Pipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

✓ 12 results · all verified · 5 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 7 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.

Foliation Holonomy and the Holonomy Groupoid

1 · Prerequisites

2 · Summary

This page develops the holonomy of a regular foliation: the way nearby leaves are compared along a leafwise path, the algebraic structures that record it, and the two global constructions that make it computable. A local transversal is the q-dimensional submanifold transverse to the leaf distribution at a point, a leafwise path is a path inside a single leaf, and along a finite chain of foliation charts one composes the plaque transports between transversals to obtain a germ of a transverse diffeomorphism, the holonomy germ. The germ is independent of the chart chain, the subdivision and the auxiliary transversals, and depends only on the leafwise homotopy class of the path relative to its endpoints; it is multiplicative under concatenation and inversion, so the germs of local diffeomorphisms of a transversal at a base point form a group in which every leaf loop yields a holonomy class. With the library's traversal-order loop product, forward holonomy is an antihomomorphism; reversing the loop gives the homomorphic holonomy representation used below.

Restricting to leaf loops defines the holonomy representation $π_{1} (L, x) \to {Diff}_{x} (T)$ ; its image is the holonomy group of the leaf, while its kernel is the subgroup whose covering corresponds to the holonomy cover of the leaf. Quotienting leafwise paths by equality of holonomy germs gives the holonomy groupoid, a quotient of the monodromy groupoid (leafwise homotopy classes of leafwise paths), and the isotropy group of the holonomy groupoid at a point is exactly the holonomy group of the leaf through it. Two further constructions make the definitions usable: the pullback foliation along a map transverse to the distribution, whose leaves are the components of intrinsic leaf preimages (the transverse fibre products), and the quotient of a foliation by a free and properly discontinuous foliated action, applied in particular to suspensions of representations $π_{1} (B, b_{0}) \to Diff (F)$ , where the holonomy germ of a base loop is the germ of the represented inverse monodromy. The closing remark explains that holonomy specifies a germ, so a global return map requires additional domain data. Countable choice ${A C}_{ω}$ is the standing choice assumption through the smooth-distribution and holonomy interface; no full axiom of choice is invoked.

3 · Logical flowchart

4 · Definitions, theorems and proofs

DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-6.1-sol)Open item page →

Local transversals to a regular foliation

Definition

Assume Countable Choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )), the standing choice assumption of the smooth-distribution setting (Smooth distributions on a manifold). Let $F$ be a regular foliation of codimension $q$ on a smooth $n$ -manifold $M$ (Regular foliation atlases); by Regular foliations and integrable distributions correspond its tangent distribution $D = T F$ is the integrable rank- $(n - q)$ smooth subbundle of $T M$ whose maximal connected integral manifolds are the leaves.

A subset $T \subseteq M$ is a local transversal to $F$ at $x \in T$ when $T$ is an embedded submanifold of $M$ of dimension $q$ containing $x$ (Embedded submanifolds and slice charts, Codimension and hypersurfaces) with

$T_{x} M = D_{x} \oplus T_{x} T .$

Equivalently, $T_{x} T$ is a linear complement of $D_{x}$ in $T_{x} M$ . Since $\dim T_{x} T = q$ and $\dim D_{x} = n - q$ , the sum condition alone already forces the sum to be direct and to fill $T_{x} M$ ; writing it as a direct sum records both clauses at once. The condition is imposed at the single point $x$ : it is a local transversality condition at $x$ , and it says exactly that $T$ meets the leaf through $x$ transversely at $x$ . In particular a codimension-one submanifold meeting a leaf tangentially at $x$ is not a local transversal to $F$ at $x$ . The definition specialises the general transversality of a submanifold to the leaf distribution $D$ ; the atlas convention and the dimension $q$ come from the foliation, so no new structure is introduced.

DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-6.1-sol)Open item page →

Leafwise paths and leafwise homotopy relative to endpoints

Definition

Let $F$ be a regular foliation of a smooth manifold $M$ with its leaves (Leaves of a regular foliation). A leafwise path for $F$ is a continuous map $a : [0, 1] \to M$ whose image is contained in a single leaf of $F$ . Thus a leafwise path from $x$ to $y$ has both endpoints in one leaf, and continuity is required in the topology of $M$ .

A leafwise homotopy relative to endpoints between leafwise paths $a, b$ with the same endpoints $x, y$ is a continuous map $H : [0, 1] \times [0, 1] \to M$ on the product space (The product set $\prod_{i \in I} X_{i}$ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space) with

$H (0, t) = a (t), H (1, t) = b (t), H (s, 0) = x, H (s, 1) = y$

for all $s, t \in [0, 1]$ , such that $t \mapsto H (s, t)$ is a leafwise path for every $s$ . This is a path homotopy relative to the endpoints in the sense of Homotopies of continuous maps, homotopies relative to a subspace, and path homotopies relative to the endpoints carrying one extra condition: every time slice lies in a single leaf. Leafwise paths $a, b$ are leafwise homotopic relative to endpoints, written $a ≃ b$ , when such an $H$ exists.

Leafwise homotopy relative to endpoints is an equivalence relation on leafwise paths with fixed endpoints. Reflexivity and symmetry are Homotopy relative to a subspace is reflexive and symmetric applied to the constant and reversed deformations, which keep every time slice leafwise when the original map does; transitivity is the concatenation of homotopies supplied by Two homotopies relative to the same subspace concatenate after piecewise-linear reparametrisation, whose piecewise-linear reparametrisation again keeps every time slice leafwise. Two leafwise paths are homotopic relative to endpoints when they are equivalent in this relation.

