C \cap Q_{0}$ , hence is nullhomotopic and has argument increment $0$ [F4]. Summing the increments over the finitely many cells, every edge of the grid that lies in the interior of $A$ occurs twice with opposite orientations and cancels by the additivity and reversal rules [F4] (the cells' boundaries are finite polygonal paths, and the common edges are traversed in opposite directions with equal image under $g$ ); what survives is the boundary of $Q_{0}$ traversed counterclockwise together with the boundaries of the squares $Q_{i}$ traversed clockwise. Therefore $0 = \deg (g ∣_{\partial Q_{0}}) - \sum_{i} \deg (g ∣_{\partial Q_{i}})$ , that is, $1 = \sum_{i} {ind}_{p_{i}} (X)$ .

6.1step 2.1step 5.1∎

By step 4.1 the sum $\sum_{i} {ind}_{p_{i}} (X)$ equals $c - s$ , where $c$ is the number of centers and $s$ the number of saddles among the nondegenerate zeros; step 5.1 gives $c - s = 1$ , so in particular $c \geq 1$ and the zeros of the characteristic line field are exactly the finitely many nondegenerate centers and saddles. The proof used the relative-genericity supplier, the chain rule, compactness and the elementary degree calculus of circle loops; all of these are choice-free and the only inherited hypothesis is the stated ${A C}_{ω}$ of the cooriented interface.

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

A finite saddle omega-graph is strongly connected and is a finite union of polycycles

Statement

Assume Countable Choice ${A C}_{ω}$ (The countable-choice principle used in the foliation pair). Let $F$ be a $C^{2}$ cooriented codimension-one foliation and let $h : D^{2} \to M$ be a characteristic disk in the relative generic position of Relative generic position for characteristic disk maps. Let $X$ be its planar characteristic vector field. Let $U \subseteq R^{2}$ be an open neighborhood of the disk and let $Y : U \to R^{2}$ be a $C^{1}$ field with $Y = X$ and $D Y = D X$ on $Γ$ . Assume the positive orbit of $y$ has compact closure $K_{0} ⋐ U$ , $Γ = ω_{Y}^{+} (y)$ contains at least one equilibrium, all equilibria in $Γ$ are nondegenerate characteristic saddles of $X$ , and $Γ$ separates two specified points of the plane. Then $Γ$ is a finite embedded directed multigraph: its vertices are those saddles and its edges are closures of distinct nonconstant trajectories with saddle alpha- and omega-limits. The directed graph is strongly connected. Consequently every edge lies in a closed directed edge walk, and finitely many such walks cover $Γ$ ; a closed directed edge walk is allowed to repeat vertices and is called a directed saddle polycycle here.

Facts & Assumptions

Given: A $C^{2}$ cooriented codimension-one foliation, a characteristic disk map $h$ in relative generic position, its planar characteristic field $X$ , a $C^{1}$ field $Y$ on a neighborhood $U$ of the disk with $Y = X$ and $D Y = D X$ on $Γ$ , and a positive orbit with compact closure $K_{0} ⋐ U$ and $ω$ -limit $Γ$ whose equilibria are finitely many nondegenerate characteristic saddles of $X$ .

[F1]

Let $Y$ be $C^{1}$ on an open $U \subseteq R^{2}$ and let $O^{+} (y)$ have compact closure $K_{0} ⋐ U$ with $K = ω^{+} (y)$ containing only finitely many equilibria. Then either $K$ is a singleton equilibrium, or $K$ is one regular periodic orbit, or $K$ is a finite set $E$ of equilibria together with at least one regular trajectory, and every regular point of $K$ lies on a nonconstant trajectory whose alpha- and omega-limit sets are points of $E$ (Local generalized Poincare-Bendixson theorem for a precompact planar orbit).

[F2]

A $C^{2}$ function on the plane with a nondegenerate indefinite Hessian at $p$ has $C^{1}$ coordinates centered at $p$ in which it equals $x y$ (A C² saddle function has C¹ Morse coordinates).

[F3]

A $C^{1}$ Euclidean field has a unique maximal flow that is jointly $C^{1}$ , its time slices are injective, each regular point has a $C^{1}$ flow box, a trajectory remaining in a compact subset of the domain has no finite maximal endpoint, and trajectories are $C^{2}$ in time (C¹ Euclidean maximal flows, variational dependence and the finite C² upgrade).

[F4]

In relative generic position the characteristic singularities of the disk map are finitely many nondegenerate points in the interior of $D^{2}$ , each a center or a saddle (Relative generic position for characteristic disk maps).

[F5]

The standing assumption of the pair is Countable Choice ${A C}_{ω}$ (The countable-choice principle used in the foliation pair).

[F6]

Closed and bounded subsets of $R^{2}$ are compact; a nested decreasing family of nonempty compact subsets has nonempty intersection; a continuous real function on a nonempty compact set attains its maximum and minimum (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line).

Proof

technique · direct

1.1givenF2F3F4

At a saddle $q \in Γ$ use the $C^{1}$ coordinates of [F2] in which the local transverse function of the disk map is $u = x y$ ; writing the pulled-back area form as $μ = m d x \land d y$ with $m$ continuous and positive and the characteristic covector as $β = a d u$ with $a$ continuous and nonzero, the field is $X = (a / m) (x \partial_{x} - y \partial_{y})$ ; replacing $(x, y)$ by $(y, x)$ if necessary (which changes $x \partial_{x} - y \partial_{y}$ to its negative) makes the coefficient $c (x, y) > 0$ , continuous on a smaller compact chart, and then $\dot{x} = c x$ , $\dot{y} = - c y$ give two incoming half-branches (both $y$ -axis rays) and two outgoing half-branches (both $x$ -axis rays) with exponential rate bounds $∣ c o o r d (t) ∣ \in [r_{0} e^{- c_{+} t}, r_{0} e^{- c_{-} t}]$ for the contracting branch and the reciprocal bounds for the expanding branch on the compact chart; since $Y = X$ on $Γ$ , every edge of $Γ$ through $q$ follows one of these four half-branches, and by uniqueness of [F3] each half-branch is contained in exactly one maximal trajectory, so at most two edges leave each vertex and there are at most twice as many edges as vertices.

2.1step 1.1F1F3

Every regular point of $Γ$ lies on a maximal trajectory whose alpha- and omega-limits are saddle points of $Γ$ : since a singleton does not separate two points of the plane, alternative (i) of [F1] fails; alternative (ii) fails because $Γ$ contains the given equilibrium; hence alternative (iii) holds; each such maximal trajectory is contained in $Γ$ by invariance and closedness of $Γ$ , has no interior equilibrium by uniqueness [F3], and its closure is one of the edge closures of step 1.1, so the edge closures together with the saddle vertices exhaust $Γ$ , distinct edges meet only at common saddle endpoints, and the four half-branches at each saddle give the local embedded-graph structure.

3.1step 2.1F1F3

With exact endpoints, $Γ$ is internally chain-transitive: fix $p, q \in Γ$ , $ϵ > 0$ and $T > 0$ ; by joint continuity of the flow [F3] on the compact set $K_{0}$ and the time interval $[T, 2 T]$ , choose $δ \in (0, ϵ / 3)$ so that $δ$ -close points have $ϵ / 3$ -close images throughout $[T, 2 T]$ ; since $Γ = ω^{+} (y)$ and the orbit tail approaches $Γ$ uniformly, choose $s$ so late that $Φ_{u} (y)$ is within $δ$ of $Γ$ for all $u \geq s$ , then choose $s$ with $Φ_{s} (y)$ within $δ$ of $p$ and $t > s + 2 T$ with $Φ_{t} (y)$ within $δ$ of $q$ ; write $t - s - T = N τ$ with an integer $N \geq 1$ and $τ \in [T, 2 T]$ ; the finitely many intermediate orbit points $Φ_{s + T + k τ} (y)$ , $0 \leq k < N$ , lie within $δ$ of $Γ$ , so finitely many nearby points $z_{k} \in Γ$ exist, and $x_{0} = p$ , $x_{1} = z_{0}$ , ..., $x_{N} = z_{N - 1}$ , $x_{N + 1} = q$ together with the times $T, τ, \dots, τ$ form an $(ϵ, T)$ -chain because each flowed image is within $ϵ / 3$ of the next orbit point and each jump is at most $ϵ / 3 + δ < ϵ$ .

4.1step 2.1step 3.1F3F6

The directed graph is strongly connected: $Γ$ is connected, being the intersection of the decreasing family of the connected closures of the orbit tails, since a separation of $Γ$ into disjoint nonempty compact pieces has positive distance and would force a sufficiently late tail closure, which is connected, into a neighbourhood of one piece and away from the other [F6]; if the graph had more than one strongly connected component, its finite condensation would have a proper terminal component $S$ , and if no edge entered $S$ from outside then no edge would leave it either, so the compact carriers of $S$ and of its complement would express the connected $Γ$ as two disjoint nonempty closed sets, which is impossible; hence some edge enters $S$ , and the compact set $N$ consisting of the vertices of $S$ , all edges internal to $S$ and a short terminal segment of every entering edge with a trimming point in its regular part satisfies $S \subseteq {int}_{Γ} N$ with no edge leaving $S$ ; points of $N$ flow strictly toward $S$ and never reach a saddle in finite time by uniqueness [F3], so $Φ_{t} (N) \subseteq N$ for $t \geq 0$ and $Φ_{T} (N) \subseteq {int}_{Γ} N$ , whence $δ_{0} : = dist (Φ_{T} (N), Γ ∖ {int}_{Γ} N) > 0$ by [F6]; a chain with $ϵ < δ_{0}$ starting at a point of $S$ stays in $N$ by induction, because $Φ_{t} (N) \subseteq Φ_{T} (N)$ for $t \geq T$ and a jump of size less than $δ_{0}$ cannot leave the $δ_{0}$ -neighbourhood of $Φ_{T} (N)$ ; choosing the terminal point $q$ in the omitted middle part of an entering edge gives $q \notin N$ , contradicting the exact-endpoint chain transitivity of step 3.1, so all vertices lie in one strongly connected component.

5.1step 4.1F5∎

Consequently, for each directed edge $e : v \to w$ strong connectivity supplies a directed path from $w$ back to $v$ , and adjoining $e$ gives a closed directed edge walk containing $e$ and repeating vertices only as allowed; there are finitely many edges by step 1.1, so finitely many such walks cover $Γ$ ; the construction used only finitely many points and paths of a finite graph plus the finite flow-box and compactness arguments, hence no choice principle, so the statement holds and its standing ${A C}_{ω}$ hypothesis is not invoked.

LemmaStatement: Literature-sourcedProof: Literature-sourcedprecheck passOpen item page →

Independence of the auxiliary form eta up to exact forms

Statement

Assume Countable Choice ${A C}_{ω}$ . Let $F$ be a transversely oriented codimension-one foliation with defining form $ω$ , and let $η, η^{'}$ be smooth $1$ -forms with $d ω = η \land ω = η^{'} \land ω$ . If $η^{'} - η = f ω$ with $f \in C^{\infty} (M)$ , then $η^{'} \land d η^{'} = η \land d η - d (f d ω) = η \land d η + d (d f \land ω)$ , and in particular $[η^{'} \land d η^{'}] = [η \land d η]$ in $H_{d R}^{3} (M; R)$ .

Facts & Assumptions

Given: A transversely oriented codimension-one foliation with defining form $ω$ and one-forms $η, η^{'}$ satisfying $d ω = η \land ω = η^{'} \land ω$ , with $η^{'} - η = f ω$ .

[F1]

With $d ω = η \land ω$ one has $d η \land ω = 0$ and $η \land d η$ is closed. (The Godbillon-Vey form eta wedge d eta is closed).

[F2]

For homogeneous smooth forms $α, β$ of degrees $p, q$ one has $d (α \land β) = d α \land β + (- 1)^{p} α \land d β$ . (The exterior derivative is a graded derivation).

[F3]

The wedge product is associative and graded-commutative, so odd-degree forms anticommute and $ω \land ω = 0$ . (The wedge product is associative and graded commutative).

[F4]

For every differential form $ω$ , $d (d ω) = 0$ . (The exterior derivative squares to zero).

Proof

technique · direct

1.1F2F4given

Write $η^{'} = η + f ω$ ; differentiating and using $d ω = η \land ω$ and $d (f ω) = d f \land ω + f d ω$ from [F2] gives $d η^{'} = d η + d f \land ω + f d ω$ , while $d (d ω) = 0$ by [F4] is compatible with $d ω = η^{'} \land ω$ .

2.1F1F3step 1.1

Expanding $η^{'} \land d η^{'} - η \land d η = (f ω) \land d η + η \land (d f \land ω + f d ω) + (f ω) \land (d f \land ω + f d ω)$ , every term containing $η \land η$ , $d η \land ω$ , $ω \land d η$ , $ω \land d ω$ or $ω \land ω$ vanishes by [F1] and the graded commutativity and $ω \land ω = 0$ of [F3], leaving $η \land d f \land ω = - d f \land d ω$ .

3.1F2F3step 2.1∎

Since $d η \land ω = 0$ by [F1] and $d η^{'} = d η + d f \land ω + f d ω$ , the identity $d (d f \land ω) = d (d f) \land ω - d f \land d ω = - d f \land d ω$ from [F2] and [F4] shows that $η^{'} \land d η^{'} = η \land d η - d (f d ω) = η \land d η + d (d f \land ω)$ , an exact modification; hence the two forms have the same de Rham class, and no choice principle is used.

LemmaStatement: Literature-sourcedProof: Literature-sourcedprecheck passOpen item page →

Rescaling the defining form changes the Godbillon-Vey form by an exact form

Statement

Assume Countable Choice ${A C}_{ω}$ . Let $F$ be a transversely oriented codimension-one foliation with defining form $ω$ and $d ω = η \land ω$ . For $ω^{'} = e^{f} ω$ with $f \in C^{\infty} (M)$ one has $d ω^{'} = η^{'} \land ω^{'}$ with $η^{'} = η + d f$ , and $η^{'} \land d η^{'} = η \land d η + d (f d η)$ ; in particular $[η^{'} \land d η^{'}] = [η \land d η]$ in $H_{d R}^{3} (M; R)$ . The same conclusion holds for any nowhere-vanishing smooth multiple $ω^{'} = g ω$ , since on each connected component $g$ is $e^{f}$ or $- e^{f}$ .

Facts & Assumptions

Given: A transversely oriented codimension-one foliation with defining form $ω$ and $d ω = η \land ω$ , and a smooth function $f$ with $ω^{'} = e^{f} ω$ .

[F1]

With $d ω = η \land ω$ one has $d η \land ω = 0$ and $η \land d η$ is closed. (The Godbillon-Vey form eta wedge d eta is closed).

[F2]

For homogeneous smooth forms $α, β$ of degrees $p, q$ one has $d (α \land β) = d α \land β + (- 1)^{p} α \land d β$ . (The exterior derivative is a graded derivation).

[F3]

For every differential form $ω$ , $d (d ω) = 0$ . (The exterior derivative squares to zero).

Proof

technique · direct

1.1F2given

Compute $d ω^{'} = d (e^{f} ω) = e^{f} d f \land ω + e^{f} d ω = e^{f} (d f \land ω + η \land ω) = (η + d f) \land ω^{'}$ by [F2], so $η^{'} = η + d f$ is admissible for $ω^{'}$ .

2.1F1F2F3step 1.1

Then $d η^{'} = d η + d (d f) = d η$ by [F3], so $η^{'} \land d η^{'} = (η + d f) \land d η = η \land d η + d (f d η)$ because $d (f d η) = d f \land d η$ by [F2]; the difference is exact and $η \land d η$ is closed by [F1], so the two forms define the same class in $H_{d R}^{3} (M; R)$ .

3.1step 2.1∎

For a general nowhere-vanishing smooth multiple $ω^{'} = g ω$ the same computation applies on each connected component with $g = \pm e^{f}$ : the sign choice leaves $d ω = η \land ω$ and the form $η \land d η$ unchanged, so the class is independent of the rescaling; no choice principle is used.

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Characteristic-disk singular images can be separated into distinct leaves relative to the boundary collar

Statement

Assume Countable Choice ${A C}_{ω}$ (The countable-choice principle used in the foliation pair). Let $F$ be a $C^{2}$ cooriented codimension-one foliation of a smooth $3$ -manifold with nowhere-vanishing defining form $ω$ , and let $h : D^{2} \to M$ be a $C^{2}$ map whose characteristic covector $h^{*} ω$ is nowhere vanishing on the closure $\overline{C}$ of a prescribed collar $C$ of $\partial D^{2}$ and has finitely many interior zeros $p_{1}, \dots, p_{N}$ , all nondegenerate, each a center or a saddle. Then for every sufficiently small prescribed $C^{2}$ neighbourhood of $h$ there is a $C^{2}$ map $h^{'}$ in that neighbourhood, together with a $C^{2}$ homotopy from $h$ to $h^{'}$ fixed on an open neighbourhood of $C$ , such that:

(i) the singular points of $h^{'}$ are exactly $p_{1}, \dots, p_{N}$ , and near each $p_{i}$ the local transverse function of $h^{'}$ differs from that of $h$ by a constant, so the transverse-coordinate Hessian and the center/saddle type are unchanged;

(ii) the images $h^{'} (p_{1}), \dots, h^{'} (p_{N})$ lie in pairwise distinct ambient leaves of $F$ .

In particular $h^{'}$ has no characteristic separatrix joining two distinct singular points; homoclinic separatrices are not excluded.

Facts & Assumptions

Given: A cooriented codimension-one $C^{2}$ foliation with defining form $ω$ , a $C^{2}$ map $h : D^{2} \to M$ regular on the closure $\overline{C}$ of a prescribed collar $C$ of $\partial D^{2}$ , and finitely many nondegenerate characteristic zeros $p_{1}, \dots, p_{N}$ in the interior of $D^{2}$ .

[F1]

In a foliation chart with transverse coordinate $z$ one has $ω = a d z$ with $a \neq 0$ , so the characteristic zero set of a map $g$ in the chart is the critical set of $u = z \circ g$ , and adding a constant to $u$ on an open set does not change the critical points or the Hessian there (Transversely oriented codimension-one foliations, The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ).

[F2]

Let $F$ be a $C^{2}$ foliation on a second-countable manifold, $L$ a leaf and $τ : J \to Q$ a vertical transverse interval in a foliation box; then $τ^{- 1} (L)$ is at most countable (A C² leaf meets a local box transversal in at most countably many points).

[F3]

For $a < b$ both $[a, b]$ and $(a, b)$ are uncountable, so no countable subset of an interval equals the interval (Every nondegenerate interval of $R$ is uncountable).

[F4]

For a compact set $K$ contained in an open set $W$ of a smooth manifold there is a smooth bump $ρ$ equal to $1$ on a neighbourhood of $K$ and supported in $W$ (A manifold bump for a compact set inside an open set).

