Reeb Stability and Global Foliation Constructions

Let $T$ be a one-dimensional manifold equipped with a $C^1$ atlas: its charts are homeomorphisms onto open subsets of $\mathbb{R}$ whose transition maps are $C^1$ with $C^1$ inverses, and "of class $C^1$" for maps of $T$ means of class $C^1$ in these charts, which is exactly the Euclidean notion of Continuously differentiable maps, local inverses, and local diffeomorphisms. Fix $x\in T$.

A $C^1$ local diffeomorphism of $T$ at $x$ fixing $x$ is a map $f:U\to T$ defined on an open neighbourhood $U\subseteq T$ of $x$ such that $f(U)$ is open, $f:U\to f(U)$ is a bijection, $f(x)=x$, and both $f$ and $f^{-1}:f(U)\to U$ are of class $C^1$. Two such local diffeomorphisms $f:U\to T$ and $g:V\to T$ define the same $C^1$ germ at $x$ when they agree on some neighbourhood of $x$ contained in $U\cap V$; write $f{\sim }_{x}g$ for this relation.

The equivalence classes of ${\sim }_x$ are the $C^1$ germs of local diffeomorphisms of $T$ at $x$, and their set is denoted ${\mathrm{Diff}}_x^1(T)$. Composition of representatives induces a binary operation ${\mathrm{Diff}}_x^1(T)\times {\mathrm{Diff}}_x^1(T)\to {\mathrm{Diff}}_x^1(T),$ the class of $f\circ g$ being independent of the chosen representatives, and with this operation ${\mathrm{Diff}}_x^1(T)$ is a group (Group and abelian group) whose identity is the germ of ${\mathrm{i}\mathrm{d}}_T$. Well-definedness of the operation, associativity, the two-sided identity and two-sided inverses are verified in C¹ germs of local diffeomorphisms form a group ↗.

Finally suppose an orientation of a neighbourhood of $x$ is fixed, represented by a chart $t$ at $x$ with $t(x)=0$; write $\tilde{f}$ for the coordinate expression of a representative $f$. The sign of the derivative ${\tilde{f}}^{\mathrm{\prime }}(0)$ is independent of the positively oriented chart and of the representative of the germ, so the germs whose representatives have ${\tilde{f}}^{\mathrm{\prime }}(0)>0$ in such a chart are well defined. They form a subgroup ${\mathrm{Diff}}_x^{1,+}(T)\le {\mathrm{Diff}}_x^1(T)$ (Subgroup), the orientation-preserving $C^1$ germs at $x$; both the invariance of the sign and the subgroup property are proved in C¹ germs of local diffeomorphisms form a group ↗.

Only the case $\dim T=1$ is used in this pair, and there only for the transverse coordinate of a codimension-one foliation, where the sign of the derivative is the transverse orientation datum.

Let $M$ be a smooth $n$-manifold, $n\ge 1$ (Smooth manifolds and their smooth charts). A $C^1$ codimension-one foliation atlas on $M$ is a family of charts ${\phi }_{\alpha }=(x_{\alpha },t_{\alpha }):U_{\alpha }\to {\mathbb{R}}^{n-1}\times \mathbb{R},$ with the $U_{\alpha }\subseteq M$ open covering $M$, each ${\phi }_{\alpha }$ a homeomorphism onto its image whose inverse is of class $C^1$ and whose components are of class $C^1$ (Continuously differentiable maps, local inverses, and local diffeomorphisms), such that for every $\alpha ,\beta$ with $U_{\alpha }\cap U_{\beta }\mathrm{\ne }\mathrm{\varnothing }$ the transition map, wherever defined, has the form $(x_{\beta },t_{\beta })=(g_{\beta \alpha }(x_{\alpha },t_{\alpha }),h_{\beta \alpha }(t_{\alpha })),$ where $g_{\beta \alpha }$ is of class $C^1$ and $h_{\beta \alpha }$ is a one-dimensional $C^1$ local diffeomorphism of intervals (Continuously differentiable maps, local inverses, and local diffeomorphisms). Thus the second coordinate of a chart depends only on the old second coordinate, and charts change along the first coordinates arbitrarily inside the slices of constant second coordinate.