Leaf topology under Countable Choice

Under ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )), every continuous map from a locally connected space into $M$ whose image lies in one leaf is continuous into that leaf's intrinsic manifold topology. Here is the needed local argument. The leaf $L$ is a second-countable injectively immersed manifold with plaque charts (Regular foliations and integrable distributions correspond, Existence and uniqueness of maximal connected integral manifolds). In any foliation chart $U$ its distinct plaques in $L$ are disjoint nonempty open subsets of $L$ : plaque inclusions and intrinsic leaf charts are locally diffeomorphic since their tangent images both equal $T F$ and the smooth inverse function theorem applies (The smooth inverse function theorem on manifolds). An enumerated basis of $L$ (Second countability: an at most countable basis for the topology) assigns to each such plaque the least index of a nonempty basic open set contained in it, so there are at most countably many plaques and at most countably many transverse coordinate values. If $f : C \to U \cap L$ is continuous and $C$ is connected, every transverse coordinate of $f$ is constant: two distinct values would force all intermediate values by connectedness (otherwise the two open half-lines at a missing value separate $C$ ), contradicting Every nondegenerate interval of $R$ is uncountable. The connected image then lies in one connected component of that level set, hence in one plaque. For a general locally connected domain, take connected open neighborhoods inside $f^{- 1} (U)$ . On each such neighborhood $f$ maps continuously into the embedded plaque, whose topology is its intrinsic leaf-chart topology. This proves the assertion. It applies to intervals and squares, so the paths and homotopies above agree with intrinsic leaf paths and homotopies; in particular $π_{1} (L, x)$ uses the intrinsic leaf topology.

DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-6.1-sol)Open item page →

Germs of local diffeomorphisms at a point

Definition

Let $M$ and $N$ be smooth manifolds (Smooth manifolds and their smooth charts) and let $x \in M$ and $y \in N$ . A germ of local diffeomorphisms from $(M, x)$ to $(N, y)$ is an equivalence class of local diffeomorphisms $f : U \to V$ (Diffeomorphisms and local diffeomorphisms of manifolds) with $U$ open in $M$ , $x \in U$ , $V$ open in $N$ , $y \in V$ and $f (x) = y$ , two such maps $f : U \to V$ and $f^{'} : U^{'} \to V^{'}$ being equivalent when they agree on some open neighbourhood $W \subseteq U \cap U^{'}$ of $x$ . This is the germ-of-maps relation of The germ of a smooth function at a point, read for local diffeomorphisms instead of functions; the class of $f$ is written ${germ}_{x} (f)$ , or simply $germ (f)$ when $x$ is understood.

When $M = N$ and $x = y$ , write ${Diff}_{x} (M)$ for the set of germs of local diffeomorphisms $(M, x) \to (M, x)$ . For two germs $[f], [g] \in {Diff}_{x} (M)$ represented by local diffeomorphisms $f : U \to V$ and $g : U^{'} \to V^{'}$ with $f (x) = g (x) = x$ , the composite $f \circ g$ is defined on $g^{- 1} (U) \cap U^{'}$ , an open neighbourhood of $x$ , and is again a local diffeomorphism fixing $x$ ; the germ of this composite is declared to be the product $[f] \cdot [g]$ . The germ of ${i d}_{M}$ is declared to be the identity, and the germ of a local inverse of a representative is declared to be its inverse. That these declarations are well defined and satisfy the group axioms is the content of Germs of local diffeomorphisms at a point form a group ↗; in particular ${Diff}_{x} (M)$ is a group under this operation, and it is the group of germs used for transverse diffeomorphisms on this page.

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

Germs of local diffeomorphisms at a point form a group

Statement

With the notation of Germs of local diffeomorphisms at a point: composition of representatives induces a well-defined binary operation on ${Diff}_{x} (M)$ ; this operation is associative, the germ of the identity is a two-sided identity, and every germ has a two-sided inverse. Hence ${Diff}_{x} (M)$ is a group.

Facts & Assumptions

Given: A smooth manifold $M$ and a point $x \in M$ , with ${Diff}_{x} (M)$ the set of germs at $x$ of local diffeomorphisms $(M, x) \to (M, x)$ .

[F1]

A germ of local diffeomorphisms at $x$ is an equivalence class of local diffeomorphisms $f : U \to V$ with $x \in U$ , $f (x) = x$ , two representatives being equivalent when they agree on a neighbourhood of $x$ ; the product of two germs is represented by the composite on the common domain, the identity germ is that of ${i d}_{M}$ , and the inverse germ is that of a local inverse (Germs of local diffeomorphisms at a point).

[F2]

A local diffeomorphism is a smooth map every point of which has an open neighbourhood mapped diffeomorphically onto an open set; in particular each local diffeomorphism has a smooth local inverse (Diffeomorphisms and local diffeomorphisms of manifolds).

[F3]

A group is a monoid in which every element is invertible, i.e. a set with an associative binary operation, a two-sided identity, and two-sided inverses (Group and abelian group, Subgroup).

Proof

technique · direct

1.1F1F2

Well-definedness. Let $f : U \to V$ and $f^{'} : U^{'} \to V^{'}$ be equivalent representatives of one germ at $x$ , and $g : W \to Z$ and $g^{'} : W^{'} \to Z^{'}$ equivalent representatives of another, all fixing $x$ . Choose an open neighbourhood $N \subseteq U \cap U^{'}$ of $x$ with $f ∣_{N} = f^{'} ∣_{N}$ . Then $g^{- 1} (N) \cap W$ and $g^{' - 1} (N) \cap W^{'}$ are open neighbourhoods of $x$ , because $g (x) = g^{'} (x) = x$ and both are smooth, hence continuous (Smooth maps are continuous), and on the intersection $P : = (g^{- 1} (N) \cap W) \cap (g^{' - 1} (N) \cap W^{'}) \cap Q$ , where $Q$ is an open neighborhood of $x$ on which $g = g^{'}$ , one has $f \circ g = f^{'} \circ g^{'}$ , since $g = g^{'}$ near $x$ and then $f = f^{'}$ on the common image. So the composite germ $[f] \cdot [g] : = [f \circ g]$ does not depend on the representatives.

1.2F1

Associativity and identity. Representatives of three germs all fix $x$ ; on a sufficiently small common neighbourhood the composites $(f \circ g) \circ h$ and $f \circ (g \circ h)$ agree, because composition of functions is associative. Likewise $f \circ i d = i d \circ f = f$ near $x$ , so the germ of ${i d}_{M}$ is a two-sided identity. Hence the operation is associative with a two-sided identity.