Proof

technique · direct

1.1givenF1F3

The compact set $\overline{C}$ contains no zero of $h^{*} ω$ . Thus each $p_{i}$ lies in the open set $int D^{2} ∖ \overline{C}$ ; choose pairwise disjoint open disks $V_{i}$ with closures in that set, with $p_{i} \in V_{i}$ , and slightly smaller compact cores $W_{i} \subset V_{i}$ around $p_{i}$ . Since the $p_{i}$ are the only zeros, and a nondegenerate zero is isolated, we may choose the $V_{i}$ so that $h^{*} ω \neq 0$ on ${\overline{V}}_{i} ∖ W_{i}$ . Shrinking the $V_{i}$ further, arrange that each $h ({\overline{V}}_{i})$ lies in a single foliation chart $Q_{i}$ with transverse coordinate $z_{i}$ , and write $u_{i} : = z_{i} \circ h$ . Process the indices in the order $i = 1, \dots, N$ .

2.1step 1.1F4

A bump on each disk and its margin. By [F4] choose for each $i$ a smooth bump $ρ_{i}$ with $0 \leq ρ_{i} \leq 1$ , equal to $1$ on a neighbourhood of $W_{i}$ and supported in $V_{i}$ . On the compact set $supp (d ρ_{i})$ , which is disjoint from $W_{i}$ , the form $d u_{i}$ is nowhere zero; by compactness there is $η_{i} > 0$ with $∣ d u_{i} ∣ \geq η_{i}$ there.

3.1step 2.1F2F3

Avoiding the finitely many earlier singular leaves. Write $(Y_{i}, u_{i}) = χ_{i} \circ h$ . The curve $τ_{i} (t) = χ_{i}^{- 1} (Y_{i} (p_{i}), u_{i} (p_{i}) + t)$ for $∣ t ∣ < ε_{i}$ parametrizes a vertical transverse interval $J_{i}$ through $h (p_{i})$ in the box $Q_{i}$ . For each $j < i$ , the map $h$ has already been replaced near $p_{j}$ by a map whose singular image $q_{j}$ is fixed in the $j$ -th stage; its leaf $L_{j}$ meets the interval $J_{i}$ in at most countably many points by [F2]. The finitely many countable sets ${z_{i} (q_{j}^{'}) : q_{j}^{'} \in J_{i} \cap L_{j}}$ have a countable union, while $J_{i}$ is uncountable [F3]; choose $δ_{i}$ with $∣ δ_{i} ∣$ so small that $u_{i} (p_{i}) + δ_{i}$ avoids all the earlier singular leaves and the endpoint of $J_{i}$ , and $∣ δ_{i} ∣ \sup ∣ d ρ_{i} ∣ < η_{i} / 2$ . Only finitely many such choices are made in the whole construction.

4.1step 2.1step 3.1F1

The local modification preserves the singular set. Define the modified map $h_{i}$ on $V_{i}$ by $h_{i} = χ_{i}^{- 1} (Y_{i}, u_{i} + δ_{i} ρ_{i})$ , where $(Y_{i}, u_{i}) = χ_{i} \circ h$ are the foliation-box coordinates, and let $h_{i} = h_{i - 1}$ outside $V_{i}$ , with $h_{0} = h$ ; since $ρ_{i}$ is compactly supported in the interior of $V_{i}$ , all derivatives agree across $\partial V_{i}$ , so $h_{i}$ is a $C^{2}$ map of the disk equal to $h$ near the collar. On the neighbourhood of $W_{i}$ where $ρ_{i} = 1$ the transverse function is $u_{i} + δ_{i}$ , whose critical set and Hessian agree with those of $u_{i}$ by [F1], so the singular point $p_{i}$ survives with its type unchanged. On $supp (d ρ_{i})$ one has $d (u_{i} + δ_{i} ρ_{i}) = d u_{i} + δ_{i} d ρ_{i}$ with $∣ d u_{i} ∣ \geq η_{i} > ∣ δ_{i} ∣ \sup ∣ d ρ_{i} ∣$ , so the differential does not vanish and no new zero appears; outside $V_{i}$ the map is unchanged and its zeros are the previously handled ones.

5.1step 3.1step 4.1

Distinct leaves. At the end of stage $i$ the singular image of $p_{i}$ is $h_{i} (p_{i})$ ; in the coordinates of the box $Q_{i}$ its transverse coordinate is $u_{i} (p_{i}) + δ_{i}$ , and by step 3.1 this value avoids the leaves $L_{j}$ of the singular images of all $j < i$ ; since the later modifications are supported in $V_{j}$ with $j > i$ and $V_{i} \cap V_{j} = \emptyset$ , they do not move the image of $p_{i}$ . Applying this for every $i$ , the final images $h^{'} (p_{1}), \dots, h^{'} (p_{N})$ lie in pairwise distinct leaves.

6.1step 2.1step 3.1step 4.1step 5.1F1

Assembling the homotopy. Define $H : D^{2} \times [0, 1] \to M$ by $H (x, s) = χ_{i}^{- 1} (Y_{i} (x), u_{i} (x) + s δ_{i} ρ_{i} (x))$ for $x \in V_{i}$ , and $H (x, s) = h (x)$ outside $⋃_{i} V_{i}$ . The disks are disjoint, so this is well defined; every $ρ_{i}$ vanishes on an open neighbourhood of $\partial V_{i}$ , so the local formulas equal $h (x)$ there for every $s$ and glue to a jointly $C^{2}$ map by the chain rule. Thus $H (\cdot, 0) = h$ and $H (\cdot, 1) = h^{'}$ , and the complement of the finite compact union $⋃_{i} supp ρ_{i}$ is an open neighbourhood of $\overline{C}$ fixed throughout. In chart coordinates the $C^{2}$ norm of each endpoint modification is bounded by $∣ δ_{i} ∣ ∥ ρ_{i} ∥_{C^{2}}$ ; composition with the fixed $C^{2}$ chart inverse is continuous in $C^{2}$ on a compact chart neighbourhood, as follows by applying the chain rule twice and uniform continuity of its derivatives there. Choose the $δ_{i}$ within the finitely many chart margins and norm bounds as well as the inequalities of step 3.1, so all interpolated chart points remain in $Q_{i}$ and $h^{'}$ lies in the prescribed $C^{2}$ neighbourhood. For $N = 0$ take $H (x, s) = h (x)$ . Step 4.1 gives (i), and step 5.1 gives (ii).

7.1step 5.1step 6.1F2∎

Finally, a characteristic separatrix joining two distinct singular points is a trajectory of the characteristic field on which the local transverse coordinate is constant, so its endpoints are two singular images lying in one leaf; assertion (ii) therefore excludes such a separatrix, while a homoclinic separatrix begins and ends at the same singularity and is not excluded. The only infinite selection in the proof is the countable union in step 3.1, which uses exactly the stated ${A C}_{ω}$ through the leaf-intersection lemma [F2]; every other choice is finite.

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

A C² first-integral period annulus has a C² leaf product

Statement

Assume ${A C}_{ω}$ . Let $A \subseteq R^{2}$ be a nonempty connected open set carrying a $C^{2}$ first-integral atlas: each chart has a $C^{2}$ submersion $u$ whose connected levels are the leaves, and overlap transverse coordinates differ by $C^{2}$ local diffeomorphisms. Suppose all leaves are simple compact $C^{2}$ circles whose bounded Jordan domains are strictly nested, with consistent orientation. Then $A$ is an open annulus, and there is a $C^{2}$ diffeomorphism $Ψ : S^{1} \times (0, 1) \to A$ taking each circle onto one leaf and increasing in the nested leaf order. For any nowhere-zero $C^{1}$ tangent generator $X$ , orient $θ$ so $Ψ_{*}^{- 1} X = a (s, θ) \partial_{θ}$ with a positive $C^{1}$ function $a$ . No $θ$ -independent speed, $C^{2}$ flow of $X$ , or $C^{2}$ coefficient $a$ is asserted. The atlas applies to $u = z \circ h$ for $C^{2}$ characteristic maps on their regular annulus.

Facts & Assumptions

Given: A connected open planar set $A$ with a $C^{2}$ first-integral atlas whose leaves are simple compact $C^{2}$ circles with strictly nested bounded Jordan domains and consistent orientation, together with the induced codimension-one $C^{2}$ foliation of the surface $A$ .

[F1]

Every finite plaque transport between $C^{2}$ local transversals is a $C^{2}$ local diffeomorphism germ, and a finite family of $C^{2}$ trace maps agreeing on open overlap collars glues to a $C^{2}$ trace map (C² plaque transport and finite transverse fences preserve C² regularity).

[F2]

[F3]

A $C^{2}$ map with invertible derivative at a point has a $C^{2}$ local inverse; a $C^{2}$ equation with nonzero normal derivative has a unique local $C^{2}$ root (C² inverses and scalar return roots).

[F4]

Assuming ${A C}_{ω}$ , every second countable space is Lindelöf (Assuming countable choice, every second countable space is Lindelöf).

[F5]

A continuous real function on an order-convex interval has a primitive there, unique up to an additive constant (Every continuous function on an interval has a primitive; two primitives differ by a constant; and $\int_{a}^{b} f = G (b) - G (a)$ for any primitive $G$ ).

[F6]

For a surjection $q : X \to Y$ the quotient topology on $Y$ is the finest topology making $q$ continuous, so a subset of $Y$ is open exactly when its preimage is open and $q$ is continuous (The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection).

[F7]

[F8]

The standing assumption of the pair is Countable Choice ${A C}_{ω}$ (The countable-choice principle used in the foliation pair).

Proof

technique · direct

1.1givenF2F7

A compact C² section drawn inside one first-integral box and shrunk so that it is transverse to the line field everywhere meets each leaf at most once: orient every circle as the boundary of its bounded Jordan domain; the determinant of a positively oriented leaf tangent and the tangent of the section is continuous and nowhere zero along the section and the transverse coordinate is locally determined, so its sign is locally constant along the connected section, and on the compact section a leaf meeting would have finitely many intersection points, since they are isolated by transversality and a compact set covered by isolating neighbourhoods is finite by [F7]; between consecutive intersections the section subarc is a connected arc avoiding the leaf, hence lies in one component of the complement of that Jordan curve by [F2], whereas the crossing direction at the two ends would have to pass from the bounded to the unbounded component and back, contradicting the constant sign.

2.1step 1.1F1F7

The compact leaf $C$ admits a finite cyclic chain of C² foliated rectangles covering it, and finite plaque transport around the chain defines a C² return map $H$ on a smaller interval of a transverse section; the returned point lies on the same global leaf as the starting point and on the section, so step 1.1 gives $H (t) = t$ , and hence the transported transverse parameter is a well-defined C² first integral on a saturated neighbourhood of $C$ , with the finitely many pieces agreeing exactly on the open overlap collars by [F1].

3.1step 2.1F1F3

Finite phase gluing produces a C² product over a neighbourhood of $C$ : choose a C² once-around parametrization $γ_{0}$ of $C$ and a finite cyclic cover by plaque arcs whose enlarged arcs lie in the rectangles of step 2.1, refined so that only adjacent enlarged arcs overlap and each overlap lies in one common rectangle; holding the transported transverse coordinate fixed at $t$ and the reference leaf coordinate of $γ_{0}$ fixed gives C² candidates $γ_{i}$ on the enlarged arcs, and on an overlap the two candidates are blended in the common plaque coordinate by $x (t, θ) = χ (θ) x_{i} (t, θ) + (1 - χ (θ)) x_{i + 1} (t, θ)$ with a C² cutoff $χ$ equal to one on an open collar at one end and zero at the other; at $t = 0$ both candidates equal $γ_{0}$ , so $\partial_{θ} x > 0$ on the finitely many closed overlaps after shrinking the transverse interval once, while on the two open collars the formula equals a single candidate exactly, and the cyclic product closes because $H$ is the identity; the gluing rule [F1] and the C² local diffeomorphism and open-mapping properties of [F3] then give a C² regular circle map $γ (t, \cdot)$ of degree one, whose images are onto the connected compact leaves, and $(t, θ) \mapsto γ (t, θ)$ is a C² local diffeomorphism because the $θ$ -block is positive and $t$ is a submersion, hence a C² product over that neighbourhood.

4.1step 3.1F6F7

Let $B$ be the quotient of $A$ by its circle leaves with the quotient topology; the local products of step 3.1 make the quotient map open and give increasing C² interval charts, so the images of a countable Euclidean basis of $A$ form a countable basis of $B$ , and $B$ is connected as a continuous image of the connected $A$ and has no endpoints; disjoint compact leaves have disjoint saturated product neighbourhoods, because disjoint compact subsets of the plane have positive distance and each leaf has arbitrarily small saturated product neighbourhoods by step 3.1, so $B$ is Hausdorff and the nested leaf order agrees with its interval-chart topology; consequently a bounded nonempty subset $S \subseteq B$ has a supremum, since otherwise the set of points below some element of $S$ and the set of points above every element of $S$ would be disjoint nonempty open sets covering the connected $B$ .

5.1step 4.1

Every closed order segment $[a, b] \subseteq B$ is compact: for an open cover let $T$ be the set of points $x$ with $[a, x]$ finitely covered; a cover member at $a$ makes $T$ nonempty, and if $c = \sup T < b$ then a cover member containing $c$ extends a finite subcover past $c$ , a contradiction, while $c = b$ means that same member completes a finite subcover of $[a, b]$ .

6.1step 5.1F4F5F7

Under the single application of ${A C}_{ω}$ in [F4], select countably many local product charts with precompact interval cores; their finite order hulls exhaust $B$ , and replacing the exhaustion by the strictly expanding one that at each step takes the least later finite hull in the fixed enumeration extending both endpoints of the previous hull gives compact shells; on each shell take the least finite subcover in that same fixed enumeration, attach explicit C² bumps $η ((t - t_{i}) / r_{i})$ with $η (v) = e^{- 1 / (1 - v^{2})}$ supported inside the next shell, and normalize the locally finite positive sum to a C² partition of unity; for increasing local coordinates $t_{i}$ the form $α = \sum_{i} ρ_{i} d t_{i}$ is positive of class $C^{1}$ , and by [F5] applied on each interval chart it has C² local primitives whose differences on connected overlaps are constants, so continuing across the compact order segments of step 5.1 defines a strictly increasing C² local diffeomorphism $B \to J$ onto an open interval, which an explicit increasing smooth reparametrization carries to $(0, 1)$ .

7.1step 6.1F3F4

Choose a countable locally finite chain of compact base slabs inside the selected product intervals with consecutive open overlap collars, select the local products, seams and orientation-preserving transition diffeomorphisms $g_{s}$ together with the slabs under the same ${A C}_{ω}$ application before gluing, and lift each transition to $G (s, θ + 2 π) = G (s, θ) + 2 π$ fixing one seam value; extending over the next slab by $G_{e x t} (s, θ) = χ (s) G (s, θ) + (1 - χ (s)) G (s_{*}, θ)$ with a fixed C² cutoff equal to one on the old-side open collar and zero before the overlap ends has $θ$ -derivative a convex combination of positive derivatives, and composing with the already-built lift makes the recursion deterministic; local finiteness and exact collar agreement give a global C² product, and fiberwise bijectivity with the local C² inverses of [F3] makes it a C² diffeomorphism $Ψ : S^{1} \times (0, 1) \to A$ .

8.1step 7.1given

Each circle $S^{1} \times {s}$ is carried onto one leaf and the coordinate increases in the nested leaf order by construction, so $A$ is an open annulus; for a nowhere-zero C¹ tangent generator $X$ the map $X \circ Ψ$ is C¹ and $D Ψ^{- 1}$ is C¹, so the coefficient $a (s, θ)$ extracted from $Ψ_{*}^{- 1} X = a \partial_{θ}$ is a nowhere-zero C¹ function, and reversing the orientation of $θ$ if necessary, which is a single global choice because the family of circles is connected, makes it positive.

9.1step 8.1F8∎

Therefore $A$ is an open annulus with a C² diffeomorphism $Ψ$ taking circles onto leaves in increasing nested order and writing $Ψ_{*}^{- 1} X = a (s, θ) \partial_{θ}$ with $a > 0$ of class C¹; the construction used the single ${A C}_{ω}$ selection of countable chart, hull and slab data in steps 6.1 and 7.1 and no other choice, and it asserts neither a $θ$ -independent speed nor a C² coefficient.

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

A fixed cap product glues by unique transverse flow roots

Statement

Assume Countable Choice ${A C}_{ω}$ (The countable-choice principle used in the foliation pair). Let $F$ be a $C^{2}$ codimension-one regular foliation of a manifold $M$ , let $W$ be a compact disk, let $B : W \to L$ be a $C^{2}$ map into one leaf $L$ , and let $V$ be a fixed smooth vector field, positively transverse to $F$ , on a neighbourhood of the compact image $B (W)$ , with flow $Φ$ . Suppose the cap continuation is finite and holonomy-trivial: finitely many flat foliation boxes cover $B (W)$ , a finite cell subdivision of $W$ carries each closed cell into one box, and plaque continuation of a fixed positively oriented $C^{2}$ transversal $τ$ through $B (x_{*})$ along $B$ -paths is independent of the path near $t = 0$ , with endpoint $T_{x} (t)$ .

Then there are a uniform open interval $J = (- r, r)$ about $0$ and a jointly $C^{2}$ map $P : W \times J \to M$ with $P (x, 0) = B (x)$ , such that every slice $P (\cdot, t)$ lies in a single leaf, every track $P (x, \cdot)$ is positively transverse to $F$ , and $d P^{- 1} (T F) = T W = \ker (d t) .$ Moreover, if $C \subseteq W$ is a collar region carrying a $C^{2}$ trace $f_{C}$ with $f_{C} (x) = T_{x} (t_{C} (x))$ for a section $t_{C} : C \to J$ and if each point of $f_{C}$ is obtained by projecting along the short $V$ -orbit segment of $B (x)$ used in the construction, then $P (x, t_{C} (x)) = f_{C} (x)$ pointwise on $C$ .

Facts & Assumptions

Given: A $C^{2}$ codimension-one foliation $F$ , a compact disk $W$ with a $C^{2}$ cap $B : W \to L$ , a fixed smooth positively transverse field $V$ near $B (W)$ , and a finite holonomy-trivial cap continuation with transported transversals $T_{x} (t)$ as in the statement.

[F2]

For a $C^{2}$ field on an open set of $R^{n}$ the maximal flow is jointly $C^{2}$ , its time slices are local $C^{2}$ diffeomorphisms, and at a regular point the flow box is $C^{2}$ (C¹ Euclidean maximal flows, variational dependence and the finite C² upgrade).

[F3]

If $g (s, t)$ is $C^{2}$ near $(s_{0}, t_{0})$ , $g (s_{0}, t_{0}) = 0$ and $g_{t} (s_{0}, t_{0}) \neq 0$ , then there is a unique local $C^{2}$ root $t = T (s)$ with $T^{'} = - g_{s} / g_{t}$ ; the same inverse-function argument gives a $C^{2}$ root depending jointly on additional $C^{2}$ parameters (C² inverses and scalar return roots).

[F4]

Every finite plaque transport between $C^{2}$ local transversals of a $C^{2}$ foliation atlas is a $C^{2}$ local diffeomorphism germ (C² plaque transport and finite transverse fences preserve C² regularity).