Charts of such an atlas are foliation charts. For a foliation chart ${\phi }_{\alpha }$ and $c\in \mathbb{R}$ the set of points of $U_{\alpha }$ whose second coordinate equals $c$ is a slice, and each of its connected components is a plaque of the atlas. Two points of $M$ are said to be plaque-chain equivalent when they can be joined by a finite chain $P_0,\dots ,P_k$ of plaques with $P_{i-1}\cap P_i\mathrm{\ne }\mathrm{\varnothing }$ for $1\le i\le k$; this is an equivalence relation. Its equivalence classes are the leaves of the atlas, and the leaf through a point $p$ is written $L_p$. A $C^1$ codimension-one foliation $F$ of $M$ is the leaf decomposition determined by one such atlas; by C¹ foliation charts preserve plaque equivalence and transverse orientation ↗ the leaf decomposition is unchanged under chart refinement or replacement by a compatible foliation atlas (cross-transitions locally preserve slices). Plaques belong to charts and generally become smaller under refinement. Equip each leaf with the topology generated by relatively open subsets of plaques, and with the $C^1$ plaque coordinates. The certificate below verifies compatibility and Hausdorffness of this intrinsic leaf topology; for a compact leaf a finite plaque-chart cover also gives second countability.

The foliation $F$ is transversely oriented, or co-oriented, when the transverse coordinates can be signed consistently. Concretely, $F$ is transversely oriented when there is a foliation atlas for $M$ presenting $F$ in which every transition germ $h_{\beta \alpha }$ is orientation-preserving: $h_{\beta \alpha }^{\mathrm{\prime }}(t)>0$ at every $t$ in its domain, equivalently each $h_{\beta \alpha }$ is increasing. The signed form of the condition allows one to choose, for every chart of a foliation atlas, a locally constant sign ${\epsilon }_{\alpha }\in \{+,-\}$ and to replace the transverse coordinate $t_{\alpha }$ by ${\epsilon }_{\alpha }{t}_{\alpha }$; all transitions become increasing precisely when, on every nonempty overlap, the two sign choices compensate the sign of $h_{\beta \alpha }^{\mathrm{\prime }}$. The equivalence of these formulations, and the fact that transverse orientability is likewise independent of the atlas, are proved in C¹ foliation charts preserve plaque equivalence and transverse orientation ↗.

In a transversely oriented foliation the two sides of each plaque carry a consistent sign, and the local transversals to the plaques are ordered in a way respected by all plaque transports; this is the only use made of transverse orientation in the C¹ stability block. The remaining regular-foliation items use smooth foliations and the separately stated smooth coorientation definition.

The countable choice principle used throughout this pair, written ${\mathrm{A}\mathrm{C}}_{\omega }$, is the assertion:

For every sequence $(A_n{)}_{n\in \mathbb{N}}$ of nonempty sets there is a function $c$ with domain $\mathbb{N}$ such that $c(n)\in A_n$ for every $n\in \mathbb{N}$.

Here "sequence" means a function on $\mathbb{N}$ (A function is a relation $f$ with $(a,b)\in f$ and $(a,c)\in f$ implying $b=c$; $f:A\to B$, the value $f(a)$, domain and codomain). We call the displayed selector $c$ an indexed choice function for the sequence. Its domain is the index set $\mathbb{N}$, whereas a choice function in Choice function has as domain the family of sets themselves. These notions must be distinguished when factors repeat. A family choice function $g$ on $\{A_n:n\in \mathbb{N}\}$ gives an indexed selector by $c(n)=g(A_n)$; the equivalence of the two existence assertions is explained in The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$).

This is the sequence formulation of countable choice. The equivalent nonempty-product formulation, that ${\prod }_{n\in \mathbb{N}}{A}_n$ is nonempty for every sequence $(A_n{)}_{n\in \mathbb{N}}$ of nonempty sets, is verified in Countable choice is equivalent to nonempty countable products ↗; that lemma is recorded as the well-definedness certificate of the present definition.

In this pair ${\mathrm{A}\mathrm{C}}_{\omega }$ is a stated hypothesis of the foliation theorems and of the items whose proof selects countably many plaque data. It is not assumed where a proof does not use it, and the items that consume it state the hypothesis explicitly.

This pair-local carrier is retained deliberately: it restates the published The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) in exactly the sequence form consumed by the foliation items, and the published definition supplies the same principle. The retention and its cross-batch consequences are recorded in the Step-3 report of this pair (finding F4).

Let $F$ be a regular foliation of a smooth manifold $M$ (Regular foliation atlases) and let $L$ be a leaf of $F$ (Leaves of a regular foliation).

An open set $U\subseteq M$ is saturated, or invariant, for $F$ when it is a union of leaves of $F$; equivalently, when for every $x\in U$ the whole leaf of $F$ through $x$ is contained in $U$. The equivalence is immediate from the definitions: a union of leaves has the pointwise property, and conversely the set of leaves meeting $U$ covers $U$ by the pointwise property, so $U$ is their union.