1.3F1F2construct

Inverses. Let $f : U \to V$ represent a germ in ${Diff}_{x} (M)$ , so $f (x) = x$ . By [F2] there is an open neighbourhood $A \subseteq U$ of $x$ such that $f ∣_{A}$ is a diffeomorphism onto the open set $f (A)$ ; the inverse $g : = (f ∣_{A})^{- 1} : f (A) \to A$ is a local diffeomorphism with $g (x) = x$ . Its germ $[g]$ satisfies $[f] \cdot [g] = [i d]$ and $[g] \cdot [f] = [i d]$ , because $f \circ g$ and $g \circ f$ are the identity on neighbourhoods of $x$ .

2.1F3step 1.1step 1.2step 1.3∎

Conclusion. The product is well defined (step 1.1), associative with two-sided identity (step 1.2), and every element is invertible (step 1.3). By [F3] the set ${Diff}_{x} (M)$ with this operation is a group.

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

A leafwise path determines a germ of a transverse diffeomorphism

Statement

Assume Countable Choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $F$ be a regular foliation of a smooth manifold $M$ , let $a$ be a leafwise path from $x$ to $y$ (Leafwise paths and leafwise homotopy relative to endpoints), let $T$ be a local transversal to $F$ at $x$ and $T^{'}$ a local transversal to $F$ at $y$ (Local transversals to a regular foliation). Suppose $0 = t_{0} < \dots < t_{N} = 1$ and foliation charts $U_{1}, \dots, U_{N}$ satisfy $a ([t_{i - 1}, t_{i}]) \subseteq U_{i}$ , and choose local transversals $T_{i}$ at $a (t_{i})$ for $i = 1, \dots, N - 1$ with $T_{0} = T$ and $T_{N} = T^{'}$ . Then the chart-wise transports along plaques compose to a germ of a local diffeomorphism $h_{a} : (T, x) \to (T^{'}, y)$ , the holonomy germ of $a$ along the displayed data (Germs of local diffeomorphisms at a point). Every leafwise path admits such a finite chart chain, so every leafwise path together with its endpoint transversals determines at least one such germ.

Facts & Assumptions

Given: A regular foliation $F$ of $M$ with tangent distribution $D = T F$ , a leafwise path $a : [0, 1] \to M$ from $x$ to $y$ , local transversals $T$ at $x$ and $T^{'}$ at $y$ , a subdivision $0 = t_{0} < \dots < t_{N} = 1$ , foliation charts $U_{1}, \dots, U_{N}$ with $a ([t_{i - 1}, t_{i}]) \subseteq U_{i}$ , and local transversals $T_{i}$ at $a (t_{i})$ with $T_{0} = T$ , $T_{N} = T^{'}$ .

[F1]

A regular foliation has an atlas of foliation charts $φ = (x, y) : U \to R^{k} \times R^{q}$ whose overlaps preserve the transverse coordinates; the connected components of the level sets $y = c$ in $U$ are the plaques of the chart, and they are integral manifolds of $D$ (Regular foliation atlases, Plaques of a flat chart, Flat charts for a distribution, Leaves of a regular foliation).

[F2]

The tangent distribution $D = T F$ is an integrable smooth distribution whose maximal connected integral manifolds are the leaves; in particular plaques are local integral manifolds of $D$ (Regular foliations and integrable distributions correspond).

[F3]

A local transversal $T$ to $F$ at $p$ is an embedded submanifold of dimension $q$ with $T_{p} M = D_{p} \oplus T_{p} T$ (Local transversals to a regular foliation).

[F4]

If $d G_{p}$ is an isomorphism of tangent spaces, then $G$ restricts to a diffeomorphism from an open neighbourhood of $p$ onto an open neighbourhood of $G (p)$ (The smooth inverse function theorem on manifolds).

[F5]

A connected open Euclidean slice is polygonally connected (For an open subset of $R^{n}$ , connectedness, path-connectedness and polygonal connectedness are equivalent).

[F6]

$[0, 1]$ is a compact metric space in its usual topology by 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; every open cover of a compact metric space has a Lebesgue number (Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right, 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).

[F7]

A germ of local diffeomorphisms from $(M, x)$ to $(N, y)$ is represented by a local diffeomorphism between open neighbourhoods, two representatives being equivalent when they agree near $x$ (Germs of local diffeomorphisms at a point).

[F8]

Under the assumed Countable Choice, a leaf of $F$ is a maximal connected integral manifold of $D$ , with its intrinsic second-countable smooth manifold structure. Every connected integral manifold contained in it factors smoothly through it (Regular foliations and integrable distributions correspond, Existence and uniqueness of maximal connected integral manifolds).

[F9]

Every nondegenerate real interval is uncountable (Every nondegenerate interval of $R$ is uncountable).

Proof

technique · direct

1.1F1F3F6givenconstruct

A finite chart chain exists. The sets $a^{- 1} (U)$ , over foliation charts, cover $[0, 1]$ . By [F6] they have a Lebesgue number $δ > 0$ . Choose $N$ with $1 / N < δ$ and put $t_{i} = i / N$ . Each parameter interval $[t_{i - 1}, t_{i}]$ has diameter $1 / N$ , so it is contained in some $a^{- 1} (U_{i})$ ; equivalently its image is contained in $U_{i}$ . At an intermediate point a small slice ${x = x (a (t_{i}))}$ in a product foliation box is an embedded $q$ -dimensional transversal, since its tangent space is complementary to $D$ . Thus the required intermediate transversals exist. The subsequent argument applies also to any displayed admissible chain.