[F5]

The holonomy germ of a leafwise path is independent of the foliation chart chain (The holonomy germ is independent of the foliation chart chain).

[F6]

The standing hypothesis is Countable Choice ${A C}_{ω}$ (The countable-choice principle used in the foliation pair).

Proof

technique · direct

1.1given

Use the fixed smooth field $V$ supplied in the statement. Choose finitely many smaller foliation boxes covering $B (W)$ , with compact cores and larger boxes still inside the domain of $V$ . Sign their transverse coordinates so $d z_{j} (V) > 0$ on the larger boxes. No foliation-coordinate field is asserted to be smooth, and $V$ is not replaced.

1.2givenF4F5

The transported transversals. Use the supplied positively oriented $C^{2}$ transversal $τ$ at $B (x_{*})$ and a finite subdivision of $W$ into closed cells each mapped by $B$ into one box of the cover. Along the tree of cells, plaque continuation of $τ$ defines for every cell $c$ a $C^{2}$ map $(x, t) \mapsto T_{x} (t)$ on $c \times J_{0}$ , for some interval $J_{0}$ about $0$ , with $T_{x} (0) = B (x)$ ; this uses the finite holonomy-trivial continuation hypothesis. Path independence of the germ away from the basepoint is [F5], and the regularity of each finite transport is [F4], so the assignment is $C^{2}$ in $(x, t)$ on each cell and the assigned pieces agree on overlaps. A finite intersection of the finitely many domains of definition gives one uniform interval $J_{0} = (- δ_{0}, δ_{0})$ on which all these transports are defined.

2.1F2step 1.1

In smooth ambient charts, [F2] supplies the local flow of $V$ ; uniqueness glues the finitely many formulas near the compact image $B (W)$ . Shrink one common time interval so these flow segments stay in the relevant larger boxes. The flow is jointly $C^{2}$ , and $s \mapsto z_{j} (Φ_{s} (B (x)))$ is strictly increasing on each such short segment. Completeness is unnecessary.

3.1step 2.1step 1.2given

Reduction of the root equation to one cell. Fix a cell $c$ contained in a box $U$ with transverse coordinate $z$ , and put $G (s, x, t) : = z (Φ_{s} (B (x))) - ζ (x, t)$ , where $ζ (x, t) : = z (T_{x} (t))$ is jointly $C^{2}$ on $c \times J_{0}$ . Since $B$ maps $c$ into one plaque of $U$ , the value $z (B (x))$ is constant on $c$ , and $ζ (x, 0) = z (B (x))$ ; since $T_{x} (t)$ is transported along a positive transverse direction, $\partial_{t} ζ > 0$ on $c \times J_{0}$ . Finally $\partial_{s} G (0, x, t) = d z (V) (B (x)) > 0$ , because $V$ is positively transverse.

4.1F3step 2.1step 3.1

At $(s, x, t) = (0, x, 0)$ the root equation has value zero and $\partial_{s} G > 0$ . Apply [F3] there, with $x, t$ as parameters, and cover each compact cell by finitely many of the resulting parameter neighborhoods. Shrink the common t-interval and the flow-time bound so every root remains in the short segment where $\partial_{s} G > 0$ . Uniqueness then pastes these local root functions into one jointly $C^{2}$ function $s_{c} (x, t)$ on an open neighborhood of each cell times one interval $J$ . Take the finite intersection of all such intervals.

5.1givenF4step 2.1step 1.2step 4.1

On an overlap use a common smaller box around $B (x)$ . The continuation data assign the same local plaque there, not just the same global leaf: transition of the transverse coordinate sends the label in one chart to the label in the other. For variable x in this overlap the central plaque label is fixed, so the equality of transverse transition germs holds on one neighborhood in x and one short interval in t; finite compact covers of the cell faces give a common interval. Both root points lie on the same short V-segment in this box and have this identical plaque label. Strict monotonicity in step 2.1 therefore gives equal flow times. Since the formulas hold on open cell neighborhoods, [F4] pastes $P (x, t) = Φ_{s_{c} (x, t)} (B (x))$ jointly $C^{2}$ on $W \times J$ .

6.1step 5.1given

Properties of $P$ . Clearly $P (x, 0) = Φ_{s_{c} (x, 0)} (B (x))$ and $s_{c} (x, 0) = 0$ because $G (0, x, 0) = z (B (x)) - ζ (x, 0) = 0$ and the root is unique; hence $P (x, 0) = B (x)$ . For fixed $t$ all points $P (x, t)$ lie in the single leaf $L_{t}$ containing $τ (t)$ , since they lie in the leaf containing $T_{x} (t)$ and each $T_{x} (t)$ is obtained from $τ (t)$ by plaque continuation; hence every slice lies in one leaf, and $\partial_{x} P$ is tangent to $F$ . For the $t$ -direction, differentiating $G (s_{c} (x, t), x, t) = 0$ gives $\partial_{t} s_{c} = - \partial_{t} G / \partial_{s} G = \partial_{t} ζ / \partial_{s} G > 0$ , so $\partial_{t} P = \partial_{t} s_{c} \cdot V (Φ_{s_{c}} (B (x)))$ is a positive multiple of the positively transverse field $V$ ; hence every track is positively transverse and $d P^{- 1} (T F) = T W = \ker (d t)$ .

6.2step 4.1step 5.1

Exact collar factorization. Let $C \subseteq W$ and $f_{C} (x) = T_{x} (t_{C} (x))$ with $t_{C} (C) \subseteq J$ be as in the statement, and suppose each $f_{C} (x)$ is obtained by projecting along the short $V$ -orbit segment of $B (x)$ used in the construction, so that $f_{C} (x) = Φ_{σ (x)} (B (x))$ for some $σ (x)$ in the same short flow-time domain. Then $f_{C} (x)$ lies in the leaf containing $T_{x} (t_{C} (x))$ , and on the $V$ -orbit of $B (x)$ the equation $z (Φ_{s} (B (x))) = ζ (x, t_{C} (x))$ is satisfied at $s = σ (x)$ ; by uniqueness of the root in step 4.1, $σ (x) = s (x, t_{C} (x))$ and therefore $P (x, t_{C} (x)) = Φ_{s (x, t_{C} (x))} (B (x)) = f_{C} (x)$ pointwise on $C$ .

7.1step 6.1step 6.2F6∎

The construction used finitely many boxes, finitely many cells, finitely many bumps and finitely many local roots, so it makes only finitely many choices; the root and flow theorems of [F2] and [F3] are choice-free, and the standing hypothesis [F6] is not used beyond the pair's interface. This proves the statement.

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)Open item page →

A one-quadrant homoclinic disk contains a center

Statement

Let $X$ be the characteristic $C^{1}$ field of a $C^{2}$ characteristic disk with finitely many nondegenerate centers and saddles. Let $K$ be the bounded source disk of a simple directed homoclinic circuit through a saddle $q$ , and suppose $K$ occupies precisely one of the four local saddle sectors of $X$ at $q$ . If $c_{K}$ and $s_{K}$ count the centers and saddles strictly inside $K$ , then $c_{K} - s_{K} = 1.$ In particular $K$ contains a center. The saddle $q$ is not included in $s_{K}$ .

Facts & Assumptions

Given: A characteristic $C^{1}$ field $X$ on a source disk with finitely many nondegenerate centers and saddles, a saddle $q$ , and the bounded source disk $K$ of a simple directed homoclinic circuit through $q$ occupying one local saddle sector at $q$ .

[F1]

A characteristic field is, in a foliation chart with transverse function $u$ , of the form $X = f J \nabla u$ with $f \neq 0$ and $J$ the quarter turn; hence on the regular part the trajectories of $X$ are exactly the level sets of $u$ , the zeros of $X$ are the critical points of $u$ , and a nondegenerate zero of definite Hessian is a center while an indefinite Hessian gives a saddle (Transversely oriented codimension-one foliations, Relative generic position for characteristic disk maps, The characteristic disk has one more center than saddle).

[F2]

Near a nondegenerate saddle of a $C^{2}$ function $u$ there are $C^{1}$ coordinates $(x, y)$ with $u - u (q) = x y$ (A C² saddle function has C¹ Morse coordinates); in these coordinates the local stable and unstable branches of $X$ are the two coordinate axes and the four local sectors are the four quadrants.

[F3]

A simple closed piecewise- $C^{2}$ regular plane curve with finitely many corners and distinct one-sided tangents at each corner bounds exactly two components, one bounded and one unbounded (A finitely cornered regular plane curve separates without choice).

[F4]

For a simple closed piecewise- $C^{2}$ regular plane curve bounding a positively oriented disk region, the directed unit tangent is a circle loop and its rotation index is $1$ : the total signed turning of the tangent equals $2 π$ (Hopf turning-tangent theorem with ordinary corners, Rotation index of a regular closed plane curve).

[F5]

For an oriented loop in $S^{1}$ the degree adds under composition with the antipodal map trivially: the antipodal map of $S^{1}$ has degree $1$ , so a loop $t \mapsto - T (t)$ has the same degree as $t \mapsto T (t)$ (The degree of a based circle loop, Degree of identity constant reflection and antipodal sphere maps).

Proof

technique · direct

1.1givenF1F2F5

The saddle q is the only zero on the homoclinic boundary: a nonconstant trajectory cannot pass through another zero. Choose orientation-preserving C¹ Morse coordinates near q, and replace X by −X if necessary, so the sector occupied by K is $x, y \geq 0$ and $X = c (x, y) (x, - y)$ with $c > 0$ . Replacing X by its negative does not change local or boundary degrees. Choose a small quarter disk in this sector whose closure contains no other zero.

2.1F2F3step 1.1construct

Cut off that quarter disk by the arc $x^{2} + y^{2} = ε^{2}$ , directed from $(0, ε)$ to $(ε, 0)$ . Its endpoints are on the zero-level separatrices, and its interior is inside K. Its inverse image in the original plane is a regular C¹ arc. On the arc, both its directed tangent and X have positive x-component and negative y-component in the open quadrant; at each endpoint they are perpendicular rather than opposite. After applying the invertible derivative of the coordinate change they remain never opposite. Replace this compact C¹ arc by a sufficiently C¹-close regular C² arc with the same endpoints, staying inside the sector and retaining that nonopposition. Such an approximation is elementary in finitely many graph charts: convolve each C¹ graph with a smooth compactly supported kernel, whose function and derivative converge uniformly; finite endpoint corrections fix the endpoints and tangent directions, and a thin graph strip preserves embedding. This gives a simple piecewise-C² curve C′ formed with the retained orbit arc. Its bounded region K′ lies in K and removes q and no interior zero.

3.1F4step 2.1

Orient C′ by the homoclinic direction and the new cut arc; the retained region is on its left in the chosen sector, so this is its positive boundary orientation. On the orbit part, the normalized X equals the directed tangent. On the cut arc it is never opposite to that tangent. At the two corners interpolate between the one-sided tangents by their nonzero convex combinations; X is never opposite to this corner interpolation, since before the approximation the two tangents and X occupy the same closed pointed quadrant, and this persists after a sufficiently small approximation. Thus normalization of $(1 - v) X + v T$ gives a homotopy from X/|X| along C′ to its tangent loop with the prescribed short corner turns. By the turning theorem that tangent loop has degree one.

4.1F1step 1.1step 2.1step 3.1

To compute the index sum, approximate C′ inside a zero-free thin collar by a simple inscribed polygon, using finitely many local graph strips; projection in those strips gives a boundary homotopy through nonzero fields. Choose disjoint small squares around every interior zero. Choose a direction with distinct projections of all outer-polygon and square vertices. Between consecutive projections the boundary edges are ordered affine graphs; the zero-free region is a finite union of bands between consecutive graphs. Subdivide their vertical walls at all edge intersections, and split each convex triangle or quadrilateral band into triangles, as in the finite polygonal subdivision of Every simple polygon admits a triangulation. On each zero-free cell the normalized field extends across the cell, so its boundary degree is zero. Summing the boundary degrees cancels every common oriented edge, and gives outer degree equal to the sum of the small-square degrees. The local calculation in The characteristic disk has one more center than saddle, in its local-degree paragraph, gives +1 for a center and −1 for a saddle; this calculation and the cancellation argument do not require its outer-boundary alternatives once the outer degree has been computed directly. Hence $c_{K} - s_{K} = \deg (X / ∣ X ∣ on C') = 1$ .

5.1step 2.1step 4.1∎

All original interior zeros are inside K′ and q was cut off, so the count in step 4.1 is exactly the stated strict-interior count. It implies $c_{K} \geq 1$ . Only finitely many charts, approximations and polygonal cells were used.

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

Finite general position for a leafwise loop

Statement

Assume ${A C}_{ω}$ . Let $S$ be a $C^{2}$ surface, $x \in S$ , and $f : (S^{1}, 0) \to (S, x)$ a continuous based loop. Then $f$ is based-homotopic to a regular $C^{2}$ immersed loop with finitely many transverse double points and no triple points. The homotopy fixes the basepoint throughout. For a foliation leaf, all maps and homotopies remain in that leaf with its intrinsic plaque topology.

Facts & Assumptions

Given: The surface, point, loop and countable choice in the statement.

[F1]

For a leaf use the intrinsic topology generated by its plaques. Plaque coordinates give surface charts: on overlapping plaque components their transitions are the leaf-coordinate components of the given $C^{2}$ foliated atlas and are $C^{2}$ local diffeomorphisms. The leaf is the plaque-chain set of Leaves of a regular foliation; closed bounded Euclidean sets are compact (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line).

[F2]

$C^{1}$ equations with invertible differential have $C^{1}$ local inverses; $C^{2}$ equations have $C^{2}$ inverses (The Euclidean inverse function theorem, C² inverses and scalar return roots).

[F3]

Sard's theorem holds for a $C^{r}$ Euclidean map of source and target dimensions $a, b$ when $r > \max {a - b, 0}$ (Morse-Sard for Euclidean maps). Lower-dimensional $C^{1}$ images are null, finite or countable null unions are null, and a null set has dense complement (The image of a lower-dimensional $C^{1}$ manifold is null, Countable unions and subsets of manifold null sets are null, A null set has dense complement in a positive-dimensional manifold).

[F4]

Smooth source-circle bumps with prescribed compact cores and supports exist (A manifold bump for a compact set inside an open set). A derivative bounded away from zero in one coordinate gives monotonicity and injectivity (The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ ).

[F5]

Countable choice is The countable-choice principle used in the foliation pair. No smooth atlas on the merely $C^{2}$ target is presumed: all target constructions use its actual $C^{2}$ coordinate changes.

Proof

1.1F1givenconstruct

Cover the compact image of $f$ by finitely many surface charts with convex coordinate disks, and subdivide the parameter circle finely enough that every segment maps into one such disk. Coordinate straight-line interpolation replaces each segment by its endpoint chord, through a homotopy fixing all segment endpoints. Near the base parameter, work in a chart centered at $x$ . Replace a sufficiently short central interval by a nonconstant straight segment through $x$ , with endpoints $x - δ v$ and $x + δ v$ for a fixed nonzero coordinate vector $v$ ; join those endpoints to the old endpoints in that same convex disk. Linear interpolation on this interval fixes its middle value $x$ throughout. Subdivide again if needed. This gives a based polygonal loop that is regular and straight on a fixed base collar; constant original loops are included by inserting this small based detour.

2.1F2F3step 1.1constructalgebra

Vary the remaining finitely many vertices in small coordinate disks, fixing that entire base collar, to make every edge nonconstant and the incoming and outgoing tangent vectors at each other vertex noncollinear. Here is the finite genericity justification: for a fixed vertex and nonzero outgoing vector, varying its incoming neighboring vertex changes the incoming coordinate direction through an open subset of the plane, followed by an invertible coordinate-change differential. Collinearity is therefore a regular scalar equation in the joint vertex parameters, off the already excluded zero-edge locus; the zero-edge equations have codimension two. At a fixed base-collar endpoint the other neighboring vertex remains adjustable and gives the same scalar rank test. Insert an extra adjustable vertex if necessary so this holds at every corner. F2 makes these bad loci $C^{1}$ submanifolds of positive codimension; F3 makes their parameter images null. Choose arbitrarily small parameters outside their finite union. Each vertex movement and its edge-chord adjustment is a homotopy in a convex chart, still fixing the base collar.

3.1F2step 2.1constructalgebra

Round the finitely many corners without losing regularity. In a vertex chart first make each incident edge exactly linear near the vertex: its Taylor remainder is $O (t^{2})$ with derivative $O (t)$ , so a cutoff on an interval of length $δ$ changes its derivative by $O (δ)$ and keeps it nonzero. If the two resulting directed velocities are $a, b$ , choose a smooth function $ρ$ on $[- δ, δ]$ , zero and one on end collars, with $ρ (- t) = 1 - ρ (t)$ . Define the replacement from the incoming endpoint by integrating $(1 - ρ (t)) a + ρ (t) b$ . Since the integrals of $ρ$ and $1 - ρ$ are both $δ$ , the replacement reaches the outgoing endpoint and agrees with both straight edges on end collars. Its derivative never vanishes: noncollinearity excludes zero from the segment $[a, b]$ . Shrink the chart and interval so its image stays in the convex disk. Coordinate straight-line homotopy relative to the two endpoints realizes the replacement; no corner is rounded at the basepoint, where the fixed collar was already straight. The result is a regular $C^{2}$ loop $g_{0}$ , based-homotopic to $f$ .

4.1F1F4step 3.1construct

There is a uniform source separation scale $η > 0$ for this immersion, stable under sufficiently small $C^{1}$ perturbations. Indeed cover the parameter circle by finitely many small intervals on which a target-coordinate projection of $g_{0}$ has derivative of one sign bounded away from zero. F4 gives injectivity on those intervals for all close maps. A Lebesgue scale for that finite cover excludes all coincidences with source distance below $η$ . Choose nested closed base collars $J \subset int J^{+}$ inside the fixed straight interval, with the diameter of $J^{+}$ below $η / 3$ . Keep the entire $J$ fixed. The compact pair configurations with distance at least $η$ contain at most one parameter in $J^{+}$ ; triple configurations with all pair distances at least $η$ also contain at most one. Thus each relevant pair has at least one fully adjustable parameter outside $J^{+}$ , and each triple has at least two.

5.1F1F2F4step 4.1constructalgebra

Build one finite parameter family $g_{v}$ using source bumps supported off $J$ , independent two-coordinate target translations in the actual surface charts, and composition of these translation factors. For each possible pair or triple coincidence choose disjoint source intervals around its adjustable parameters, with bumps equal to one there and target-chart margins containing their compact images. Finitely many configuration neighborhoods cover the compact pair and triple coincidence sets. At $v = 0$ , moving one adjustable image spans the two normal directions to the pair diagonal; moving two adjustable images spans the four normal directions to the triple diagonal, even when the third image is fixed. These are transverse-to-diagonal assertions, not a false full submersion claim for all values of a family with a fixed branch. Uniform smallness and the finite compact covers preserve the rank tests near all possible coincidences. Off those neighborhoods the original value configurations miss the closed diagonals by a positive margin, so small perturbations produce no new incidences there. Every member remains immersed and retains the separation estimate of step 4.1.