A saturated neighbourhood of $L$ is a saturated open set $U\subseteq M$ with $L\subseteq U$. If $U$ is saturated then $F$ restricts to a regular foliation of $U$: restrict the foliation charts to their intersections with $U$ and take connected components of their slices as plaques. These restricted charts form an atlas of the open submanifold $U$ (Regular foliation atlases). Since $U$ contains every leaf meeting it, every plaque chain in such a leaf remains in $U$, so the restricted leaves are exactly the leaves of $F$ meeting $U$ (Leaves of a regular foliation).

The neighbourhood conclusion of Reeb stability below is stated as the existence, for every neighbourhood $W$ of $L$ in $M$, of a saturated neighbourhood $U\subseteq W$ of $L$; this property of $L$ is recorded separately below as stability of the leaf.

Let $\phi :G\to H$ be a group homomorphism (Monoid homomorphism and group homomorphism). Then:

1. if $G$ is finitely generated (Finitely generated groups), then its image $\mathrm{im}\phi \le H$ (The kernel and image of a group homomorphism) is finitely generated;
2. if $G$ is finite (The cardinality $\mid A\mid$ of a finite set), then $\mathrm{im}\phi$ is finite.

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The countable-choice principle used in the foliation pair). Let $W$ be a smooth $n$-manifold with boundary, $n\ge 1$, with boundary charts modelled on the half-space ${\mathbb{H}}^n={\mathbb{R}}^{n-1}\times [0,\mathrm{\infty })$ and smooth structure as in Smooth charts, atlases, and structures with boundary and Topological manifolds with boundary; thus $\mathrm{\partial }W$ is a closed embedded smooth $(n-1)$-manifold (The boundary of a positive-dimensional manifold is a closed embedded smooth (n-1)-manifold).

Fix $1\le q\le n$ and split ${\mathbb{H}}^n={\mathbb{R}}^{n-q}\times {\mathbb{H}}^q$. A regular foliation of $W$ tangent to $\mathrm{\partial }W$ of codimension $q$ is a regular foliation atlas for the smooth structure of $W$ (Regular foliation atlases) whose charts are relatively open subsets of ${\mathbb{H}}^n$ and whose transition maps are compatible with the standard decomposition of ${\mathbb{H}}^n$ into the model plaques ${\mathbb{R}}^{n-q}\times \{y\}$, $y\in {\mathbb{H}}^q$; that is, on every overlap the coordinates of the second factor depend only on the second coordinates of the first factor, exactly as for a foliation atlas on a boundaryless manifold, and the plaque decomposition restricts to the half-space.

The model plaques meet $\mathrm{\partial }{\mathbb{H}}^n={\mathbb{R}}^{n-q}\times \mathrm{\partial }{\mathbb{H}}^q$ in the sets ${\mathbb{R}}^{n-q}\times \{y\}$ with $y\in \mathrm{\partial }{\mathbb{H}}^q$. Consequently, near a boundary point of $W$ the leaves of $F$ are the intersections of the model plaques with the half-space; the boundary $\mathrm{\partial }W$ is a union of leaves of $F$, and each such boundary plaque lies in the induced regular foliation of $\mathrm{\partial }W$ of codimension $q-1$; the tangent distribution $TF$ of the foliation (Smooth distributions on a manifold) is tangent to the boundary along $\mathrm{\partial }W$ in the sense that $T_p\mathrm{\partial }W$ contains $D_p=T{F}_p$ for $p\in \mathrm{\partial }W$. Thus the leaf directions have zero normal component there. A leaf of the foliation restricted to the interior $\mathrm{Int}W$ is a leaf of $F$, and it need not meet $\mathrm{\partial }W$.

The pair uses this vocabulary only for $q=1$: the Reeb foliations of the solid torus and its gluing are tangent to the boundary, and the boundary torus of a Reeb component is a single leaf.

Let $F$ be a regular foliation of a smooth manifold $M$ (Regular foliation atlases) and let $L$ be a leaf of $F$.

The leaf $L$ is stable when every open neighbourhood $W$ of $L$ in $M$ contains a saturated neighbourhood of $L$ (Saturated neighbourhoods of a leaf); equivalently, when the saturated neighbourhoods of $L$ form a fundamental system of neighbourhoods of $L$.