1.2F1F2F3F4F5construct

Transport inside one chart. Let $p, p^{'}$ lie in one plaque of a foliation chart $U$ with coordinates $(x, y)$ , and let $S, S^{'}$ be local transversals at those points. The restriction of $y$ to either transversal has invertible differential: its kernel is the intersection of the transversal tangent space with $D = \ker d y$ , which is zero, and its source and target have dimension $q$ . By [F4], shrink to neighborhoods $A \subseteq S$ and $B \subseteq S^{'}$ on which these restrictions are diffeomorphisms with the same open image about $y (p) = y (p^{'})$ . Then $h = (y ∣_{B})^{- 1} \circ (y ∣_{A})$ is a diffeomorphism matching transverse coordinates. It matches points in the same plaque of $U$ after shrinking $A, B$ : by [F5], connect $x (p)$ to $x (p^{'})$ by a polygonal path in the open slice at $y (p)$ . Its compact image has a finite cover by product boxes contained in the chart image; sufficiently small changes of the transverse value keep this path in those boxes. Short segments in the endpoint boxes connect it to $x (u)$ and $x (h (u))$ for $u$ near $p$ . Thus $u$ and $h (u)$ lie in one connected level-set component. If the leaf dimension is zero, the plaque is a singleton and the same conclusion holds in a small transverse box.

2.1F4F7F8F9step 1.2givenconstruct

The chain composes. To establish the single-plaque assertion, give $L$ the intrinsic structure of [F8]. Each plaque of $U_{i}$ contained in $L$ is open in $L$ : its inclusion factors smoothly through $L$ , with invertible differential since both tangent images equal $D$ , so [F4] applies. Distinct plaques are disjoint, and an enumerated basis of $L$ assigns to each plaque the least index of a nonempty basic set contained in it. Thus $L$ meets at most countably many plaques of $U_{i}$ and has at most countably many transverse values there. Each transverse coordinate of the connected continuous image $a ([t_{i - 1}, t_{i}])$ is constant: two distinct values would force a nondegenerate interval of values by connectedness, contradicting [F9]. The image is therefore a connected subset of one level set and lies in one connected component, hence in one plaque. Step 1.2 gives a transport germ $h_{i} : (T_{i - 1}, a (t_{i - 1})) \to (T_{i}, a (t_{i}))$ . Shrink representatives successively so every composite is defined near its source point; then $h_{N} \circ \dots \circ h_{1}$ is a local diffeomorphism near $x$ with value $y$ . Its germ is the holonomy germ along the displayed data.

3.1F7step 1.1step 2.1∎

Conclusion. By steps 1.1 and 2.1 every leafwise path with its endpoint transversals and a chosen finite chart chain produces a germ $h_{a} : (T, x) \to (T^{'}, y)$ of a local diffeomorphism, namely the composition of the chart-wise plaque transports. Since a finite chart chain always exists by step 1.1, every leafwise path determines at least one such germ.

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

The holonomy germ is independent of the foliation chart chain

Statement

Assume Countable Choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). In the situation of A leafwise path determines a germ of a transverse diffeomorphism, the germ $h_{a} : (T, x) \to (T^{'}, y)$ depends only on the leafwise path $a$ and the endpoint transversals $T, T^{'}$ : it is unchanged by passing to a refinement of the chart chain, by changing the subdivision points, and by changing the auxiliary intermediate transversals $T_{1}, \dots, T_{N - 1}$ . Consequently $h_{a} = h_{a} (T^{'}, T)$ is a well-defined germ of a local diffeomorphism from $(T, x)$ to $(T^{'}, y)$ .

Facts & Assumptions

Given: A leafwise path $a$ from $x$ to $y$ in a regular foliation $F$ of $M$ , endpoint transversals $T$ at $x$ and $T^{'}$ at $y$ , and two finite chart chains as in A leafwise path determines a germ of a transverse diffeomorphism, together with the chart-wise transport germs of that lemma.

[F1]

In a foliation chart $φ = (x, y) : U \to R^{k} \times R^{q}$ the plaques are the connected components of the level sets of $y$ , the plaques are integral manifolds of $D = T F$ , and a leafwise path segment contained in $U$ lies in a single plaque; the transport between local transversals inside $U$ matches points with equal transverse coordinates (Regular foliation atlases, Flat charts for a distribution, Plaques of a flat chart, A leafwise path determines a germ of a transverse diffeomorphism).

[F3]

A local transversal $T$ at $p$ satisfies $T_{p} M = D_{p} \oplus T_{p} T$ and meets each nearby plaque in exactly one nearby point, so the transport germ across a plaque between two transversals is well defined (Local transversals to a regular foliation).

[F4]

Two representatives of a germ of local diffeomorphisms agree on some neighbourhood of the source point (Germs of local diffeomorphisms at a point).

[F5]

Every open cover of the compact metric space $[0, 1]$ has a Lebesgue number, so a sufficiently fine subdivision has every subinterval mapped into a member of the cover (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).

[F6]

A smooth map with invertible differential is a diffeomorphism on sufficiently small neighborhoods (The smooth inverse function theorem on manifolds).

Proof

technique · direct

1.1F1F3F4

Inside a single chart the transport is a coordinate matching. Let $U$ be a foliation chart with coordinates $(x, y)$ containing the image of a subinterval, and let $S, S^{'}$ be local transversals at the two subinterval endpoints $p, p^{'}$ , which lie in a common plaque of $U$ . Then the transport germ $S \to S^{'}$ constructed in the single-chart case of A leafwise path determines a germ of a transverse diffeomorphism is the map $u \mapsto$ (the point of $S^{'}$ with the same transverse coordinate as $u$ ) near $p$ . Consequently it is unchanged if an intermediate transversal $S^{''}$ at an interior point $q$ of the subinterval is inserted or replaced: the composite of the transports $S \to S^{''}$ and $S^{''} \to S^{'}$ matches transverse coordinates in the same chart and hence agrees near $p$ with the direct transport, since all three maps send a point to the point with the same $y$ -coordinate.

2.1F1F3F4F5F6step 1.1

Comparison on small overlaps. Around each point of a path segment contained in $U \cap U^{'}$ , choose a smaller product foliation box whose closure need not be fixed, with domain contained in $U \cap U^{'}$ . There the transition has transverse part $y^{'} = r (y)$ by [F1]. Its differential is invertible: the full transition differential is block triangular and invertible, so its transverse diagonal block is invertible. By [F6], after shrinking $r$ is injective near the transverse value, and therefore matching $y$ is equivalent to matching $y^{'}$ for transversals with endpoints in this small box. The transports computed in $U$ and $U^{'}$ consequently agree as germs on that piece. Compactness and [F5] give a finite subdivision of the common segment into these boxes; composing and using step 1.1 proves equality over the entire segment. This comparison uses the transverse transitions on neighborhoods, not only equality of the central plaque germs.