6.1F2F3step 4.1step 5.1algebra

In a common target chart, the pair difference equation has two independent parameter derivatives. F2 makes its total zero set a $C^{2}$ manifold of dimension $P + 2 - 2 = P$ , where $P$ is the parameter dimension. Critical values of its parameter projection are null by F3; at a regular parameter, elementary linear algebra identifies projection regularity with surjectivity of the two-source-parameter difference derivative. The slice coincidences are therefore transverse isolated pairs. The corresponding triple difference equation has four independent parameter derivatives; its zero manifold has dimension $P + 3 - 4 = P - 1$ , so its parameter image is null by F3 and good slices have no triples. These statements are applied on open separated-configuration neighborhoods; their finite or countable coordinate covers are covered by F3's null-union clause. If desired also exclude another branch hitting the fixed image $x$ : outside the fixed base collar the same adjustable-value equation has source dimension one and target codimension two, giving zero-manifold dimension $P - 1$ and a null parameter image; the fixed straight collar has only its specified preimage of $x$ . Choose one arbitrarily small parameter outside these null exceptional sets. No unsupported four-direction rank test using only one moved branch is used.

7.1F1F3F4F5step 3.1step 4.1step 6.1∎

For this parameter the double-pair set is closed in the compact separated pair configuration space and discrete by step 6.1, hence finite; no near-diagonal coincidence exists by step 4.1. The loop is regular and has no triple image. The homotopy $g_{t v}$ fixes every point of $J$ , in particular the basepoint $x$ , and joins $g_{0}$ to this final loop. Composing it with the based homotopies of steps 1.1–3.1 proves the statement. The source bumps all vanish near the basepoint; their cores were never required to cover that fixed collar. Target chart operations stay in $S$ , hence in the given leaf when $S$ is a leaf. All chart and parameter families are finite; only the explicitly cited Sard/null machinery inherits countable choice.

DefinitionDefinition: Literature-sourcedProof: Not applicableOpen item page →

The Godbillon-Vey class of a codimension-one foliation

Definition

Assume Countable Choice ${A C}_{ω}$ . Let $F$ be a transversely oriented codimension-one foliation of a smooth manifold $M$ , let $ω$ be a nowhere-vanishing defining $1$ -form with $T F = \ker ω$ , and let $η$ be a smooth $1$ -form with $d ω = η \land ω$ , whose existence is guaranteed by Frobenius divisibility: d omega equals eta wedge omega. The Godbillon-Vey class of $F$ is $G V (F) : = [η \land d η] \in H_{d R}^{3} (M; R)$ . It is well defined: $η \land d η$ is closed (The Godbillon-Vey form eta wedge d eta is closed), and the class is unchanged by replacing $η$ by $η + f ω$ (Independence of the auxiliary form eta up to exact forms) and by rescaling $ω$ (Rescaling the defining form changes the Godbillon-Vey form by an exact form). Only the de Rham class over $R$ is named; no integral refinement is defined on this page.

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One-sided trivial-holonomy classes form a normal subgroup

Statement

Assume Countable Choice ${A C}_{ω}$ . Let $F$ be a transversely oriented codimension-one foliation, $L$ a leaf, $x \in L$ , and $j$ one of its two sides. The classes in $π_{1} (L, x)$ whose one-sided holonomy germ is the identity form a normal subgroup $N_{j} (L, x) ◃ π_{1} (L, x)$ . Consequently the quotient $P_{j} (L, x) = π_{1} (L, x) / N_{j} (L, x)$ used to define ordinary one-sided limit cycles is a group.

Facts & Assumptions

Given: A transversely oriented codimension-one foliation $F$ , a leaf $L$ , a base point $x \in L$ , one side $j$ of $L$ , and the standing countable choice assumption.

Proof

technique · direct

1.1given

By the definition of the holonomy representation, each class in $π_{1} (L, x)$ is assigned the germ of the return map along its reversed representative, on the chosen side (the library homomorphism convention), and the transverse orientation makes these germs side-preserving, so the assignment is a group homomorphism from $π_{1} (L, x)$ to the group of side-preserving germs of local diffeomorphisms of a half-transversal of the given side (The holonomy representation and the holonomy group of a leaf, Germs of local diffeomorphisms at a point).

2.1step 1.1∎

The classes in $π_{1} (L, x)$ whose one-sided holonomy germ is the identity are exactly the kernel of that homomorphism; the kernel of a group homomorphism is a normal subgroup, so the indicated classes form $N_{j} (L, x) ◃ π_{1} (L, x)$ , and the quotient $P_{j} (L, x) = π_{1} (L, x) / N_{j} (L, x)$ used to define ordinary one-sided limit cycles is a group; no choice principle beyond the standing assumption is used.

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passOpen item page →

A compact leafwise nullhomotopy persists under a transverse deformation

Statement

Assume Countable Choice ${A C}_{ω}$ (The countable-choice principle used in the foliation pair). Let $F$ be a $C^{2}$ codimension-one regular foliation, and let $H : S^{1} \times (- δ, δ) \to M$ be a $C^{2}$ trace annulus such that each loop $H_{s} = H (\cdot, s)$ lies in a single leaf and every point track $s \mapsto H (θ, s)$ is transverse to $F$ (Smooth maps transverse to a regular foliation). If $H_{s_{0}}$ is null-homotopic in its leaf by a compact continuous disk map, then $H_{s}$ is null-homotopic in its leaf for all $s$ in some open interval about $s_{0}$ .

Facts & Assumptions

Given: A $C^{2}$ codimension-one regular foliation $F$ , a $C^{2}$ trace annulus $H$ with leafwise loops and transverse tracks, a parameter $s_{0}$ , and a compact continuous disk map $u : D^{2} \to L_{s_{0}}$ with $u ∣_{\partial D^{2}} = H_{s_{0}}$ .

[F1]

A $C^{2}$ foliation atlas is a $C^{1}$ foliation atlas with the transition form $(x^{'}, t^{'}) = (g (x, t), h (t))$ , $h$ a $C^{1}$ local diffeomorphism (C¹ codimension-one regular foliations and transverse orientation, Regular foliation atlases).

[F2]

In a flat chart the plaques are the connected components of the level sets of the transverse coordinate; a leafwise path segment contained in a flat chart lies in a single plaque, and plaque transport between local transversals inside that chart matches points with equal transverse coordinate (Flat charts for a distribution, Plaques of a flat chart, Leaves of a regular foliation).

[F3]

Holonomy germs of leafwise paths between fixed endpoint transversals are invariant under leafwise homotopies relative to endpoints (Holonomy depends only on leafwise homotopy relative to endpoints); the holonomy representation is the homomorphism on leafwise homotopy classes of The holonomy representation and the holonomy group of a leaf; in particular a leafwise loop that is null-homotopic relative to its basepoint has identity holonomy germ, and the constant loop contributes the identity.

[F4]

Every open cover of the compact metric square $[0, 1]^{2}$ has a Lebesgue number (Every open cover of a compact metric space has a Lebesgue number: a $δ > 0$ such that every nonempty subset of diameter less than $δ$ lies inside a single member of the cover).

[F5]

[F7]

The standing hypothesis is Countable Choice ${A C}_{ω}$ (The countable-choice principle used in the foliation pair).

Proof

technique · direct

1.1givenF1F2F4

Subdivide the continuous cap u into finitely many sufficiently small parameter triangles, using the pullback of nested foliation boxes and [F4]. Choose their images inside convex plaque-coordinate cores with larger boxes available. Refine near shared faces if necessary so each edge and both adjacent triangles have a common small box inside their larger boxes. All charts and positive margins are finite. Include the basepoint $θ_{0}$ as a boundary vertex and use the transversal $τ (t) = H (θ_{0}, s_{0} + t)$ .

2.1F1F2F3F5step 1.1

Choose a spanning tree in this finite triangulation. Continue τ along the u-image of its tree paths to fixed short transversals at all vertices. Each edge, combined with the two tree paths, gives a based loop in the disk, whose u-image is nullhomotopic in the leaf. Its holonomy is the identity by [F3]. There are finitely many such relations, so choose one interval J where all transported vertex points satisfy them. Consequently vertices of each triangle lie in the same local plaque of that triangle's box. At a boundary vertex prescribe the point $H (θ, s_{0} + t)$ : continuing along the boundary gives the same plaque label as the tree path by those relations. The equality here is of local transverse labels; nearby points in one global leaf are not automatically in one plaque. The finite-chart proof of [F3] applies verbatim to the C² atlas; it is also the homotopy argument of Holonomy of a C¹ foliation is a representation into C¹ transverse germs, with orientation irrelevant.

3.1F1F2step 1.1step 2.1construct

Give each interior edge a single chosen continuous plaque path between its transported endpoints in its small common box, for example its coordinate chord. Use the prescribed path $H_{s}$ on each boundary edge. The finitely many nested boxes and a sufficiently fine original subdivision ensure these paths stay in the larger boxes of their adjacent triangles: each edge's small box has closure inside those larger boxes, and its convex plaque core contains the endpoint paths after one common shrink of J. Plaque-label compatibility in step 2.1 therefore puts the complete boundary of each triangle in one convex plaque core. Unlike independent coordinate formulas on cells, these edges are defined once and used by both incident faces.

4.1F2step 2.1step 3.1

Fill each triangle by coning its already chosen boundary path to one point of that convex plaque core, in the plaque coordinates. This is a continuous disk with exactly the chosen edge paths on its boundary. The finitely many disks agree on every shared edge, and closed pasting produces a continuous map of the original disk into one leaf, with boundary exactly $H_{s}$ . Continuity is in the intrinsic plaque topology because each piece lies in a single plaque and the pasting has finitely many pieces. This is a nullhomotopy of $H_{s}$ for every $s \in s_{0} + J$ . No differentiability of the original continuous cap has been asserted.

5.1F7step 4.1∎

The construction used only finitely many boxes, triangles, tree transports and germ relations, and one finite common interval. It proves the stated persistence under the declared ACω hypothesis, without limits of changing filling disks.

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

A fixed leafwise cap gives a joint transverse product with exact collar data

Statement

Assume Countable Choice ${A C}_{ω}$ (The countable-choice principle used in the foliation pair). Let $F$ be a $C^{2}$ cooriented codimension-one regular foliation of a $3$ -manifold $M$ , let $W$ be a compact disk, and let $B : W \to L$ be a $C^{2}$ map into one leaf. Fix $x_{*} \in W$ and a $C^{2}$ transversal $τ : (- δ, δ) \to M$ through $B (x_{*})$ . Plaque continuation of $τ$ along $B \circ q$ , for a path $q$ from $x_{*}$ to $x$ , is independent of $q$ near $t = 0$ because $B$ maps the simply connected disk into one leaf; write $T_{x} (t)$ for the endpoint in the transported transversal at $B (x)$ .

Then there are a uniform interval $J = (- r, r)$ and a jointly $C^{2}$ map $P : W \times J \to M$ with $P (x, 0) = B (x)$ , leaf-valued slices $P (\cdot, t)$ and transverse tracks $P (x, \cdot)$ , such that $d P^{- 1} (T F) = T W = \ker (d t) .$ The uniform interval is constructed from the fixed cap before any actual section range is checked. Let $C \subseteq W$ be a prescribed collar region with a $C^{2}$ trace $f_{C}$ and its actual holonomy-trivialized section $t_{C}$ , so that $f_{C} (x) = T_{x} (t_{C} (x))$ and each $f_{C} (x)$ is obtained from $B (x)$ by projection along the short flow segments of a fixed smooth positively transverse field $V$ near $B (W)$ . If a smaller closed collar $C_{0} \subseteq C$ has compact section range $S = t_{C} (C_{0}) ⋐ J$ , choose an open interval $I$ with $S ⋐ I ⋐ J$ . Then the restriction of $P$ to $W \times I$ satisfies $P (x, t_{C} (x)) = f_{C} (x)$ pointwise on $C_{0}$ and on its interior. The boundary section may be nonzero. If $C$ is a collar of $\partial W$ and $t_{C} = 0$ there, continuity gives such a $C_{0}$ as a special case.

Facts & Assumptions

Given: A $C^{2}$ codimension-one foliation $F$ of a $3$ -manifold $M$ , a compact disk $W$ , a $C^{2}$ cap $B : W \to L$ , a $C^{2}$ transversal $τ$ through $B (x_{*})$ , and a collar region $C \subseteq W$ with trace $f_{C}$ and section $t_{C}$ as in the statement.

[F1]

A compact disk is simply connected, and a based loop in a simply connected space is null-homotopic relative to its basepoint (Simply connected topological spaces, Based loops and the fundamental group).

[F2]

Holonomy germs of leafwise paths between fixed endpoint transversals depend only on the leafwise homotopy class relative to endpoints (Holonomy depends only on leafwise homotopy relative to endpoints), and the holonomy representation is the homomorphism into transverse germs of The holonomy representation and the holonomy group of a leaf.

[F3]

A finite plaque transport between $C^{2}$ local transversals of a $C^{2}$ foliation atlas is a $C^{2}$ local diffeomorphism germ (C² plaque transport and finite transverse fences preserve C² regularity).

[F4]

A fixed leafwise cap together with a fixed smooth positively transverse field and a finite holonomy-trivial continuation admits a jointly $C^{2}$ transverse product obtained by unique short flow roots, and the exact collar factorization holds when the trace is expressed in the transported coordinate with range in the uniform interval and is obtained by projecting along the flow orbits (A fixed cap product glues by unique transverse flow roots).

[F5]

If $K \subseteq M$ is compact inside an open $W \subseteq M$ in a smooth manifold, there is a smooth bump equal to $1$ near $K$ with support in $W$ (A manifold bump for a compact set inside an open set).

[F6]

Plaques of a flat chart are the connected components of the level sets of the transverse coordinate, and leaves are the plaque-chain sets (Flat charts for a distribution, Plaques of a flat chart, Leaves of a regular foliation, Regular foliation atlases).

[F7]

Closed bounded subsets of $R^{3}$ are compact (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line).

[F8]

The standing hypothesis is Countable Choice ${A C}_{ω}$ (The countable-choice principle used in the foliation pair).

Proof

technique · direct

1.1givenF1F2F3

Independence of the path. Let $q, q^{'}$ be two paths in $W$ from $x_{*}$ to $x$ ; then $q * q^{' - 1}$ is a based loop at $x_{*}$ , null-homotopic in the disk $W$ by [F1]. Composing the null-homotopy with the $C^{2}$ map $B$ gives a leafwise homotopy in $L$ relative to endpoints between the corresponding leafwise paths, so by [F2] the holonomy germs agree: the transported transversal at $B (x)$ is independent of $q$ near $t = 0$ . By [F3] each finite transport is a $C^{2}$ local diffeomorphism germ, so in each fixed local endpoint transversal the finite chart formulas are $C^{2}$ . Subdivide the compact parameter disk into finitely many cells mapping into boxes, and use their finitely many edge-loop relations to choose one common interval $J_{0} = (- δ_{0}, δ_{0})$ ; label transitions agree on open cell neighborhoods as in [F4].

2.1givenF5F6step 1.1

Use the fixed smooth field V of any prescribed collar data. When no collar data are prescribed, construct such a field near the compact image: in smooth ambient charts choose constant fields with positive transverse evaluation on smaller domains, and sum finitely many nonnegative compact-set bumps from [F5] whose cores cover the image. Positivity is an open convex condition and coorientation fixes its sign. A field obtained this way is smooth in the ambient smooth charts; no C² foliation-coordinate field is called smooth. If τ is negatively oriented, apply the positive-transversal construction to $t \mapsto τ (- t)$ and reverse the parameter again in the final product.

3.1step 1.1step 2.1F6

Finite continuation data. Because $B (W)$ is compact and covered by finitely many flat boxes, a finite subdivision of the disk $W$ into closed cells carries each cell into one box; combined with step 1.1 this is exactly the finite holonomy-trivial cap continuation required as a hypothesis of [F4].

4.1step 2.1step 3.1F4

The product. Applying [F4] to the compact disk $W$ , the cap $B$ , the field $V$ of step 2.1 and the continuation data of step 3.1 produces a uniform interval $J = (- r, r) \subseteq J_{0}$ and a jointly $C^{2}$ map $P : W \times J \to M$ with $P (x, 0) = B (x)$ , leaf-valued slices, transverse tracks and $d P^{- 1} (T F) = T W = \ker (d t)$ ; the interval $J$ is fixed by the finitely many flow-root data of the cap before any collar section is examined.

5.1step 4.1F4F7

Exact collar range bookkeeping. Let $C_{0} \subseteq C$ be a closed collar with $S = t_{C} (C_{0}) ⋐ J$ . The set $S$ is compact, because $C_{0}$ is compact and $t_{C}$ is continuous, and $S$ is contained in the open interval $J$ with positive distance from its endpoints; hence there is an open interval $I$ with $S ⋐ I ⋐ J$ . For $x \in C_{0}$ one has $t_{C} (x) \in S \subseteq I \subseteq J$ , so $P (x, t_{C} (x))$ is defined, and $f_{C} (x) = T_{x} (t_{C} (x))$ lies in the transported leaf with label $t_{C} (x)$ ; the exact collar clause of [F4], whose projection hypothesis on $f_{C}$ is part of the data, therefore gives $P (x, t_{C} (x)) = f_{C} (x)$ for every $x \in C_{0}$ , hence also on the interior of $C_{0}$ .

6.1step 5.1F7

Boundary-zero special case. If $C$ is a collar of $\partial W$ and $t_{C} = 0$ on the boundary collar, then by continuity of $t_{C}$ the section range of a sufficiently thin closed collar $C_{0} \subseteq C$ around $\partial W$ is arbitrarily close to $0$ ; since $J$ is an open interval about $0$ , such a $C_{0}$ satisfies $t_{C} (C_{0}) ⋐ J$ , so step 5.1 applies.

7.1step 4.1step 5.1step 6.1F8∎

The construction of $P$ and of the range interval used finitely many boxes, cells, bumps and local roots, so no choice beyond the standing hypothesis [F8] is invoked; steps 4.1–6.1 prove the statement.

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)Open item page →

A flat transverse drift realizes the period-annulus frontier as an omega-limit set

Statement

Let $X$ be a $C^{1}$ planar vector field on an open neighborhood of a compact disk $D$ , and let $Ψ : S^{1} \times (0, 1) \to A \subseteq int D$ be a given $C^{2}$ leaf product as supplied by A C² first-integral period annulus has a C² leaf product, with $X = a (s, θ) \partial_{θ}$ and a positive $C^{1}$ coefficient $a$ . Assume the periodic curves $C_{s} = Ψ (S^{1} \times {s})$ bound nested Jordan domains $D_{s}$ and that $Γ = \partial ⋃_{0 < s < 1} int (D_{s})$ is compact. Then there is a $C^{1}$ vector field $Y$ on a neighborhood of $D$ that equals $X$ on $Γ$ and off an outer subannulus and has a positive orbit $y$ with $ω_{Y}^{+} (y) = Γ$ . The construction uses no choice principle.