More generally, a subset $B\subseteq M$ is stable in the sense of Reeb when for every open neighbourhood $W$ of $B$ there is an open neighbourhood $W^{\mathrm{\prime }}\subseteq W$ of $B$ such that every leaf of $F$ meeting $W^{\mathrm{\prime }}$ is contained in $W$. For a leaf $B=L$ this is equivalent to stability of $L$: if such a $W^{\mathrm{\prime }}$ exists, then its saturation $U=\bigcup \{L^{\mathrm{\prime }}:L^{\mathrm{\prime }}\text{ a leaf and }{L}^{\mathrm{\prime }}\cap W^{\mathrm{\prime }}\mathrm{\ne }\mathrm{\varnothing }\}$ is open and saturated (each box carries an open set to its open plaque saturation, and finite plaque transport carries this property along every leaf), contains $L$, and satisfies $U\subseteq W$ by the property of $W^{\mathrm{\prime }}$, so $U$ is a saturated neighbourhood of $L$ inside $W$; conversely a saturated neighbourhood $U\subseteq W$ of $L$ is itself a neighbourhood $W^{\mathrm{\prime }}\subseteq W$ of $L$ every leaf meeting which is contained in $U\subseteq W$.

Stability of a leaf is a neighbourhood property: it depends only on the germ of the foliation along $L$, since both quantifiers involve neighbourhoods of $L$. This is the property that the local and global Reeb stability theorems below establish under their finiteness hypotheses.

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The countable-choice principle used in the foliation pair, Smooth partitions of unity exist on manifolds). Let $F$ be a regular codimension-one foliation of a smooth manifold $M$ (Regular foliation atlases) with tangent distribution $D=TF$, a smooth rank-$(n-1)$ subbundle of $TM$ (Smooth distributions on a manifold).

The foliation $F$ is transversely oriented, or co-oriented, when there is a nowhere-vanishing smooth $1$-form $\omega$ on $M$ whose kernel is $D$: $D=\ker \omega \mathrm{.}$ Equivalently, $F$ is transversely oriented when there is a nowhere-vanishing smooth vector field $X$ on $M$ transverse to $F$, that is, $X_p\mathrm{\notin }{D}_p$ for every $p\in M$.

The two formulations are equivalent because either object is a trivialization of the same line bundle. A smooth $1$-form vanishing on $D$ is a section of the annihilator bundle $D^{\circ }\subseteq T^{\ast }M$ (The annihilator bundle of a distribution), which has rank one; vanishing of this section is an intrinsic condition, so a nowhere-vanishing $\omega$ with $D=\ker \omega$ is exactly a global frame of $D^{\circ }$, and a line bundle admits a nowhere-vanishing section exactly when it is trivial. Dually, a vector field $X$ transverse to $F$ descends to a nowhere-vanishing section of the normal line bundle $TM\mathrm{/}D$, with the normalizations $\omega (X)=1$ identifying the two trivializations pointwise. Thus transverse orientability is exactly triviality of the normal line bundle $TM\mathrm{/}D$. When $M$ is closed this triviality is a genuine restriction, related to orientability of $M$ and of the foliation (Orientable manifolds).

On a transversely oriented codimension-one foliation the local transversals to $F$ are ordered: in a foliation chart the sign of $\omega$ orients the one-dimensional transverse coordinate, and this orientation is respected by all plaque transports, so the transverse direction is globally coherent along each leaf. This definition concerns smooth foliations; the C¹ block uses C¹ codimension-one regular foliations and transverse orientation instead.

The passage between the two smooth global objects uses only the declared ${\mathrm{A}\mathrm{C}}_{\omega }$ partition-of-unity input (Smooth partitions of unity exist on manifolds). Given $\omega$, locally choose smooth transverse fields $X_i$ with $\omega (X_i)=1$ and patch them with a subordinate partition ${\rho }_i$; then $X={\sum }_i{\rho }_i{X}_i$ satisfies $\omega (X)=1$. Conversely, given transverse $X$, choose local annihilator forms ${\omega }_i$ normalized by ${\omega }_i(X)=1$ and patch them the same way. Their sum annihilates $D$ and evaluates to one on $X$, so its kernel is exactly $D$. This supplies the lift from the normal line to actual smooth fields/forms rather than treating the lift as automatic.

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The countable-choice principle used in the foliation pair). Let $X$ be a nonempty compact connected one-dimensional topological manifold without boundary (Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right, Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets, Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not). Then $X$ is homeomorphic to the circle $S^1=\{(x,y)\in {\mathbb{R}}^2:x^2+y^2=1\}$ (Euclidean spheres and closed balls as subspaces of ${\mathbb{R}}^n$). If in addition $X$ carries a smooth structure making it a smooth one-manifold without boundary, then $X$ is diffeomorphic to this circle (Diffeomorphisms and local diffeomorphisms of manifolds).