3.1F5step 1.1step 2.1

Two chains compute the same germ. Let two finite chart chains with subdivisions be given. By step 1.1 the computed germ changes neither when a subinterval is subdivided inside one of the given charts nor when intermediate transversals are inserted, so we may refine both chains. The images of sufficiently small subintervals of a common refinement lie in a single chart of the first chain and a single chart of the second chain simultaneously; by [F5] finitely many such subintervals suffice to cover $[0, 1]$ , and by step 2.1 the transport over each such subinterval is the same germ whichever of the two charts is used to compute it. Composing the germs over the common refinement, both chains give the same composite germ from $(T, x)$ to $(T^{'}, y)$ .

3.2F3step 1.1step 2.1

Changing intermediate transversals. At a subdivision point $a (t_{i})$ both adjacent charts $U_{i}, U_{i + 1}$ contain $a (t_{i})$ (each contains the closed subinterval adjacent to the point), so $U_{i} \cap U_{i + 1}$ is a neighbourhood of $a (t_{i})$ ; shrinking the subdivision around $t_{i}$ and applying steps 1.1 and 2.1 to the transport across the resulting small subinterval shows that the composite is unchanged when the auxiliary transversal $T_{i}$ at $a (t_{i})$ is replaced by another local transversal.

4.1F4step 3.1step 3.2∎

Conclusion. By steps 3.1 and 3.2 the composed germ does not depend on the displayed chain, the subdivision or the auxiliary transversals; only the leafwise path and the endpoint transversals remain. Hence $h_{a} = h_{a} (T^{'}, T)$ is a well-defined germ of a local diffeomorphism $(T, x) \to (T^{'}, y)$ , as claimed.

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

Holonomy depends only on leafwise homotopy relative to endpoints

Statement

Assume Countable Choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $F$ be a regular foliation of $M$ , let $a, b$ be leafwise paths from $x$ to $y$ that are leafwise homotopic relative to endpoints (Leafwise paths and leafwise homotopy relative to endpoints), and let $T, T^{'}$ be local transversals at $x$ and $y$ (Local transversals to a regular foliation). Then $h_{a} (T^{'}, T) = h_{b} (T^{'}, T)$ as germs. In particular the holonomy germ of a leafwise path depends only on its leafwise homotopy class relative to endpoints and on the endpoint transversals.

Facts & Assumptions

Given: A leafwise homotopy $H : [0, 1]^{2} \to M$ relative to endpoints from the leafwise path $a$ to the leafwise path $b$ , with $a, b$ leafwise paths from $x$ to $y$ , and local transversals $T$ at $x$ and $T^{'}$ at $y$ .

[F1]

$H$ is continuous, $H (0, \cdot) = a$ , $H (1, \cdot) = b$ , $H (s, 0) = x$ , $H (s, 1) = y$ , and every slice $t \mapsto H (s, t)$ is a leafwise path (Leafwise paths and leafwise homotopy relative to endpoints).

[F2]

The holonomy germ of a leafwise path is well defined: it is unchanged by passing to a refinement of the chart chain, by changing the subdivision points, and by changing the auxiliary intermediate transversals, so it depends only on the leafwise path and the endpoint transversals (The holonomy germ is independent of the foliation chart chain).

[F3]

$[0, 1]^{2}$ is a compact metric space by 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; every open cover has a Lebesgue number, so a sufficiently fine rectangular grid has every cell $[s_{j - 1}, s_{j}] \times [t_{k - 1}, t_{k}]$ mapped into a member of a given open cover of $[0, 1]^{2}$ (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).

[F4]

The germ $h_{a} (T^{'}, T)$ is, by construction, the germ of the composite of the chart-wise plaque transports along a finite chart chain of $a$ : for a subdivision $0 = t_{0} < \dots < t_{N} = 1$ , foliation charts $U_{1}, \dots, U_{N}$ with $a ([t_{i - 1}, t_{i}]) \subseteq U_{i}$ and local transversals $T_{i}$ at $a (t_{i})$ , the chart-wise transport inside $U_{i}$ matches points of the transversals at $a (t_{i - 1})$ and $a (t_{i})$ with equal transverse coordinates, and these germs compose (A leafwise path determines a germ of a transverse diffeomorphism).

[F5]

Under the assumed Countable Choice, leaves are maximal connected integral manifolds, with their intrinsic second-countable smooth structure; connected integral manifolds factor smoothly through them (Regular foliations and integrable distributions correspond, Existence and uniqueness of maximal connected integral manifolds).

[F6]

A smooth map with invertible differential is locally a diffeomorphism (The smooth inverse function theorem on manifolds); a nondegenerate real interval is uncountable (Every nondegenerate interval of $R$ is uncountable).

Proof

technique · direct

1.1F1given

The homotopy image lies in one leaf. Since $H (s, 0) = x$ for every $s$ and every slice $t \mapsto H (s, t)$ is a leafwise path, the point $H (s, t)$ lies in the leaf $L$ through $x$ for every $(s, t)$ . Consequently for every curve $c : [0, 1] \to [0, 1]^{2}$ the composite $H \circ c$ is a leafwise path in $L$ , and if $c$ runs from $(0, 0)$ to $(1, 1)$ then $H \circ c$ runs from $x$ to $y$ .