Facts & Assumptions

Given: A $C^{1}$ planar field $X$ near a compact disk $D$ , a $C^{2}$ leaf product $Ψ : S^{1} \times (0, 1) \to A \subseteq int D$ with $X = a (s, θ) \partial_{θ}$ and $a > 0$ of class $C^{1}$ , nested Jordan domains $D_{s}$ bounded by $C_{s} = Ψ (S^{1} \times {s})$ , and the compact frontier $Γ = \partial ⋃_{0 < s < 1} int (D_{s})$ .

[F1]

The product $Ψ$ is a $C^{2}$ diffeomorphism onto $A$ with $C^{2}$ inverse, and $W : = Ψ_{*} \partial_{s}$ is a $C^{1}$ field on $A$ transverse to $X$ (A C² first-integral period annulus has a C² leaf product).

[F2]

[F3]

A $C^{1}$ Euclidean field has a unique maximal flow that is jointly $C^{1}$ , each regular point has a $C^{1}$ flow box, and a trajectory remaining in a compact subset of the domain has no finite maximal endpoint (C¹ Euclidean maximal flows, variational dependence and the finite C² upgrade).

Proof

technique · direct

1.1givenF1F2

Set $Ω_{s} = int D_{s}$ and $Ω = ⋃_{0 < s < 1} Ω_{s}$ , so that $Γ = \partial Ω$ ; strict nesting gives $cl Ω_{s} \subseteq Ω_{t}$ and $C_{s} \subseteq Ω_{t}$ for $s < t$ , and the sets $T_{r} = cl ⋃_{s \geq r} C_{s}$ are nonempty compact subsets of $D$ decreasing in $r$ , so their tail intersection $K_{\infty}$ lies in $cl Ω$ , meets no $Ω_{t}$ because a ball about a point of $Ω_{t}$ is avoided by all $C_{s}$ with $s > t$ , and therefore lies in $Γ$ ; if arbitrarily late $C_{s}$ had points at distance at least $ϵ$ from $Γ$ the nested compact sets $T_{r} \cap {dist (\cdot, Γ) \geq ϵ}$ would have a common point, so $\sup_{x \in C_{s}} dist (x, Γ) \to 0$ by [F2], while conversely for fixed $p \in Γ$ and $ϵ > 0$ a point $x \in Ω \cap B_{ϵ / 2} (p)$ and an index $t$ with $x \in Ω_{t}$ force every $C_{s}$ with $s > t$ to meet the segment from $x$ to $p$ , and a finite cover of the compact $Γ$ by such balls makes $Γ$ everywhere within $ϵ$ of $C_{s}$ ; hence $C_{s} \to Γ$ in Hausdorff distance and $Γ \subseteq cl A$ .

2.1step 1.1F1F2

Fix $s_{1} \in (0, 1)$ and set $m (s) = \min_{θ} a (s, θ) > 0$ , $L (s) = \max_{θ} ∥ \partial_{s} Ψ (s, θ) ∥$ and $q (s) = \min {(1 - s)^{2}, (1 - s)^{2} m (s) / (1 + L (s))}$ , which are continuous and positive on compact subintervals; with $τ (s) = \log ((1 - s_{1}) / (1 - s))$ and the explicit bump $η (t) = e^{- 1 / (1 - t^{2})}$ for $∣ t ∣ < 1$ and $η = 0$ otherwise, the functions $ρ_{n} (s) = η (τ (s) - n - \frac{1}{2}) / \sum_{j \geq 0} η (τ (s) - j - \frac{1}{2})$ form a smooth locally finite partition of $[s_{1}, 1)$ with positive sum, uniformly finite overlap, compact supports in $(0, 1)$ and active indices tending to infinity as $s \to 1$ ; for each support the compact set $B_{n} = Ψ (S^{1} \times supp ρ_{n})$ is disjoint from $Γ$ with positive distance $d_{n}$ , while $q_{n} = \min_{supp ρ_{n}} q > 0$ and $M_{n} = 1 + \sup_{B_{n}} (∣ ρ_{n} W ∣ + ∥ D (ρ_{n} W) ∥) < \infty$ are attained finite extrema of continuous functions on nonempty compacta by [F2], so these are uniquely specified real numbers and no sequence of witnesses is selected.

3.1step 2.1F2

With $c_{n} = 2^{- n - 1} \min (q_{n}, d_{n} / M_{n})$ and $b_{0} = \sum_{n \geq 0} c_{n} ρ_{n}$ one has $b_{0} > 0$ and $b_{0} \leq q$ , while the fields $V_{n} = c_{n} ρ_{n} W$ satisfy $∣ V_{n} (z) ∣ \leq 2^{- n - 1} dist (z, Γ)$ and $∥ D V_{n} (z) ∥ \leq 2^{- n - 1} d_{n}$ on $B_{n}$ and vanish elsewhere, because $c_{n} M_{n} \leq 2^{- n - 1} d_{n}$ and points of $B_{n}$ have distance to $Γ$ at least $d_{n}$ ; near $Γ$ only indices $n \geq N$ contribute for $N$ arbitrarily large, so $V = b_{0} W$ satisfies $∣ V (z) ∣ \leq 2^{- N} dist (z, Γ)$ and $∥ D V (z) ∥ \leq 2^{- N} \sup_{n \geq N} d_{n}$ , giving $V = o (dist (z, Γ))$ and $D V \to 0$ at $Γ$ ; therefore the extension of $V$ by zero across $Γ$ is $C^{1}$ with zero derivative there, and multiplying $b_{0}$ by one fixed smooth cutoff flat at $s_{1}$ , positive for $s > s_{1}$ and equal to one near $1$ , produces a $C^{1}$ function $b$ with $0 \leq b \leq q$ that extends the drift by zero across the inner edge.

4.1step 3.1F1

Define $Y = X + b (s) W$ on the outer subannulus ${s > s_{1}}$ and $Y = X$ elsewhere on a neighborhood of $D$ ; since $b$ is a $C^{1}$ function of the $C^{2}$ leaf coordinate $s$ and $W$ is $C^{1}$ , the field $Y$ is $C^{1}$ , agrees with $X$ off the outer subannulus and on $Γ$ , and in product coordinates reads $Y = a (s, θ) \partial_{θ} + b (s) \partial_{s}$ with $a > 0$ and $b \geq 0$ , so no new zero is created in the drift region.

5.1step 4.1F1F3

Let $y$ be the maximal $Y$ -trajectory starting at $Ψ (0, s_{0})$ with $s_{1} < s_{0} < 1$ : along it $\dot{s} = b (s)$ and $\dot{θ} = a (s, θ)$ , so $s$ increases strictly, $d s / d θ \leq (1 - s)^{2} / (1 + L (s))$ , and $\int_{s_{2}}^{s} d u / b (u) \geq \int_{s_{2}}^{s} d u / (1 - u)^{2} \to \infty$ as $s \to 1$ , so $s$ tends to $1$ only at infinite time with $θ (s) - θ (s_{2}) \geq \int_{s_{2}}^{s} (1 + L (u)) / (1 - u)^{2} d u \to \infty$ ; during each full phase turn starting at parameter $s_{k}$ the parameter increases by at most one fixed normalization constant times $(1 - s_{k})^{2}$ , and $\int_{s_{k}}^{s} L (u) d u = \int L (s (θ)) (d s / d θ) d θ \leq (1 - s_{k})^{2}$ up to the same constant, so the corresponding fixed-phase ambient displacement from the leaf $C_{s_{k}}$ is bounded by that integral and every complete turn stays uniformly within that distance of the whole reference circle, while every phase is visited during the turn; as $k \to \infty$ the leaves $C_{s_{k}}$ converge to $Γ$ in Hausdorff distance by step 1.1, so every point of $Γ$ is a limit of the orbit and the orbit tail approaches $Γ$ , giving $ω_{Y}^{+} (y) = Γ$ ; the orbit remains in the compact set $cl Ω$ , never meets $Γ$ because $C_{s} \cap Γ = \emptyset$ for all $s < 1$ , and is defined for all positive times by [F3].

6.1step 5.1∎

Consequently $Y$ is a $C^{1}$ field on a neighborhood of $D$ that equals $X$ on $Γ$ and off the outer subannulus and has the positive orbit $y$ with $ω_{Y}^{+} (y) = Γ$ ; every selection in the construction was an explicit band function or a uniquely determined extremum of a continuous function on a compact set, so no choice principle is used.

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

A separated characteristic disk has a minimal nonidentity simple cycle

Statement

Assume Countable Choice ${A C}_{ω}$ (The countable-choice principle used in the foliation pair). A relative generic characteristic disk with closed transverse boundary and with distinct singular images in distinct ambient leaves (Characteristic-disk singular images can be separated into distinct leaves relative to the boundary collar) has an inclusion-minimal simple regular or homoclinic characteristic cycle whose ambient holonomy germ is nonidentity (The holonomy representation and the holonomy group of a leaf). Its bounded source domain minimizes area among such cycles. The minimum-selection statement alone does not assert inward identity at a three-sector homoclinic cycle.

Facts & Assumptions

Given: A relative generic characteristic disk $h : D^{2} \to M$ with closed transverse boundary, characteristic field $X$ with finitely many nondegenerate centers and saddles, and distinct singular ambient leaves.

[F1]

Every regular point of the disk lies on a characteristic trajectory; the one-sided limit sets of an orbit with precompact closure are nonempty, compact, connected and invariant, and are either a singleton equilibrium, a single periodic orbit, or consist of finitely many equilibria and saddle-to-saddle connecting trajectories (Local generalized Poincare-Bendixson theorem for a precompact planar orbit).

[F2]

A nondegenerate $C^{2}$ saddle first integral has $C^{1}$ coordinates in which it is $u v$ (A C² saddle function has C¹ Morse coordinates). Its zero level has four regular $C^{1}$ half-branches, two incoming and two outgoing for the characteristic direction. A connecting orbit has a compact $C^{1}$ extension through each saddle endpoint.

[F3]

Every bounded regular-cycle disk of the characteristic field contains a center, and strictly inside a one-sector homoclinic disk there is one more center than saddle (The characteristic disk has one more center than saddle, A one-quadrant homoclinic disk contains a center).

[F4]

In a short regular flow box an everywhere-transverse section meets any simple periodic characteristic circle at most once. Indeed orient that circle by the field. Its local intersection signs with the section are all equal, whereas successive crossings of a Jordan circle along the section must alternate between its inside and outside. The same argument applies to a simple $C^{1}$ homoclinic cycle away from its corner, using [F8]. This argument does not presume a period annulus between nonidentity cycles.

[F5]

The maximal flow of the $C^{1}$ field $X$ is jointly $C^{1}$ and depends uniformly on initial data on compact time intervals (C¹ Euclidean maximal flows, variational dependence and the finite C² upgrade).

[F6]

Closed bounded plane sets are compact (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line). A decreasing sequence of nonempty compact sets has nonempty intersection: otherwise their open complements cover the first compact set and a finite subcover makes a later set empty.

[F7]

Planar Lebesgue measure is sigma-finite, monotone and finite on bounded Borel sets; a Jordan domain is Borel, and a strict inclusion of Jordan domains leaves a nonempty open region of positive measure Finite first measure also gives continuity from above (Continuity from above when one set has finite measure). (Lebesgue measure is sigma-finite, and every metrically bounded subset of $R^{n}$ has finite outer measure, Measures are monotone, A box in $R^{n}$ with parameters $a_{i} \leq b_{i}$ is Lebesgue measurable of measure $\prod_{i < n} (b_{i} - a_{i})$ , whichever of its faces are included, Assuming countable choice, every Borel subset of $R^{n}$ is Lebesgue measurable).

[F8]

A simple closed piecewise- $C^{1}$ curve with finitely many corners and distinct one-sided tangents at each corner bounds exactly one bounded component (A finitely cornered regular plane curve separates without choice).

[F9]

Finite plaque transports have a common transverse interval after finitely many shrinkings, and their germs are invariant under leafwise homotopies (Holonomy of a C¹ foliation is a representation into C¹ transverse germs, The holonomy representation and the holonomy group of a leaf).

Proof

technique · direct

1.1givenF1

Call a characteristic cycle simple when it is either a regular periodic orbit or a homoclinic orbit through one saddle whose source closure is a simple closed curve, and let its domain be the bounded component of the complement of that curve. Its ambient holonomy is the holonomy germ of the based leafwise loop obtained by following the cycle once around, in the sense of The holonomy representation and the holonomy group of a leaf. Extend X C¹ to a neighborhood of the disk by C¹ planar fields on a closed disk extend to a neighbourhood, and choose the sign of the characteristic field so it points inward on the transverse boundary; its normal component is nonzero and has one sign there. Begin with an orbit entering the boundary outside the finitely many stable separatrices of the saddles: its positive orbit stays in the compact disk, so [F1] applies to its limit set.

2.1F1F2F4F5F9step 1.1

A nonidentity simple cycle exists. The entering orbit cannot converge to a saddle, since it was chosen off the finitely many stable separatrices; nor can a center belong to its limit set, since small invariant center circles cannot be crossed. By [F1], the limit is a periodic orbit or a connected finite saddle graph. Every connecting edge maps into one ambient leaf, and the singular leaves are distinct, so a connected saddle graph has just one vertex, with one or two homoclinic edges by [F2]. Trim each edge by short transverse ports and use one Morse box at the saddle. In that box $u v$ is constant along an orbit; on either fixed nonzero sign, its quadrant arcs pair the incoming ports with the outgoing ports deterministically. Together with the at most two edge strips, this gives a return itinerary of one or two edges. A sufficiently late entering orbit follows one such itinerary repeatedly; it avoids the axes and cannot change its side without crossing a separatrix. The compact edge strips and single target box define one ambient transverse return map H on an interval about the limiting port. At the source port its pulled-back transverse function is a local diffeomorphism, so the source return map is conjugate to H on that approached side. If H were the identity on a neighborhood of the limiting parameter, sufficiently late returns would be periodic, contradicting the nonclosed entering orbit. Thus the limiting word has nonidentity germ. In the two-edge case its word is the product of the two homoclinic lobe words with fixed transport conjugations; if both were identity, so would be their product. Therefore a simple periodic or homoclinic cycle has nonidentity ambient holonomy.

3.1step 2.1F3F7

The area infimum and the nested family. Let $A$ be the infimum of the areas of the bounded domains of all simple cycles with nonidentity ambient holonomy in the unchanged source disk. There are only finitely many singular simple cycles, because each is a homoclinic orbit based at one of the finitely many saddles and each saddle has four local half-branches; if any simple cycle attains $A$ , it is already a minimizer and a strictly contained nonidentity cycle would have smaller area by [F7], so it is inclusion-minimal. Otherwise choose regular cycles with domains of area tending to $A$ ; by ${A C}_{ω}$ choose one for each $n$ with area below $A + 1 / n$ . By [F3] each bounded regular-cycle disk contains a center, and there are only finitely many centers, so an infinite subsequence of these cycles contains one fixed center $p$ . Two regular characteristic circles surrounding $p$ are disjoint and nested, because trajectories do not cross and their bounded Jordan domains both contain $p$ ; each selected area is strictly above A, and the areas of this subsequence tend to A; recursively take the least later index of strictly smaller area. Nesting then gives closed disks $K_{1} \supseteq K_{2} \supseteq \dots$ with boundaries $C_{n} = \partial K_{n}$ , areas decreasing to $A$ , and each $C_{n}$ of nonidentity holonomy.

4.1F3F6F7F9step 3.1

A positive-area limit. Choose a small invariant center disk $U$ about $p$ whose image lies in one ambient foliation box. Every characteristic circle in $U$ maps into one plaque and has identity ambient holonomy, so none of the $C_{n}$ lies there. Nor can $C_{n}$ cross its invariant boundary, by uniqueness of trajectories. Since $U$ is connected, contains $p$ and is disjoint from $C_{n}$ , it lies in the bounded Jordan domain of $C_{n}$ . Thus $K = ⋂_{n} K_{n}$ contains $U$ and has positive area. Each cycle boundary is a finite union of compact C¹ arcs, including its saddle endpoints by F2, and has area zero: on each Lipschitz parametrized arc a partition into n pieces covers it by n squares of side O(1/n), with total area O(1/n). Hence the closed and open cycle domains have the same area. Each $K_{n}$ is a compact Borel set of finite area; continuity from above in F7 gives $area (K) = \lim_{n} area (K_{n}) = A$ . No nonidentity-germ property was inferred from bare continuity.

5.1F1F2F4F5F6step 4.1

The nested boundary limit is a finite graph or circle. The whole sequence $C_{n}$ converges in Hausdorff distance to $G = \partial K$ . A limit of points of late $C_{n}$ lies in every $K_{j}$ and cannot lie in $int K$ , because a ball contained in $K$ is disjoint from every boundary $C_{n}$ . Conversely, for $z \in \partial K$ and any small ball about $z$ , choose $w$ in that ball outside $K$ . It is outside some $K_{j}$ and hence all later $K_{n}$ , whereas $z \in K_{n}$ ; the segment from $z$ to $w$ meets every later $C_{n}$ in the ball. Finite covers and compactness give both uniform Hausdorff bounds. A Hausdorff limit of these connected compact circles is connected: two separated compact pieces of $G$ would give disjoint small neighborhoods which every late connected $C_{n}$ must meet while staying in their union. Flow dependence F5 makes $G$ invariant. At a regular point, each nearby $C_{n}$ crosses a fixed short section at most once by F4; its limit therefore has exactly one transverse coordinate there and is one local flow arc. A periodic component is consequently open as well as closed in $G$ , so connectedness makes it all of $G$ . Otherwise a regular orbit in $G$ cannot accumulate at a regular point: repeated visits to its flow box would give distinct intersections with that short section, contradicting the same local one-arc property. Its compact connected alpha- and omega-limits are therefore single equilibria by F1. Centers are excluded by step 4.1 and their invariant local disks. Hence every regular edge ends at saddles, and the finitely many half-branches of F2 give only finitely many edges. Thus $G$ is a regular circle or a finite connected saddle graph. This uses no unsupported hyperspace selection or period-annulus hypothesis.

6.1F2F4F9step 2.1step 5.1

One saddle and one fixed return germ. Trim the regular edges of $G$ and put one short transverse section on each. Hausdorff convergence and flow-box projection make every late $C_{n}$ cross each section; F4 makes it cross exactly once. It stays in a small neighborhood of the finite graph and crosses its finitely many saddle ports, so only finitely many directed itineraries occur. Fix one itinerary on a subsequence. Its closed walk traverses every edge, hence the graph is strongly connected. Each edge maps, including its $C^{1}$ saddle endpoints, to one ambient leaf: subdivide that compact tangent path into ambient foliation boxes and use the constant transverse coordinate. Distinct singular ambient leaves therefore force a single saddle vertex. Its two outgoing half-branches allow one or two homoclinic edges. Now fix one target foliation box at that saddle, and finite target boxes along the trimmed edges. A passage near the saddle remains in its one box at one transverse level and can be replaced, relative to its endpoints, by a reference plaque path in that level. Finite transport and homotopy invariance F9 give a single return map $H$ on a common interval about zero at a fixed regular target section. The based loop $h (C_{n})$ has the germ of this same $H$ at its section parameter $t_{n} \to 0$ ; it is not a sequence of unrelated return maps. If $H$ were the identity on an interval about zero, it would have identity germ at every sufficiently late $t_{n}$ , contradicting the chosen nonidentity holonomy of $C_{n}$ . Hence the limiting fixed word is nonidentity at zero. For two lobes its word factors as their holonomy words, with the fixed transport conjugations; if both lobe germs were identity their composite would be identity, so at least one lobe is nonidentity.