The empty manifold is excluded by the hypothesis: it is compact and connected under the conventions of this library, but it is not homeomorphic to a circle. The hypothesis "without boundary" is likewise essential: the closed interval $[0,1]$ is compact and connected but has boundary points and is not a circle.

Assume the full Axiom of Choice. For every sequence $(A_n{)}_{n\in \mathbb{N}}$ of nonempty sets there is a sequence $(a_n{)}_{n\in \mathbb{N}}$ with $a_n\in A_n$ for every $n\in \mathbb{N}$. Thus the Axiom of Choice implies the pair-local countable choice principle ${\mathrm{A}\mathrm{C}}_{\omega }$ (The countable-choice principle used in the foliation pair).

For a $C^1$ foliation atlas as in C¹ codimension-one regular foliations and transverse orientation, the plaque-chain relation is an equivalence relation that is independent of chart refinement and of the choice of foliation atlas presenting the same slices, and the transverse orientation condition is independent of compatible atlas changes and of the chosen signed transverse coordinates. The intrinsic plaque topology is Hausdorff and locally Euclidean, and a compact leaf is second countable.

Composition of representatives induces a well-defined group operation on ${\mathrm{Diff}}_x^1(T)$; the germ of the identity is a two-sided unit, every germ has a two-sided inverse, and the germs of positive derivative in an oriented coordinate form the subgroup ${\mathrm{Diff}}_x^{1,+}(T)$.

Let $F$ be a $C^1$ codimension-one foliation of a smooth Hausdorff manifold $M$ (Smooth manifolds and their smooth charts, Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not) and let $L$ be a leaf of $F$ that is compact in its intrinsic leaf-manifold topology (Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right). Then the inclusion $L\hookrightarrow M$ is a $C^1$ embedding, so $L$ is a compact embedded $C^1$ hypersurface of $M$.

For every sequence $(A_n{)}_{n\in \mathbb{N}}$ of nonempty sets, there exists a indexed choice function $c$ with domain $\mathbb{N}$ and $c(n)\in A_n$ for every $n\in \mathbb{N}$ if and only if the product ${\prod }_{n\in \mathbb{N}}{A}_n$ is nonempty. Thus the sequence formulation of The countable-choice principle used in the foliation pair and the nonempty-product formulation of the same principle are equivalent.

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The countable-choice principle used in the foliation pair). Let $W_1,W_2$ be smooth $n$-manifolds with nonempty boundary (Smooth manifolds and their smooth charts, Smooth charts, atlases, and structures with boundary) and let $\phi :\mathrm{\partial }{W}_1\to \mathrm{\partial }{W}_2$ be a diffeomorphism (Diffeomorphisms and local diffeomorphisms of manifolds). Then the quotient $W:=W_1\bigsqcup W_2/\{x\sim \phi (x):x\in \mathrm{\partial }{W}_1\}$ carries a smooth structure for which the two inclusions $W_i\hookrightarrow W$ are smooth embeddings onto their images, making $W$ a smooth $n$-manifold without boundary. Fixing collars fixes this smooth structure; the smooth gluing type is independent of the collar choices, up to diffeomorphism. If moreover each $W_i$ carries a regular codimension-$q$ foliation $F_i$ tangent to $\mathrm{\partial }{W}_i$ (Regular foliation atlases) and $\phi$ carries the foliation of $\mathrm{\partial }{W}_1$ induced by $F_1$ to that induced by $F_2$ (The boundary of a positive-dimensional manifold is a closed embedded smooth (n-1)-manifold), and their tangent plane fields have matching smooth jets in signed collar coordinates, then the $F_i$ glue to a regular codimension-$q$ foliation $F$ of $W$ restricting to $F_i$ on $W_i$. Here matching jets means the following explicit local condition. Identify the collars with $\mathrm{\partial }{W}_1\times (-\epsilon ,\epsilon )$ using $\phi$ and opposite normal signs. Let $E$ be the common rank-$(n-q)$ tangent distribution on the seam, and choose a complement to $E$ there, extended constantly in the signed collar. Each side's nearby plane field is the graph of a smooth linear map $A_{-}(z,s)$ or $A_{+}(z,s)$ from $E_z$ to this complement. Require ${\mathrm{\partial }}_s^j{A}_{-}(z,0)={\mathrm{\partial }}_s^j{A}_{+}(z,0)$ for every $j\ge 0$. Equality of boundary foliations alone does not imply this condition and does not suffice for smooth foliated gluing.