2.1F1F3step 1.1construct

Staircase paths in the square. The sets $H^{- 1} (U)$ , over foliation charts of $F$ , form an open cover of the square; by [F3] there are grids $0 = s_{0} < \dots < s_{m} = 1$ and $0 = t_{0} < \dots < t_{n} = 1$ such that $H$ maps every cell $R_{j k} = [s_{j - 1}, s_{j}] \times [t_{k - 1}, t_{k}]$ into a single foliation chart. Consider the monotone lattice paths from $(0, 0)$ to $(1, 1)$ built from the unit steps $E$ (increasing $s$ ) and $N$ (increasing $t$ ). Starting from the path $σ_{0} = E^{m} N^{n}$ , the bottom edge followed by the right edge, bubble the $N$ steps to the left: each of the $n$ steps $N$ crosses each of the $m$ steps $E$ once, in $m n$ successive interchanges of an adjacent pair $E N$ into $N E$ , until the path $σ_{m n} = N^{n} E^{m}$ , the left edge followed by the top edge, is reached. Parametrise the paths $σ_{0}, \dots, σ_{m n}$ so that $σ_{ℓ}$ and $σ_{ℓ + 1}$ coincide outside a subinterval on which they run from the common start of the interchanged steps to their common end along the two L-routes (the two two-segment side paths) of the cell spanned by those steps. Then each $P_{ℓ} : = H \circ σ_{ℓ}$ is, by step 1.1, a leafwise path from $x$ to $y$ , with $P_{0}$ a reparametrisation of the concatenation of the constant path at $x$ with $b$ , and $P_{m n}$ a reparametrisation of the concatenation of $a$ with the constant path at $y$ .

2.2F1F2F4F5F6step 1.1construct

Adding one interchange changes nothing. Fix $ℓ$ , let $R$ be the cell spanned by the interchanged steps, mapped by $H$ into a foliation chart $U$ , and let $C_{1}, C_{2}$ be the common start and the common end of the two interchanged steps, so that $σ_{ℓ}$ and $σ_{ℓ + 1}$ agree outside one parameter interval on which they run from $C_{1}$ to $C_{2}$ along the two L-routes of $R$ . The image $H (R)$ is connected and lies in $U \cap L$ by step 1.1, and it lies in one plaque as follows. By [F5], give $L$ its intrinsic second-countable manifold structure. Each plaque of $U$ in $L$ is intrinsically open: its inclusion factors smoothly through $L$ with invertible differential, since both tangent images equal $T F$ , and [F6] applies. Distinct plaques are disjoint, so assigning the least index of a nonempty basic open set contained in each plaque injects this family into an enumerated basis. Thus the transverse values of $L \cap U$ are countable. Every transverse coordinate of the connected continuous image $H (R)$ is constant, since two values would force a nondegenerate interval of values, contrary to [F6]. The image lies in one connected level-set component, hence one plaque. In particular $p : = H (C_{1})$ and $q : = H (C_{2})$ lie in a common plaque of $U$ , and both routes have images in $U$ . Choose local transversals $S_{p}$ at $p$ and $S_{q}$ at $q$ , a subdivision of $[0, 1]$ that contains the two parameter values belonging to $C_{1}$ and $C_{2}$ and has no further subdivision point between them, foliation charts equal to $U$ on the middle interval and covering the common outer parts of the two paths, and intermediate transversals accordingly: this subdivision, these charts and these transversals satisfy the admissibility condition of [F4] for $P_{ℓ}$ and for $P_{ℓ + 1}$ , because outside the middle interval the two paths coincide and inside it both routes have images in $U$ with endpoints in a common plaque. Every chart-wise transport of [F4] is determined by its chart and its two transversals alone, so $P_{ℓ}$ and $P_{ℓ + 1}$ receive one and the same composite germ; by [F4] that germ is a germ of each of the two paths, and by [F2] it is the intrinsic holonomy germ of each. Hence $h_{P_{ℓ}} (T^{'}, T) = h_{P_{ℓ + 1}} (T^{'}, T)$ .

3.1F1F2F4step 2.1step 2.2∎

Conclusion. Chaining step 2.2 over $ℓ = 0, \dots, m n - 1$ gives $h_{P_{0}} (T^{'}, T) = h_{P_{m n}} (T^{'}, T)$ . By step 2.1 the paths $P_{0}$ and $P_{m n}$ are reparametrisations of $c_{x} * b$ and of $a * c_{y}$ , where $c_{x}, c_{y}$ denote the constant paths at $x, y$ ; reparametrising a chart chain changes only its subdivision points, so by [F2] it suffices to compare the germs of the two concatenations. Apply [F4] to $c_{x} * b$ with a chart chain whose subdivision contains the junction, whose chart on the constant piece is a foliation chart around $x$ , and whose intermediate transversal at the junction is $T$ itself: the transport along the constant piece matches equal transverse coordinates at the single point $x$ , so it is the identity germ of $(T, x)$ , while the composite along the remaining pieces is a chain composite of $b$ and therefore equals $h_{b} (T^{'}, T)$ by [F2]; hence $h_{P_{0}} (T^{'}, T) = h_{b} (T^{'}, T)$ . The same argument applied to $a * c_{y}$ with intermediate transversal $T^{'}$ at $y$ gives $h_{P_{m n}} (T^{'}, T) = h_{a} (T^{'}, T)$ . Therefore $h_{a} (T^{'}, T) = h_{b} (T^{'}, T)$ for leafwise homotopic paths with the same endpoints, and the holonomy germ depends only on the leafwise homotopy class relative to endpoints and on the endpoint transversals.

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

Holonomy respects path concatenation and reversal

Statement

Assume Countable Choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $a$ be a leafwise path from $x$ to $y$ , let $b$ be a leafwise path from $y$ to $z$ , and let $T, S, R$ be local transversals at $x, y, z$ (Local transversals to a regular foliation). Then, with $a * b$ the concatenation (traverse $a$ , then $b$ ), $h_{a * b} (R, T) = h_{b} (R, S) \circ h_{a} (S, T) .$ Also $h_{a^{- 1}} (T, S) = (h_{a} (S, T))^{- 1}$ , where $a^{- 1}$ is the reversed leafwise path.

Facts & Assumptions

Given: Leafwise paths $a$ from $x$ to $y$ and $b$ from $y$ to $z$ in a regular foliation $F$ of $M$ , local transversals $T$ at $x$ , $S$ at $y$ , $R$ at $z$ , and the concatenation $a * b$ and reversal $a^{- 1}$ of leafwise paths.

[F1]

The holonomy germ $h_{a} (T^{'}, T) : (T, x) \to (T^{'}, y)$ of a leafwise path is well defined, independent of the chart chain, the subdivision and the auxiliary transversals, and in a single foliation chart it is the germ matching points of the transversals with equal transverse coordinates (The holonomy germ is independent of the foliation chart chain, Plaques of a flat chart, Regular foliation atlases).