7.1step 6.1F7F8

The limiting cycle and area minimality. If $G$ is a regular circle or a simple homoclinic loop, then $G$ is a simple cycle with nonidentity word $H$ and domain of area $area (K) = \lim_{n} area (K_{n}) = A$ , so it attains the infimum. If $G$ has two edges, at least one of the two simple lobes has nonidentity word by the factorization in step 2.1; let $L$ be that lobe. Its boundary lies in $G = \partial K$ , and its entire bounded Jordan domain is contained in every $K_{n}$ , because $L \subseteq K \subseteq int K_{n}$ by strict nesting, while the connected unbounded exterior of $K_{n}$ lies in the unbounded component of the complement of $L$ . The boundary $C_{n}$ lies in that same component by its exterior collars, so the bounded lobe cannot leave $K_{n}$ ; passing to the limit its area is at most $\lim_{n} area (K_{n}) = A$ , while by definition of the infimum it is at least $A$ , so it equals $A$ and $L$ attains the infimum.

8.1step 7.1F7∎

Inclusion-minimality. Every simple cycle with nonidentity word strictly interior to the minimizer would bound a strictly smaller Jordan domain, since two distinct simple closed curves leaving a nonempty open source region between them give a strict measure inequality [F7]; such a cycle would have area strictly below $A$ , contradicting the definition of $A$ . Hence the minimizer is inclusion-minimal among the nonidentity simple cycles of the unchanged source disk. The construction used only the countable area-minimizing selection of step 3.1, which uses exactly the stated ${A C}_{ω}$ , and no modification of the disk; all remaining selections are finite.

CorollaryStatement: Literature-sourcedProof: Literature-sourcedprecheck passOpen item page →

Closed defining forms have vanishing Godbillon-Vey class

Statement

Assume Countable Choice ${A C}_{ω}$ . Let $F$ be a transversely oriented codimension-one foliation of a smooth manifold $M$ defined by a closed nowhere-vanishing $1$ -form $ω$ with $T F = \ker ω$ (so that $\ker ω$ is integrable by Closed constant-rank one-forms define integrable hyperplane fields). Then $G V (F) = 0$ in $H_{d R}^{3} (M; R)$ .

Facts & Assumptions

Given: A transversely oriented codimension-one foliation $F$ of a smooth manifold $M$ defined by a closed nowhere-vanishing one-form $ω$ with $T F = \ker ω$ , and the standing countable choice assumption.

[F1]

For a defining form $ω$ and a one-form $η$ with $d ω = η \land ω$ , the Godbillon-Vey class is the de Rham class $G V (F) = [η \land d η]$ . (The Godbillon-Vey class of a codimension-one foliation).

[F2]

For a transversely oriented codimension-one foliation with nowhere-vanishing defining form $ω$ there is a smooth one-form $η$ with $d ω = η \land ω$ . (Frobenius divisibility: d omega equals eta wedge omega).

Proof

technique · direct

1.1F2given

For the closed defining form $ω$ the choice $η = 0$ satisfies $d ω = 0 = 0 \land ω$ , so it is one of the forms whose existence the divisibility lemma [F2] guarantees.

2.1F1step 1.1∎

The Godbillon-Vey form of this choice is $η \land d η = 0 \land 0 = 0$ , so the class defined in [F1] is the class of the zero form, namely $G V (F) = 0$ in $H_{d R}^{3} (M; R)$ ; this applies in particular to fibre foliations of bundles over $S^{1}$ defined by pullbacks of volume forms on the circle, and no choice principle is used.

TheoremStatement: Literature-sourcedProof: Literature-sourcedprecheck passOpen item page →

Godbillon-Vey invariance under smooth foliated concordance

Statement

Assume Countable Choice ${A C}_{ω}$ . Let $M$ be a closed smooth manifold and let $F_{0}, F_{1}$ be smoothly foliated-concordant transversely oriented codimension-one foliations of $M$ (Smooth foliated concordance of codimension-one foliations). Then $G V (F_{0}) = G V (F_{1})$ in $H_{d R}^{3} (M; R)$ .

Facts & Assumptions

Given: A closed smooth manifold $M$ , smoothly foliated-concordant transversely oriented codimension-one foliations $F_{0}, F_{1}$ of $M$ , a concordance $(G, ω)$ on $W = M \times [0, 1]$ , and the standing countable choice assumption.

[F1]

The Godbillon-Vey class of a transversely oriented codimension-one foliation with defining form $ω$ and $d ω = η \land ω$ is the de Rham class $[η \land d η]$ . (The Godbillon-Vey class of a codimension-one foliation).

[F2]

A codimension-one foliation of a manifold with boundary transverse to the boundary restricts to a codimension-one foliation of the boundary, and if $d ω = η \land ω$ then $d (ω ∣_{\partial W}) = (η ∣_{\partial W}) \land (ω ∣_{\partial W})$ . (Restriction of a foliation transverse to the boundary).

[F3]

Smoothly homotopic maps induce the same map on de Rham cohomology. (Smoothly homotopic maps induce the same de rham map).

[F4]

The de Rham complex, wedge identities and natural pullback extend to smooth manifolds with boundary (The de Rham complex and pullback extend to manifolds with boundary).

Proof

technique · direct

1.1F1given

Let $(G, ω)$ be the concordance on $W = M \times [0, 1]$ and choose $η$ with $d ω = η \land ω$ by the divisibility lemma, so that $[η \land d η] = G V (G) \in H_{d R}^{3} (W; R)$ according to [F1].

2.1F2step 1.1

By [F2] the inclusions $i_{j} : M \to W$ , $i_{j} (x) = (x, j)$ , pull the defining data back to defining data of $F_{j}$ : $i_{j}^{*} ω$ is a defining form for $F_{j}$ and $d (i_{j}^{*} ω) = (i_{j}^{*} η) \land (i_{j}^{*} ω)$ , so $i_{j}^{*} (η \land d η) = (i_{j}^{*} η) \land d (i_{j}^{*} η)$ represents $G V (F_{j})$ , that is $i_{j}^{*} G V (G) = G V (F_{j})$ .

3.1F3F4step 2.1∎

For completeness the endpoint equality holds on the boundary cylinder as follows. Write the closed form $α = η \land d η$ on $M \times [0, 1]$ as $α_{s} + d s \land γ_{s}$ . By [F4], $d α = 0$ gives $\partial_{s} α_{s} = d_{M} γ_{s}$ . Integrating the smooth coefficients in s yields $i_{1}^{*} α - i_{0}^{*} α = d_{M} \int_{0}^{1} γ_{s} d s$ . Thus the endpoint forms represent the same de Rham class, and step 2.1 identifies them with the two Godbillon–Vey classes. This is the homotopy identity of [F3], here derived explicitly on the cylinder.

RemarkRemark: Literature-sourcedProof: Not applicableOpen item page →

The supplied smooth Godbillon-Vey construction does not cover merely C1 foliations

Statement

Assume Countable Choice ${A C}_{ω}$ . The construction in The Godbillon-Vey class of a codimension-one foliation is stated for smooth foliations and smooth defining data. It supplies no cohomological Godbillon–Vey class for merely $C^{1}$ foliations.

Remarks

This is a limitation of the supplied construction, not a necessary regularity threshold for every Godbillon–Vey theory. Hurder–Katok, §7, Proposition 7.1, constructs a natural Godbillon–Vey invariant for transversally $C^{1, α}$ codimension-one foliations of closed oriented $3$ -manifolds when $α > 1 / 2$ , extending the $C^{2}$ invariant. Their argument uses a distributional pairing; it is not the smooth differential-form construction supplied here.

The usual finite-regularity theory defines the Godbillon–Vey class for $C^{2}$ foliations and the Godbillon measure for $C^{1}$ foliations. Hurder–Langevin, §3, printed p.10, states this distinction, then explicitly specializes §3.1 to smooth foliations and refers elsewhere for the required finite-regularity modifications. Those modifications are not proved on this page. In particular one cannot obtain the $C^{2}$ -atlas theory merely by replacing “smooth” by “ $C^{2}$ ” in the smooth-form proof: the associated defining form may have lower regularity, and comparison with smooth de Rham cohomology requires additional work.

DefinitionDefinition: Literature-sourcedProof: Not applicableOpen item page →

Limit cycles of a leaf

Definition

Assume Countable Choice ${A C}_{ω}$ . Let $F$ be a transversely oriented codimension-one foliation, $L$ a leaf, and $x \in L$ . For a chosen side $j$ of $L$ , let $N_{j} (L, x) ◃ π_{1} (L, x)$ be the subgroup of classes whose one-sided normal- fence holonomy germ is the identity. The quotient $P_{j} (L, x) = π_{1} (L, x) / N_{j} (L, x)$ is Novikov’s one-sided limit-cycle group. A class $[α]$ is a limit cycle on side $j$ precisely when its image in $P_{j} (L, x)$ is nonidentity, equivalently its one-sided holonomy germ is nonidentity. The two groups $P_{+} (L, x)$ and $P_{-} (L, x)$ record ordinary right and left limit cycles.

Remarks

The separate limitwise-nullhomotopy subgroup $Π_{1}^{j} (L, x)$ is defined later on this page as a set of classes in $N_{j} (L, x)$ whose sufficiently small displaced loops are nullhomotopic in their leaves. Its containment in $N_{j} (L, x)$ is part of that definition; it is distinct from the ordinary limit-cycle quotient $P_{j} (L, x)$ defined here.

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passOpen item page →

A center period annulus has an orbit or polycycle frontier

Statement

Assume Countable Choice ${A C}_{ω}$ (The countable-choice principle used in the foliation pair). Let $F$ be a $C^{2}$ cooriented codimension-one foliation of a $3$ -manifold, and let $h : D^{2} \to M$ be a disk map in the relative generic position of Relative generic position for characteristic disk maps. Assume the boundary is either leafwise or a closed transversal, as in cases (b) and (a) of that supplier. The connected family of regular closed characteristic trajectories surrounding any center has a maximal period annulus. Its outer frontier is a regular closed orbit; or a finite connected strongly connected directed saddle-separatrix graph whose edges are nonconstant saddle-to-saddle trajectories and whose edges are covered by finitely many directed saddle polycycles; or, when the disk boundary is a leafwise characteristic orbit, that boundary orbit. Loops, repeated saddle vertices, shared edges, and parallel edges are allowed in the saddle graph and polycycles. When the disk boundary is transverse to $F$ , the period annulus cannot meet it. The annulus parameter gives a $C^{2}$ transverse trace of the prescribed closed characteristic loops. The outer return holonomy is not assumed nontrivial.

Facts & Assumptions

Given: A cooriented codimension-one $C^{2}$ foliation $F$ of a $3$ -manifold with nowhere-vanishing $C^{2}$ defining form, and a $C^{2}$ disk map $h : D^{2} \to M$ whose characteristic covector is in relative generic position: its singularities are finitely many nondegenerate interior centers and saddles, its characteristic covector is nowhere vanishing on a boundary collar, and along $\partial D^{2}$ either the boundary is a closed transversal or it is mapped into a single leaf.

[F1]

In relative generic position the characteristic singularities of the disk map are finitely many nondegenerate points in the interior, each a center or a saddle; at a center the characteristic line field has a family of small closed orbits around it, and at a saddle it has the four-sector hyperbolic picture (Relative generic position for characteristic disk maps).

[F2]

A connected open set carrying a $C^{2}$ first-integral atlas whose leaves are simple compact circles with strictly nested bounded Jordan domains and consistent orientation is an open annulus with a $C^{2}$ product $Ψ : S^{1} \times (0, 1) \to A$ onto its leaves, increasing in the nested order, and a nowhere-zero $C^{1}$ tangent generator is written $a (s, θ) \partial_{θ}$ with positive $C^{1}$ coefficient after orienting $θ$ (A C² first-integral period annulus has a C² leaf product).

[F3]

Under the hypotheses of [F2] with a compact frontier $Γ = \partial ⋃_{s} int D_{s}$ , there is a $C^{1}$ field $Y$ on a neighborhood of the disk, equal to the generator on $Γ$ and off an outer subannulus, with a positive orbit whose $ω$ -limit set is $Γ$ , and no choice principle is used (A flat transverse drift realizes the period-annulus frontier as an omega-limit set).

[F4]

If $Γ = ω_{Y}^{+} (y)$ contains at least one equilibrium, all its equilibria are nondegenerate saddles, and it separates two points, then $Γ$ is a finite embedded strongly connected directed saddle multigraph covered by finitely many closed directed edge walks (A finite saddle omega-graph is strongly connected and is a finite union of polycycles).

[F5]

If a positive orbit of a $C^{1}$ planar field has compact closure in the domain and its $ω$ -limit set contains only finitely many equilibria, then it is either a singleton equilibrium, or one regular periodic orbit, or a finite equilibrium set together with nonconstant trajectories whose alpha- and omega-limits are equilibria (Local generalized Poincare-Bendixson theorem for a precompact planar orbit).

[F6]

[F7]

A piecewise- $C^{2}$ topological embedding $S^{1} \to R^{2}$ with finitely many corners, two distinct one-sided tangent rays at each corner and regular edges has a complement with exactly two connected components, one bounded and one unbounded (A finitely cornered regular plane curve separates without choice).

[F8]

A planar field $C^{1}$ up to the boundary of the closed disk has a $C^{1}$ extension to a neighborhood of the disk, with value and derivative agreeing on the disk (C¹ planar fields on a closed disk extend to a neighbourhood).

[F9]

A cooriented codimension-one foliation is given by a foliated atlas whose transverse coordinate changes are diffeomorphisms, and the transverse orientation selects the positive side of each leaf (Transversely oriented codimension-one foliations).

[F10]

The interior, closure and boundary of a set in a topological space, with $\partial A = \overline{A} ∖ int A$ (Interior, closure, boundary, exterior, derived set and isolated point in a topological space).

[F11]

The connected components of a space partition it and are closed; a component is the union of all connected subsets through any of its points (The components of a space are its maximal connected subsets, they partition it, and each of them is closed).

[F12]

The standing assumption of the pair is Countable Choice ${A C}_{ω}$ (The countable-choice principle used in the foliation pair).

Proof

technique · direct

1.1givenF1F8F9

Writing $ω$ for the defining form, the characteristic covector $β = h^{*} ω$ is $C^{1}$ with kernel exactly the characteristic line field and with the characteristic singularities as zeros, and equivalently it is obtained by summing finitely many pulled-back local transverse covectors, which differ on overlaps by positive nowhere-vanishing factors; choosing an oriented area form $μ$ and defining $X$ by $ι_{X} μ = β$ makes $X$ a $C^{1}$ planar field whose regular line foliation has a $C^{2}$ first-integral atlas with local first integrals $u = z \circ h$ , and by [F1] the singularities of $X$ in the disk are finitely many nondegenerate interior centers and saddles with nonvanishing $β$ on a boundary collar; in the transversal boundary case $β$ does not vanish on the boundary tangent and in the leafwise case $β$ does, so then $X$ is tangent to and nonvanishing along $\partial D^{2}$ and the boundary circle is one regular closed $X$ -orbit; finally [F8] extends $X$ to a $C^{1}$ field on an open neighborhood $U$ of the disk.

2.1step 1.1F1F11

Fix a center $c$ of $X$ and let $P_{c}$ be the set of points of regular periodic $X$ -orbits contained in $int D^{2}$ whose bounded Jordan interior contains $c$ ; a nonconstant periodic orbit is an embedded circle whose period set is a closed additive subgroup of $R$ and therefore has a least positive period, so the orbit is a $C^{2}$ embedded circle because an individual trajectory of a $C^{1}$ field is $C^{2}$ in time ( $x'' = D X (x) X (x)$ ), the center picture of [F1] makes $int P_{c}$ nonempty, and the local flow carries $P_{c}$ and $int P_{c}$ to themselves; the component $A$ of $int P_{c}$ containing the small center collar is open and saturated, because components are unions of connected subsets and each orbit is connected.

3.1step 2.1F7F10

The circles of $A$ are strictly nested and consistently oriented: two distinct orbits are disjoint by uniqueness of trajectories and their bounded interiors both contain $c$ ; a connected circle disjoint from a Jordan curve lies in one complementary component by [F7], so if $C_{2}$ lay in the exterior of $C_{1}$ then the bounded domain of $C_{1}$ , being connected, disjoint from $C_{2}$ and containing $c \in int D_{C_{2}}$ , would lie in the bounded domain of $C_{2}$ , giving strict nesting; and the sign in which the $X$ -orientation of an orbit agrees with the boundary orientation of its bounded domain is locally constant on the connected $A$ , hence one global sign.

4.1step 3.1F2F12

By [F2] applied to the $C^{2}$ first-integral atlas of step 1.1 on the connected set $A$ with its strictly nested consistently oriented compact circle leaves, $A$ is an open annulus with a $C^{2}$ product $Ψ : S^{1} \times (0, 1) \to A$ taking circles onto leaves and increasing in the nested order; $A$ is maximal among connected open regular circle families continuing the chosen center collar, because any such family lies in $P_{c}$ , hence in $int P_{c}$ , hence in this component.

5.1step 4.1F6F10

Write $C_{s} = Ψ (S^{1} \times {s})$ , let $Ω_{s}$ be the bounded Jordan domain of $C_{s}$ and $Ω = ⋃_{0 < s < 1} Ω_{s}$ ; the closed bounded domains lie in $D^{2}$ because their exteriors contain the connected complement of the disk, for $s < t$ one has $cl Ω_{s} \subseteq Ω_{t}$ and $C_{s} \subseteq Ω_{t}$ by strict nesting, $Γ = \partial Ω$ by [F10] is a nonempty compact subset of $D^{2}$ by [F6], and $Γ$ separates $c$ from any fixed point $q$ outside the closed disk, since $c$ lies in some $Ω_{s}$ , $q$ lies outside $cl Ω$ , and every path between them has a first exit from $Ω$ , on $Γ$ .

6.1step 5.1F6

The exhaustion $T_{r} = cl ⋃_{s \geq r} C_{s}$ shows $C_{s} \to Γ$ in Hausdorff distance and $Γ \subseteq cl A$ : the tail intersection lies in $cl Ω$ and misses $Ω$ because a ball about a point of $Ω_{t}$ is avoided by all $C_{s}$ with $s > t$ , hence lies in $\partial Ω = Γ$ ; if arbitrarily late $C_{s}$ had points at distance at least $ϵ$ from $Γ$ the nested compacta $T_{r} \cap {dist (\cdot, Γ) \geq ϵ}$ would meet, contradiction; and conversely each $p \in Γ$ and $ϵ > 0$ admit $x \in Ω \cap B_{ϵ / 2} (p)$ and $t$ with $x \in Ω_{t}$ , so every $C_{s}$ with $s > t$ meets the segment from $x$ to $p$ within $ϵ$ of $p$ , and a finite cover of the compact $Γ$ gives $Γ$ everywhere within $ϵ$ of $C_{s}$ .