[F2]

Concatenation and reversal of leafwise paths are leafwise paths: the concatenation traverses $a$ on the first half and $b$ on the second, the reversal traverses $a$ backwards; both lie in the common leaf (Leafwise paths and leafwise homotopy relative to endpoints).

[F3]

Germs of local diffeomorphisms at a point form a group under composition, so germs have inverses and composites of germs are germs; two germs are equal when representatives agree near the source point (Germs of local diffeomorphisms at a point form a group, Germs of local diffeomorphisms at a point).

Proof

technique · direct

1.1F1F2F3

Single-chart computation. Suppose that both $a$ and $b$ have images in a single foliation chart $U$ with coordinates $(x, y)$ ; then so does $a * b$ , and $x, y, z$ lie in one plaque of $U$ because a leafwise path segment in a chart stays in a plaque. By [F1] each of the three transports matches transverse coordinates in $U$ : $h_{a} (S, T)$ sends $u \in T$ to the point of $S$ with the same $y$ -coordinate, $h_{b} (R, S)$ sends that point to the point of $R$ with the same $y$ -coordinate, and $h_{a * b} (R, T)$ sends $u$ directly to the point of $R$ with the same $y$ -coordinate. The composite therefore agrees with the direct transport on a neighbourhood of $x$ , so their germs are equal by [F3]. Similarly, traversing $a$ backwards exchanges source and target and inverts the coordinate matching, so $h_{a^{- 1}} (T, S)$ is the inverse germ of $h_{a} (S, T)$ .

1.2F1F2

General chain computation. Choose a chart chain for $a$ with endpoint transversals $T, S$ and a chart chain for $b$ with endpoint transversals $S, R$ . Concatenating the two chains and the two subdivisions at the middle time gives a chart chain for $a * b$ with endpoint transversals $T, R$ and the intermediate transversal $S$ at the middle point. By definition of the holonomy germ as the composite of the chart-wise transports, the germ obtained from this concatenated chain is exactly $h_{b} (R, S) \circ h_{a} (S, T)$ ; by chain independence [F1] it equals the intrinsic germ $h_{a * b} (R, T)$ .

2.1F1F3step 1.1

Reversal. Choose a chart chain for $a$ ; reading the same charts and subdivision backwards gives a chart chain for $a^{- 1}$ with the endpoint transversals exchanged. In each chart the reversed transport is the inverse of the forward transport by step 1.1, and by the group law for germs [F3] the composite of the inverses is the inverse of the composite, so $h_{a^{- 1}} (T, S) = (h_{a} (S, T))^{- 1}$ by [F1].

3.1step 1.2step 2.1∎

Conclusion. Steps 1.2 and 2.1 give $h_{a * b} (R, T) = h_{b} (R, S) \circ h_{a} (S, T)$ and $h_{a^{- 1}} (T, S) = (h_{a} (S, T))^{- 1}$ for arbitrary leafwise paths $a, b$ and endpoint transversals.

DefinitionDefinition: AI-adaptedProof: Not applicableOpen item page →

The holonomy representation and the holonomy group of a leaf

Definition

Assume Countable Choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $F$ be a regular foliation of $M$ , let $L$ be a leaf (Leaves of a regular foliation), let $x \in L$ , and let $T$ be a local transversal to $F$ at $x$ (Local transversals to a regular foliation). Equip $L$ with the unique intrinsic smooth manifold structure of its maximal connected integral manifold of $T F$ (Regular foliations and integrable distributions correspond, Existence and uniqueness of maximal connected integral manifolds). This topology, rather than the ambient subspace topology, defines $π_{1} (L, x)$ . The leaf inclusion is smooth and continuous, so intrinsic leaf loops and endpoint-fixed homotopies are leafwise paths and homotopies in $M$ .

Since $T$ is a smooth manifold, the germs at $x$ of local diffeomorphisms $T \to T$ form the group ${Diff}_{x} (T)$ (Germs of local diffeomorphisms at a point, Germs of local diffeomorphisms at a point form a group).

The holonomy representation of $L$ at $x$ relative to $T$ is $ρ_{x} : π_{1} (L, x) \to {Diff}_{x} (T), ρ_{x} ([a]) : = h_{a^{- 1}} (T, T),$ where $a$ is a based loop in the intrinsic leaf topology (Based loops and the fundamental group, Leafwise paths and leafwise homotopy relative to endpoints). The inverse is essential: the library product $[a] [b] = [a * b]$ traverses $a$ first, whereas ordinary composition of germs applies the rightmost map first. Homotopy invariance and reversal therefore give $ρ_{x} ([a] [b]) = (h_{b} \circ h_{a})^{- 1} = h_{a}^{- 1} \circ h_{b}^{- 1} = ρ_{x} ([a]) \circ ρ_{x} ([b])$ (Holonomy depends only on leafwise homotopy relative to endpoints, Holonomy respects path concatenation and reversal). Thus $ρ_{x}$ is a homomorphism. The unreversed map $[a] \mapsto h_{a} (T, T)$ is an antihomomorphism with the same image and kernel. The holonomy group of $L$ at $x$ is the image $Hol (L, x) : = ρ_{x} (π_{1} (L, x)) \leq {Diff}_{x} (T),$ a subgroup of the group of germs in the sense of Subgroup and Group and abelian group.

Replacing $T$ by another local transversal $T_{1}$ at $x$ replaces $ρ_{x}$ by a conjugate homomorphism: with $α$ the germ of the transport across the plaque at $x$ from $T_{1}$ to $T$ , one has $ρ_{x}^{T_{1}} ([a]) = α^{- 1} \circ ρ_{x}^{T} ([a]) \circ α$ for every leaf loop $a$ . Indeed, viewing the loop $a$ as the concatenation of the constant path at $x$ , then $a$ , then the constant path at $x$ , the concatenation law gives exactly this formula, with the constant-path germs supplying $α$ and $α^{- 1}$ (Holonomy respects path concatenation and reversal). It follows that the kernel of $ρ_{x}$ and the conjugacy class of the holonomy group are intrinsic to the leaf and do not depend on the choice of the local transversal $T$ . The representative $ρ_{x}$ itself does depend on $T$ .