7.1step 6.1F1F6

No center lies on $Γ$ : the fixed center $c$ lies in the open $Ω_{t}$ , and for any other center $d$ the center picture of [F1] supplies a small saturated disk $V$ disjoint from $c$ ; every regular orbit meeting $V$ is a complete small level circle inside $V$ by uniqueness, so it does not enclose $c$ and is not in $A$ , whence $V$ is disjoint from $cl A$ and from $Γ$ ; consequently every equilibrium on $Γ$ is among the finitely many nondegenerate saddles of the disk.

8.1step 7.1F1F8

In the transversal boundary case the period annulus does not meet the boundary: in an inward collar coordinate $r \geq 0$ the radial component of $X$ is nonzero on $\partial D^{2}$ because $β$ is nonzero on the boundary tangent, its sign is constant along the connected boundary circle, and continuity gives $δ > 0$ and $k > 0$ with $∣ X r ∣ \geq k$ and one fixed sign throughout $0 \leq r \leq δ$ ; if a periodic orbit met ${r < δ}$ then its radial coordinate has a minimum below $δ$ , attained on the compact periodic curve, where its derivative along $X$ must be zero, contradicting $∣ X r ∣ \geq k$ , so no orbit of $A$ meets that collar and, in particular, the period annulus cannot meet the boundary.

9.1step 8.1F3

By [F3] applied to the product $Ψ$ and the compact frontier $Γ$ take the field $Y = X + V$ constructed by the flat-drift proof, and its positive orbit $y$ with compact closure in $D^{2}$ and $ω_{Y}^{+} (y) = Γ$ . The derivative equality needed below follows from that construction, not merely from $Y = X$ on $Γ$ : its locally finite band terms $V_{n}$ are supported on compact sets $B_{n} \subset A$ of distance $d_{n} > 0$ from $Γ$ , with $∣ V_{n} (z) ∣ \leq 2^{- n - 1} dist (z, Γ)$ and $∥ D V_{n} (z) ∥ \leq 2^{- n - 1} d_{n}$ . The fixed inner cutoff is identically one near $Γ$ . Any finite collection of band supports stays away from $Γ$ , so only $n \geq N$ contribute sufficiently near it; the $d_{n}$ are bounded by the diameter of the disk. Thus $V (z) = o (dist (z, Γ))$ and $D V (z) \to 0$ . Extend $V = 0$ on $Γ$ : for $p \in Γ$ , $dist (z, Γ) \leq ∣ z - p ∣$ proves $D V (p) = 0$ , giving $Y = X$ and $D Y = D X$ on $Γ$ . Hence $Γ$ is compact, invariant under the flow, and connected because it is the intersection of the decreasing family of connected closures of the orbit tails, while its regular $Y$ -trajectories are $X$ -trajectories by equality of the fields on the invariant set and uniqueness.

10.1step 9.1F5

Apply [F5] to the positive orbit of $y$ , whose $ω$ -limit set $Γ$ contains only the finitely many equilibria of step 7.1: alternative (i) fails because a singleton does not separate $c$ from $q$ , since the complement of one point of the plane is path connected by explicit polygonal detours, so either $Γ$ is one regular periodic orbit, or $Γ$ contains equilibria and every regular point of it lies on a nonconstant trajectory whose alpha- and omega-limits are among those saddles.

11.1step 10.1F1F4F5

In the second alternative of step 10.1, [F4] applies with the field $Y$ , its positive orbit and the separating compact $Γ$ whose equilibria are nondegenerate saddles, so $Γ$ is a finite embedded strongly connected directed saddle multigraph whose edges are the closures of the distinct nonconstant saddle-to-saddle trajectories, and finitely many closed directed edge walks cover it; in the first alternative $Γ$ is a single regular closed orbit, and if in the leafwise boundary case $Γ$ meets $\partial D^{2}$ , then the boundary circle is itself a regular closed orbit inside $Γ$ , invariance forces the whole boundary orbit into $Γ$ , and alternatives (i) and the equilibrium alternative cannot hold because the boundary carries no equilibrium, so $Γ$ equals that boundary orbit; thus the outer frontier is a regular closed orbit, the boundary orbit in the leafwise case, or a finite strongly connected saddle graph covered by finitely many polycycles, with loops, repeated vertices, shared and parallel edges allowed.

12.1step 11.1F9

Finally the annulus parameter gives the prescribed trace: for each fixed phase $θ_{0}$ the map $s \mapsto h (Ψ (θ_{0}, s))$ is $C^{2}$ because $h$ and $Ψ$ are, and it is transverse to $F$ because the pulled-back characteristic covector applied to $\partial_{s} Ψ$ is nonzero, as $\partial_{θ} Ψ$ spans the characteristic direction and $(\partial_{θ} Ψ, \partial_{s} Ψ)$ is a basis; these closed traces are exactly the prescribed loops $C_{s}$ , and no nontriviality of their return holonomy is assumed or used.

13.1step 12.1F12∎

Therefore every center of a disk map in relative generic position is surrounded by a maximal period annulus whose outer frontier is one of the listed alternatives, the transversal boundary case cannot be met by the annulus, and the annulus parameter supplies the $C^{2}$ transverse trace of the prescribed closed characteristic loops; the only countable selections in the proof are those in the product supplier of step 4.1, made under the standing ${A C}_{ω}$ of [F12], while all other steps use finitely many explicit objects.

DefinitionDefinition: Literature-sourcedProof: Literature-sourcedOpen item page →

Limitwise-nullhomotopy predicate on based loops

Definition

Assume Countable Choice ${A C}_{ω}$ . Let $F$ be a transversely oriented codimension-one foliation, $L$ a leaf, $x \in L$ , and $j$ one of its two sides. For a based loop $f : S^{1} \to L$ whose class belongs to $N_{j} (L, x)$ , and for a chosen sufficiently short normal fence on side $j$ , define the representative-level predicate $Q_{j} (f)$ to hold when every sufficiently small positive normal displacement $f_{ϵ}$ is null- homotopic in its leaf. At this stage $Q_{j}$ is a predicate on a specified loop and fence; no representative-independence is part of this definition.

For clarity, a displacement is obtained by starting at a chosen positive point of the base transversal and continuing the loop plaque by plaque through a finite chart subdivision. Within each chart its transverse label is held fixed; the fence specifies the nearby endpoint in that plaque. Since $[f] \in N_{j}$ , the return map is the identity on a sufficiently short interval on side j, so these displacements are closed loops. The quantifier means: there exists ε₀>0 such that every displacement with 0<ε<ε₀ is nullhomotopic in its own leaf. No uniform bound on the filling disks is part of the definition.

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

A finite characteristic circuit has C² regular port traces

Statement

Assume Countable Choice ${A C}_{ω}$ (The countable-choice principle used in the foliation pair). Let $F$ be a $C^{2}$ cooriented codimension-one foliation of a $3$ -manifold $M$ , let $h : D^{2} \to M$ be a $C^{2}$ disk map in the relative generic position of Relative generic position for characteristic disk maps, and let $Γ$ be the outer frontier of a maximal period annulus of the characteristic field of $h$ , so that $Γ$ is a regular closed orbit, a finite saddle-separatrix circuit, or the boundary orbit, by A center period annulus has an orbit or polycycle frontier. Choose an adjacent period annulus following the finite itinerary of $Γ$ and a regular $C^{2}$ port section through each branch of every passage of that itinerary, at positive distance from the saddle points.

Then:

(a) each such regular port section can be parameterized by the transported transverse first integral of the ambient foliation, with nonzero derivative, and the ports depend jointly $C^{2}$ on the level, including at level $0$ ;

(b) the trimmed regular edge strips and their endpoint collars at the ports are jointly $C^{2}$ down to the frontier;

(c) the nearby closed characteristic loops force the one-sided composite transverse return map of the itinerary to equal the identity on an interval.

No source strip through a saddle is asserted. The statement concerns the $C^{2}$ regularity of the port data; it does not assert a $C^{2}$ family of loops for the unmodified hyperbolic parametrizations near the saddle corners.

Facts & Assumptions

Given: A $C^{2}$ cooriented foliation $F$ of $M$ , a disk map $h$ in relative generic position, the frontier circuit $Γ$ of a chosen period annulus with its finite itinerary and chosen regular port sections, and the characteristic first integrals $u = z \circ h$ in flat charts.

[F1]

The characteristic field of $h$ is a $C^{1}$ planar field with a $C^{2}$ first-integral atlas given by the local transverse functions $u = z \circ h$ of flat charts of $F$ ; its singularities in the disk are finitely many nondegenerate interior centers and saddles (Relative generic position for characteristic disk maps, Flat charts for a distribution).

[F2]

The outer frontier of a maximal period annulus of the characteristic field is a regular closed orbit, a finite connected strongly connected saddle separatrix graph covered by finitely many directed saddle polycycles, or the boundary orbit; the annulus carries a $C^{2}$ transverse trace of its prescribed closed characteristic loops (A center period annulus has an orbit or polycycle frontier).

[F3]

If $g (s, t)$ is $C^{2}$ near $(s_{0}, t_{0})$ with $g (s_{0}, t_{0}) = 0$ and $g_{t} (s_{0}, t_{0}) \neq 0$ , then there is a unique local $C^{2}$ root $t = T (s)$ , and the same inverse-function argument gives a $C^{2}$ root depending jointly on additional $C^{2}$ parameters (C² inverses and scalar return roots).

[F4]

In a flat chart the plaque level sets are the characteristic leaves of $h$ ; a finite plaque transport between $C^{2}$ transversals is a $C^{2}$ local diffeomorphism germ, and finite families of $C^{2}$ pieces agreeing on open overlap collars glue to a $C^{2}$ map (C² plaque transport and finite transverse fences preserve C² regularity, Plaques of a flat chart, Regular foliation atlases).

[F5]

The standing hypothesis is Countable Choice ${A C}_{ω}$ (The countable-choice principle used in the foliation pair).

Proof

technique · direct

1.1givenF1F2

The circuit data are finite: by [F2] the frontier consists of finitely many saddle points and finitely many compact regular edges, and its directed polycycles form a finite cover of the edge set. Each regular edge $E$ is a nonconstant trajectory of the characteristic field, so the first integral $u$ of [F1] is constant along $E$ and $d u \neq 0$ on $E$ . Near a regular point the level sets of $u$ are the characteristic leaves, and the port sections chosen in the statement are transverse to the characteristic foliation there.

2.1step 1.1F3F4

Port parameterization. Fix a port section $Σ$ through a regular point $p$ of an edge. In a flat chart containing $Σ$ the scalar $g (ξ, t) : = u (ξ) - t$ is $C^{2}$ , $g (p, u (p)) = 0$ and $\partial_{ξ} g \neq 0$ along $Σ$ because $Σ$ crosses the level set transversally; by [F3] the level $t$ meets $Σ$ in a unique point depending jointly $C^{2}$ on $t$ near $u (p)$ . Taking $t = 0$ at the frontier level shows that the ports, including the frontier port, depend $C^{2}$ on the level. On overlaps of two flat charts the two first integrals differ by the $C^{2}$ transverse transition of $F$ , so the parameterization is chart-independent and the transversality of each port to the ambient foliation is preserved.

3.1F3F4step 1.1step 2.1construct

On each compact trimmed regular edge the C² first integral is a submersion. A finite chain of its inverse-coordinate rectangles supplies local level strips. To obtain exact overlaps, choose a reference C² parametrization of the edge; neighboring strip candidates agree on it at level zero, and in their common plaque coordinate blend them with a fixed source cutoff on an overlap, equal to the corresponding candidate on its end collars. The reference tangent has one strict sign, so after a common shrink the blended tangent keeps that sign. Its transverse label is kept fixed throughout. Finite such blends give a jointly C² regular strip, including its endpoint collars at the ports. This constructs compatible pieces before applying [F4].

4.1F1F3F4step 2.1step 3.1

At a saddle choose one target foliation box containing the images of the two sufficiently close ports and of the intervening saddle passage. For each nearby source level the passage lies in a single target plaque; matching the two port transverse coordinates in this target box therefore gives a C² local transverse transition. This concerns the ambient plaque label and the regular endpoint collars of step 3.1, and supplies no regular source strip across the saddle.

5.1step 2.1step 3.1step 4.1F2

The nearby closed loops. By [F2] the chosen period annulus carries its prescribed closed characteristic loops with a $C^{2}$ transverse trace; for every level $t$ in some one-sided interval $(0, ε)$ the corresponding loop follows the finite itinerary and closes up. The composite transverse return map of the itinerary is obtained by composing the finitely many port and strip transitions of steps 2.1–4.1 around the itinerary; it is a $C^{2}$ germ of a real function, and each closed level loop returns to its own level, so the return map fixes every $t \in (0, ε)$ .

6.1step 5.1

Fixed on an interval. A $C^{2}$ function that fixes every point of a nondegenerate interval equals the identity on that interval; hence the one-sided composite transverse return map is the identity on $(0, ε)$ , which is (c).

7.1step 2.1step 4.1step 6.1F5∎

All constructions selected finitely many charts, edges, ports and intervals; the root and transport theorems used are choice-free, so nothing beyond the standing hypothesis [F5] is invoked, and (a), (b), (c) follow from steps 2.1, 4.1 and 6.1.

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passOpen item page →

Limitwise-nullhomotopy predicate descends to a normal subgroup

Statement

Assume Countable Choice ${A C}_{ω}$ . For a transversely oriented codimension- one foliation, a leaf $L$ , a base point $x \in L$ , and a side $j$ , the predicate $Q_{j} (f)$ of Limitwise-nullhomotopy predicate on based loops is independent of the chosen normal fence and is constant on based-homotopy classes of loops in $N_{j} (L, x)$ . The set of classes in $N_{j} (L, x)$ satisfying this predicate is a well-defined normal subgroup of $π_{1} (L, x)$ . Consequently the class-level subgroup $Π_{1}^{j} (L, x)$ may be defined using any representative and any sufficiently short normal fence.

Facts & Assumptions

Given: A transversely oriented codimension-one foliation, a leaf $L$ , a base point $x \in L$ , a side $j$ , and the predicate $Q_{j}$ on based loops in $N_{j} (L, x)$ for a chosen sufficiently short normal fence.

Proof

technique · direct

1.1givenconstruct

A based leafwise homotopy of two loop representatives has compact intrinsic image. Subdivide its parameter cylinder into finitely many small rectangles inside convex plaque-coordinate boxes. Continue one positive base transversal along a finite tree of the subdivision. Face relations give identical transported labels by the finite chart homotopy argument of Holonomy of a C¹ foliation is a representation into C¹ transverse germs. The sole noncontractible circuit of the parameter cylinder is the original loop; since its class lies in $N_{j}$ , its return germ is the identity on a sufficiently short interval on side j. Thus all finitely many edge relations hold on one positive interval. Assign boundary edge paths to the two displaced loops, assign interior edges once in their common plaque cores, and fill each face by coning its boundary in one convex plaque core. As in A compact leafwise nullhomotopy persists under a transverse deformation in its shared-edge and face-filling construction, this yields a leafwise homotopy between the displaced boundary loops, with a moving basepoint. Nullhomotopy is unchanged by that basepoint change. Hence the predicate is constant on based-homotopy classes.

2.1step 1.1construct

Two short fence choices are compared at the basepoint by the local plaque transport between their transversals. It is an increasing germ sending zero to zero, so it sends all sufficiently small positive parameters into, and onto, a sufficiently small positive interval. For matching parameters their displaced loops have the same local transverse labels; the finite rectangle construction of step 1.1, applied to the constant representative homotopy and these two boundary choices, gives a leafwise homotopy between them. Therefore the condition "every sufficiently small displacement is null" is unchanged by the fence.

3.1step 1.1step 2.1construct

The constant loop satisfies the predicate, using its constant plaque-wise displacement; step 2.1 makes this true for any fence. For two loops in $N_{j}$ satisfying the predicate, choose one common short base transversal. Their displaced loops are closed and null at each sufficiently small parameter, so their concatenation is null. Displacement of the concatenation agrees with that concatenation up to the finite plaque homotopies in step 1.1. Reversal likewise gives the reversed null loop. Thus the class set contains the identity and is closed under products and inverses.

4.1step 1.1step 2.1step 3.1given∎

For any based loop g, its side-preserving transport germ sends a sufficiently small positive parameter ε to another positive parameter tending to zero. Displacing $g * f * \overset{ˉ}{g}$ gives the g-path, the displaced null loop f at that transported parameter, and the reverse g-path. This loop is null. To check that the conjugate also lies in $N_{j}$ , let $ρ$ be the reversed-loop holonomy homomorphism on the full base transversal (The holonomy representation and the holonomy group of a leaf). Coorientation makes its germs increasing and side-preserving. Restriction to side $j$ is a well-defined homomorphism $r_{j}$ into the group of local half-transversal germs: restriction commutes with composition and inversion, and agreement near the basepoint remains agreement on that side. Thus $N_{j} = \ker (r_{j} \circ ρ)$ is normal, and the conjugate belongs to it. The kernel of $ρ$ itself need not equal $N_{j}$ . Therefore the predicate-defined subgroup is normal in $π_{1} (L, x)$ . All compact subdivisions and germ relations were finite.

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A saddle polycycle has a smooth transverse family on either adjacent annulus

Statement

Assume Countable Choice ${A C}_{ω}$ (The countable-choice principle used in the foliation pair). Let $P$ be a finite saddle-separatrix circuit in a generic characteristic disk map, and choose an adjacent period annulus following its finite circuit itinerary. Then there is a $C^{2}$ immersed representative of $P$ in its ambient leaf and a jointly $C^{2}$ family $H_{s}$ , ending at that representative, such that every loop $H_{s}$ lies in a single leaf and every point track $s \mapsto H_{s} (θ)$ is transverse to $F$ . For each positive $s$ the loop $H_{s}$ is leafwise freely homotopic to the prescribed nearby characteristic level loop, by tracked plaque replacements, keeping the selected regularized port collars fixed during saddle replacement. The construction applies separately to either adjacent annulus. No $C^{2}$ convergence of the unmodified hyperbolic parametrizations through the saddle corners is asserted.

Facts & Assumptions

Given: A generic characteristic disk map with a finite saddle-separatrix circuit $P$ , an adjacent period annulus with its finite itinerary, and the regular port sections of the itinerary.

[F1]

In the generic characteristic disk, a local transverse function $u = z \circ h$ is $C^{2}$ , and the characteristic covector is a nowhere-zero scalar multiple of $d u$ ; hence $d u \neq 0$ at every regular point and its level arcs are the characteristic trajectories (Relative generic position for characteristic disk maps, Regular foliation atlases).

[F2]

A $C^{2}$ scalar equation with nonzero derivative in its unknown has a unique local $C^{2}$ root; a $C^{2}$ map with invertible derivative has a $C^{2}$ local inverse (C² inverses and scalar return roots).

[F3]

In a flat chart the plaques are the connected components of the level sets of the transverse coordinate; the transition between two charts is $(x^{'}, t^{'}) = (g (x, t), h (t))$ with $g, h$ of class $C^{2}$ , and finite compatible $C^{2}$ pieces glue to a $C^{2}$ map (Flat charts for a distribution, Plaques of a flat chart, C² plaque transport and finite transverse fences preserve C² regularity, Regular foliation atlases).

[F4]

A map is transverse to $F$ when its differential together with the leaf tangent distribution spans the ambient tangent space at every point; a curve is positively transverse when its derivative has a nonzero component in the positive transverse direction (Smooth maps transverse to a regular foliation).

[F5]

The characteristic singularities are nondegenerate centers and saddles and admit the four-sector hyperbolic picture at every saddle (Relative generic position for characteristic disk maps).

[F6]

The standing hypothesis is Countable Choice ${A C}_{ω}$ (The countable-choice principle used in the foliation pair).

Proof

technique · direct

1.1givenF1F2F5construct

Regular ports for the chosen circuit. The circuit and its itinerary have finitely many edge and saddle-passage occurrences by hypothesis, including repetitions. Trim each occurrence at regular points close to its saddle endpoints, choosing the whole intervening source saddle passage in the preimage of one convex target foliation box. At a regular port choose a short $C^{2}$ source segment $σ (ξ)$ transverse to the characteristic direction. By [F1], $(u \circ σ)^{'} \neq 0$ there, so [F2] gives a $C^{2}$ port point $ξ (t)$ for each nearby transverse label $t$ , including the limiting label. The target port trace $h (σ (ξ (t)))$ is $C^{2}$ and transverse to $F$ because its target transverse coordinate is $t$ .

2.1F1F2F3step 1.1construct

Regular strips without a maximal-center hypothesis. Cover each compact trimmed edge by finitely many source rectangles $(v, u)$ supplied by [F1] and [F2], and use $v$ to parameterize its regular level arcs. Consecutive rectangles overlap along a compact regular arc. In one common inverse-coordinate rectangle keep the level label fixed and interpolate the two longitudinal parameters with a fixed source cutoff on the overlap, agreeing with the respective parameters near its ends. At level zero choose the same reference longitudinal parameter; its derivative is nonzero, so after shrinking the level interval the interpolated derivative retains its sign. These finite interpolations produce $C^{2}$ level strips and exact overlap collars down to the limiting edge. Composing with $h$ gives compatible ambient $C^{2}$ strips and collars, even if their ambient longitudinal tangents vanish. All transverse labels on successive strips differ by the $C^{2}$ local diffeomorphisms of [F3].

3.1givenF1F2F3step 1.1step 2.1construct

One parameter and closed levels. In each chosen saddle box the incoming and outgoing ports of the prescribed passage have the same target transverse label, since the characteristic passage lies in one plaque. Their labels therefore match by a $C^{2}$ local diffeomorphism down to the limiting level, without claiming a regular source strip through the saddle. Compose these transitions and the regular-strip transitions around the finite itinerary to obtain one $C^{2}$ return map $R$ at a base port. The chosen adjacent period annulus following this itinerary supplies closed characteristic loops for every sufficiently small port parameter on its chosen side: that is the local side of the annulus at this regular port. Each such loop returns to the same point, so $R (t) = t$ on that one-sided interval, also at its limiting endpoint by continuity. Transport its base parameter through the finite transitions. Their derivatives are nonzero, so this gives compatible $C^{2}$ transverse parameters for all strips and passages and closes the final collar exactly. These conclusions use the given annulus, with no maximal-center or outer-frontier assumption.

4.1F3step 1.1step 2.1step 3.1construct

Regularize the ambient edges. The limiting images form a continuous loop in one intrinsic leaf: each compact edge segment and each passage lies in a finite chain of plaques, with matching endpoints. Their ambient images need not initially be immersed. Before fixing the ambient port collars, regularize the limiting ambient leafwise loop by finitely many plaque-coordinate chord and corner replacements, as in Finite general position for a leafwise loop, steps 1.1–3.1. The same replacements are made at each nearby transverse level, keeping the transverse coordinate fixed; on overlaps use a common plaque coordinate and fixed source cutoffs. At level zero the resulting reference edge tangents are nonzero, so after one common shrink they stay nonzero at every nearby level. Record these as the regularized edge strips and port collars. The initial replacements themselves are tracked leafwise homotopies. Thus no immersion of the original disk map along characteristic edges has been assumed.

5.1F3F4step 3.1step 4.1construct

In a single target foliation box at a saddle, write the incoming and outgoing regularized collars as $(Y_{-} (θ, t), t)$ and $(Y_{+} (θ, t), t)$ , using the transported transverse label of step 3.1. Choose a regular C² plaque joining path E that agrees exactly with $Y_{-} (θ, 0)$ and $Y_{+} (θ, 0)$ on their smaller end collars. One may build it by finite nonconstant polygonal segments and rounded nonopposite corners in the two-dimensional convex plaque disk, inserting a small detour if required. With disjoint end cutoffs $χ_{-}, χ_{+}$ equal to one on those smaller collars, put $A (θ, t) = (E (θ) + χ_{-} (θ) (Y_{-} (θ, t) - Y_{-} (θ, 0)) + χ_{+} (θ) (Y_{+} (θ, t) - Y_{+} (θ, 0)), t)$ . All interpolation occurs in the leaf coordinates. Hence every slice lies in its plaque and every track has transverse derivative one, including where both cutoffs vanish. At t=0 the tangent is E′≠0, so the family is immersed after a common shrink. It agrees exactly with the two collar families at the ends.

6.1step 5.1F3

Leafwise homotopy of a passage. On the plaque disk the patch $A (\cdot, t)$ is joined to the original saddle passage by convex interpolation in the plaque coordinates: at each $θ$ the interpolation stays in the convex disk, and for fixed $t$ it lies in the leaf of level $t$ ; hence $A (\cdot, t)$ is leafwise freely homotopic to the original passage relative to the two smaller port collars, for every small $t$ .

7.1step 4.1step 6.1F3

The rounded limit loop. Assemble the finitely many regular edge strips of the frontier edges with the finitely many joining paths $E$ of their passages; this is a compact closed curve that is regular on each piece and may have corners at the junctions. Replace it, inside the finitely many plaque disks of the junctions and of the port collars, by a finite polygonal path with nonzero edges and then round the finitely many resulting corners so that adjacent directed edges are not opposite, each replacement keeping the common port collars fixed. The result is a $C^{2}$ immersed closed curve $H_{0}$ in the frontier leaf, and each replacement is leafwise homotopic to the identity relative to the port collars, so $H_{0}$ is a $C^{2}$ immersed representative of $P$ in its ambient leaf.

8.1step 1.1step 2.1step 3.1step 5.1step 6.1step 7.1F3F4

The family for positive levels. Apply the same two operations — the passage patch of step 5.1 and the corner roundings of step 7.1 — to every prescribed level loop of the adjacent annulus in its own plaque coordinates: replace each saddle passage by $A (\cdot, t)$ and round the same finitely many corners. By steps 1.1–3.1 every ingredient depends jointly $C^{2}$ on $(θ, t)$ down to $t = 0$ , and the corner roundings are applied through the $C^{2}$ plaque coordinates supplied by the strips; hence the resulting maps $H_{t}$ form a jointly $C^{2}$ family on $S^{1} \times [0, ε)$ with $H_{t}$ closed in the leaf of level $t$ , ending at $H_{0}$ as $t \to 0^{+}$ . Every point track is transverse to $F$ because all replacements preserve the nonzero derivative of the transported transverse label, by step 5.1 and [F4].

9.1step 6.1step 8.1

Homotopy to the prescribed loops. For each $t > 0$ the loop $H_{t}$ is obtained from the prescribed characteristic level loop by finitely many operations, each of which is a leafwise free homotopy relative to the regular port collars: the passage replacement is step 6.1, and the corner roundings are performed inside plaque disks and are homotopic to the identity of the loop there. Composing the finitely many homotopies gives a leafwise free homotopy from the prescribed level loop to $H_{t}$ .

9.2step 5.1step 8.1

Scope of the regularity claim. The $C^{2}$ assertions of this lemma concern the constructed family, whose corners have been rounded and whose saddle passages have been replaced by the convex patches of step 5.1; the unmodified hyperbolic parametrizations through a saddle corner are not claimed to admit a $C^{2}$ family, and no $C^{2}$ convergence of such raw parametrizations is asserted.

10.1givenstep 1.1step 2.1step 3.1step 8.1step 9.1

Either adjacent annulus. Steps 1.1–3.1 construct the port and strip data from the chosen annulus and its actual itinerary on either side. If an adjacent period annulus exists on the other side, choose its base parameter positive toward that annulus and repeat the construction for its itinerary. No existence of a second annulus is asserted.

11.1step 9.1step 10.1F6∎

The construction selected finitely many charts, passages, corners, cutoffs and intervals; no choice beyond the standing hypothesis [F6] is used.

DefinitionDefinition: Literature-sourcedProof: Literature-sourcedOpen item page →

Limitwise-nullhomotopy subgroup of a leaf

Definition

Assume Countable Choice ${A C}_{ω}$ . For a transversely oriented codimension- one foliation, a leaf $L$ , a base point $x \in L$ , and a side $j$ , define $Π_{1}^{j} (L, x)$ to be the set of classes $[α] \in N_{j} (L, x)$ for which the predicate $Q_{j} (f)$ of Limitwise-nullhomotopy predicate on based loops holds for a based representative $f$ . By Limitwise-nullhomotopy predicate descends to a normal subgroup, this is independent of the representative and fence and is a normal subgroup of $π_{1} (L, x)$ . It is Novikov’s limitwise-nullhomotopy subgroup, distinct from the ordinary limit-cycle quotient $P_{j}$ .

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Fixed transverse fences and their finite crossing words

Statement

Assume ${A C}_{ω}$ . Let $F$ be a $C^{2}$ cooriented foliation, let $L$ be a leaf, and let $f : S^{1} \to L$ be the $C^{2}$ immersed generic finite-double-point representative of a nonzero limitwise-nullhomotopy class of $L$ supplied by Finite general position for a leafwise loop. Then:

(a) there is a sufficiently short one-field leaf-synchronized fence over $f$ : a single smooth transverse field $V$ with flow $φ$ and a jointly $C^{2}$ family $F (u, t) = φ_{τ (u, t)} (f (u))$ , $τ (u, 0) = 0$ , $τ_{t} > 0$ , whose levels are closed loops in single leaves;

(b) compact leafwise sets are separated from short nonzero $V$ -displacements: for every compact intrinsic set $K$ in a leaf contained in the domain of $V$ there is $η > 0$ with $φ_{s} (K) \cap K = \emptyset$ for $0 < ∣ s ∣ < η$ ;

(c) every actual collision at every level uses an eligible pair in one fixed finite cyclic source word. Some eligible pairs may fail to collide at a particular height. When a cut gives closed subloops on its actual common interval, each retained subword has strictly smaller cut rank.

Facts & Assumptions

Given: A $C^{2}$ cooriented foliation $F$ , a leaf $L$ , and the generic finite-double-point representative $f$ of a nonzero limitwise-nullhomotopy class of $L$ on a fixed side, with $N$ transverse double points and no triple points.

[F1]

The loop $f$ is a $C^{2}$ immersion with finitely many transverse double points, no triple points, and a finite cyclic structure of its parameter circle at the marked crossing preimages (Finite general position for a leafwise loop).

[F2]

A smooth positively transverse vector field exists near the compact loop: in finitely many smooth AMBIENT charts choose constant vectors with positive transverse evaluation and shrink their domains to retain positivity; sum them with nonnegative smooth bumps whose smaller cores cover the loop. Positivity is an open convex condition, because positivity of the transverse component is an open convex condition; a compactly supported field has a jointly $C^{2}$ flow and $C^{2}$ flow boxes (Flat charts for a distribution, C¹ Euclidean maximal flows, variational dependence and the finite C² upgrade, A manifold bump for a compact set inside an open set).

[F3]

In a flat chart the plaques are the level sets of the transverse coordinate and plaque transport matches equal transverse coordinates; finite plaque transports are $C^{2}$ local diffeomorphisms and compatible pieces glue (Plaques of a flat chart, Leaves of a regular foliation, C² plaque transport and finite transverse fences preserve C² regularity). The holonomy germ of a leafwise path is independent of the chart chain (The holonomy germ is independent of the foliation chart chain), and the holonomy group consists of the germs of leafwise loops (The holonomy representation and the holonomy group of a leaf).

[F4]

The class of $f$ is limitwise nullhomotopic on the chosen side: the predicate is well defined on classes and descends to the normal subgroup $Π$ ; in particular all sufficiently short positive normal displacements of $f$ are null-homotopic in their leaves, hence are closed loops in those leaves (Limitwise-nullhomotopy predicate on based loops, Limitwise-nullhomotopy predicate descends to a normal subgroup).

[F5]

The standing hypothesis is Countable Choice ${A C}_{ω}$ (The countable-choice principle used in the foliation pair).

Proof

technique · direct

1.1givenF2

Choose the field. By [F2] there is a smooth vector field $V$ , positively transverse to $F$ , on a neighbourhood of the compact loop $f (S^{1})$ ; fix it once and for all. This is the single field of the "one-field" fence.

2.1step 1.1F2F3construct

Separation. Choose finitely many intrinsic open leaf neighborhoods $U_{j}$ covering $K$ , each with compact closure in a single plaque of a larger flat box. For sufficiently small common flow time every point of $K \cap {\overline{U}}_{j}$ remains in that larger box and its transverse coordinate changes with strictly positive derivative. Hence $φ_{s} (p) \neq q$ for $p, q \in K \cap U_{j}$ and small nonzero $s$ , since both initial points have the same plaque coordinate. The set $A = ⋃_{j} (K \cap U_{j}) \times (K \cap U_{j})$ is open in $K \times K$ and contains its diagonal. Its complement is compact and has no pair $p = q$ ; the leaf inclusion is injective and continuous, so at time zero its image misses the closed ambient diagonal. Compactness gives a common short interval on which it still misses that diagonal. Taking the minimum of this interval and the finitely many local flow intervals proves $φ_{s} (K) \cap K = \emptyset$ for $0 < ∣ s ∣ < η$ . The same argument treats a finite union of compact sets in distinct leaves, with each $U_{j}$ chosen inside its own plaque. The compact complement is taken after an OPEN diagonal neighborhood, not after a union of closed cores.

2.2step 1.1F2F3

The fence. Cover the compact loop by finitely many flat boxes and subdivide $S^{1}$ so finely that each closed subarc is carried into one box; in each box the $V$ -flow lines are transverse to the plaques, so the flow box of [F2] and the implicit function theorem identify the nearby plaques as graphs over the corresponding loop pieces via $V$ -orbit projection. Successive plaque continuations starting at $φ_{t} (f (0))$ agree on overlaps by the chart-chain independence of [F3], and uniqueness of the flow time to a given plaque makes the projected time single-valued; after a finite subdivision of $S^{1}$ this yields a jointly $C^{2}$ family $F (u, t) = φ_{τ (u, t)} (f (u))$ with $τ (u, 0) = 0$ and $τ_{t} > 0$ for $t$ in a one-sided interval $[0, b)$ and $u \in S^{1}$ .

3.1step 1.1step 2.2F4

The levels close. By [F4] every sufficiently short positive normal displacement of $f$ is null-homotopic in its leaf, hence closed in its leaf. The levels of the fence of step 2.2 are exactly these displacements, expressed with the fixed field $V$ ; shrinking $b$ if necessary, $F (0, t) = F (1, t)$ and the time function $τ (\cdot, t)$ is periodic, so each level $F (\cdot, t)$ is a closed loop lying in a single leaf, and the fence is leaf-synchronized.

4.1step 2.1step 2.2step 3.1

Shortness and the fixed crossing pairs. Apply step 2.1 to the compact set $K = f (S^{1})$ in its leaf and shrink the fence so that $0 \leq τ (u, t) < η / 3$ for all $u, t$ . If $F (u, t) = F (v, t)$ for some level $t$ , then the flow group law gives $φ_{τ (u, t) - τ (v, t)} (f (u)) = f (v)$ , and $∣ τ (u, t) - τ (v, t) ∣ < 2 η / 3 < η$ , so step 2.1 forces $τ (u, t) = τ (v, t)$ and then $f (u) = f (v)$ . Hence every double point of every level uses one of the $N$ fixed parameter pairs $(u_{i}, v_{i})$ of $f$ .

5.1step 3.1step 4.1F2

Transversality persists. Shrink the fence once more so that all levels remain immersions and the tangent vectors at the finitely many possible crossings $(u_{i}, v_{i})$ remain nonparallel. At such a possible crossing the common time satisfies $τ (u_{i}, t) = τ (v_{i}, t)$ ; the tangent of a level is the leaf-tangential part of $D φ_{τ} f^{'} (u_{i})$ , and as $t ↓ 0$ the flow map tends to the identity uniformly on the compact loop with $C^{1}$ control, so the two images of the nonparallel vectors $f^{'} (u_{i}), f^{'} (v_{i})$ stay nonparallel for the short fence by uniform $C^{1}$ convergence on the compact source circle as $t ↓ 0$ ; the possible crossings of all levels are therefore transverse double points and no triple points occur.

6.1step 4.1step 5.1F1

The finite crossing word. If $N = 0$ , use the single cyclic arc given by the whole parameter circle and rank zero. Otherwise mark the $2 N$ crossing preimages on the parameter circle, decompose $f$ into the corresponding finite cyclic word of arcs, and read every level loop of the fence with the same marking: by step 4.1 its actual crossings are a subset of the $N$ eligible marked pairs. Thus all levels use one fixed source-word marking, although equality of the two flow times need not hold at every eligible pair at every height. Switches are used only at actual coincidences and only on intervals where the resulting subloops close. This is the fixed finite combinatorial carrier needed by the cut rank, not an assertion that all original crossings persist.

7.1step 6.1

Decreasing subword-cut rank. Define the rank $r (w)$ of the word to be the number of original switch vertices visited twice by $w$ , counting a switch vertex once for its two representative ends. A reduced word traverses some arcs at most once and switches only at marked pairs; cutting a selected crossing, which is a vertex visited twice, splits $w$ into two cyclic subwords each visiting that vertex only once and never revisiting an already used switch vertex, so each cut subword satisfies $r \leq r (w) - 1$ . Since all possible non-endpoint collisions of any level of any reduced word are at vertices counted by $r (w)$ by steps 4.1–6.1, the ranking is fixed by the original loop and is uniform over all lower levels.

8.1step 2.2step 2.1step 7.1F5∎

The construction chose finitely many boxes, arcs, marked points and uniform positive constants, so no choice beyond the standing hypothesis [F5] is used; steps 2.1, 2.2 and 6.1–7.1 establish (a), (b) and (c).

5 · Examples, counterexamples and false statements

None yet.