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

The deck group of a connected covering acts by a covering-space action

Statement

Let $p : E \to B$ be a covering map with connected total space $E$ . Then the deck group $Deck (p)$ acts on $E$ by a covering-space action: every $e \in E$ has an open neighbourhood $U$ with $h U \cap U = \emptyset$ for every nonidentity $h \in Deck (p)$ .

Facts & Assumptions

Given: A covering map $p : E \to B$ with connected total space $E$ , a point $e \in E$ , and the deck group $Deck (p)$ acting on $E$ by evaluation.

[F1]

A deck transformation is an isomorphism $h : E \to E$ over $B$ , and the deck transformations form the group $Deck (p)$ acting on $E$ by evaluation (Deck transformations and the deck-transformation group of a covering).

[F2]

For a covering with connected total space, two deck transformations agreeing at one point are equal; consequently the deck group acts freely on the total space (On a connected covering space, a deck transformation is determined by one point and the deck action is free).

[F3]

A covering map $p : E \to B$ has for every $b \in B$ an evenly covered open neighbourhood $W$ : the preimage $p^{- 1} (W)$ is a disjoint union of open sheets $V_{j}$ , and each restriction $p ∣_{V_{j}} : V_{j} \to W$ is a homeomorphism (Covering maps, evenly covered neighbourhoods, fibres, sheets, and trivial coverings).

[F4]

An action of a group on a space $E$ by homeomorphisms is a covering-space action when every $e \in E$ has an open neighbourhood $U$ with $g U \cap U = \emptyset$ for every nonidentity $g$ (Covering-space actions by disjoint translates of neighbourhoods).

Proof

technique · direct

1.1F3construct

Put $b : = p (e)$ . By [F3] choose an evenly covered open neighbourhood $W$ of $b$ and let $U$ be the sheet of $p^{- 1} (W)$ containing $e$ . Then $U$ is an open neighbourhood of $e$ and $p ∣_{U} : U \to W$ is a homeomorphism, hence injective.

2.1F1F2step 1.1

Suppose $h U \cap U$ is nonempty. Then some $u \in U$ has $h (u) \in U$ , and $p (h (u)) = p (u)$ by [F1]. Injectivity of $p ∣_{U}$ gives $h (u) = u$ . By [F2], a deck transformation fixing any point is the identity. Thus $h U \cap U = \emptyset$ for every nonidentity $h$ . No connectedness of the chosen sheet or of the evenly covered neighborhood is needed.

3.1F1F4step 2.1∎

Since $e$ was arbitrary and every deck transformation is a homeomorphism, [F4] proves that the deck group acts by a covering-space action.

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

The covering of a leaf associated with the holonomy kernel exists

Statement

Assume Countable Choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $F$ be a regular foliation, $L$ a leaf with base point $x$ , $T$ a local transversal at $x$ , $ρ_{x} : π_{1} (L, x) \to {Diff}_{x} (T)$ the holonomy representation and $K = \ker ρ_{x}$ (The holonomy representation and the holonomy group of a leaf). Then there is a connected covering $p : \hat{L} \to L$ with $p_{*} π_{1} (\hat{L}, \hat{x}) = K$ for a point $\hat{x}$ over $x$ : take the universal cover $\tilde{L} \to L$ , identify $π_{1} (L, x)$ with its deck group, and put $\hat{L} : = \tilde{L} / K$ . Moreover any two connected coverings of $L$ with image subgroup $K$ are isomorphic over $L$ .

Facts & Assumptions

Given: A leaf $L$ of a regular foliation $F$ with base point $x$ , a local transversal $T$ at $x$ , the holonomy representation $ρ_{x}$ , and $K = \ker ρ_{x} \leq π_{1} (L, x)$ .

[F1]

The leaf $L$ carries a unique smooth structure for which the inclusion is a connected injective immersion and an integral manifold of $T F$ ; in particular $L$ is a connected smooth manifold of dimension $\dim M - codim F$ (Existence and uniqueness of maximal connected integral manifolds, Immersed submanifolds, Leaves of a regular foliation).

[F2]

Every connected topological manifold is locally path connected and locally simply connected in the sense required for covering theory: it is locally Euclidean, and the images of convex open sets under charts are simply connected because convex subsets of $R^{n}$ are contractible; consequently a connected manifold is path connected. A zero-dimensional connected manifold is a singleton, so its local simple connectivity follows directly (Topological manifolds are locally compact and locally path connected, Every nonempty convex subset of $R^{n}$ is contractible).

[F3]

Every path-connected, locally path-connected, semilocally simply connected space has a universal cover, and the deck group of a universal cover is isomorphic to the fundamental group of the base, the isomorphism carrying a loop class to the deck transformation moving a chosen fibre point to the corresponding lifted endpoint (Every nonempty path-connected locally path-connected semilocally simply connected space has a universal cover, Universal covering spaces, For a path-connected locally path-connected semilocally simply connected base, the deck group of a universal cover is isomorphic to the fundamental group).

[F4]

The deck group of a covering with connected total space acts by a covering-space action, and the orbit map of a covering-space action is a covering map with deck group exactly the acting group when the total space is path-connected (The deck group of a connected covering acts by a covering-space action, The orbit map of a covering-space action is a covering, with the acting group equal to the deck group when the total space is path-connected, Covering-space actions by disjoint translates of neighbourhoods).

[F5]

A covering $q : \hat{L} \to L$ induces an injection $q_{*}$ on fundamental groups, and the image subgroup has index equal to the number of sheets; two connected coverings of $L$ with the same image subgroup are isomorphic over $L$ by the lifting criterion, applied using the universal cover's dominating property (A covering map induces an injective homomorphism on fundamental groups, For a nonempty path-connected total space, a covering fibre is in bijection with the right cosets of the induced fundamental-group subgroup, Lifting criterion for maps from path-connected locally path-connected spaces, For a path-connected locally path-connected base, a universal cover maps uniquely over the base to every connected covering, and any two universal covers are uniquely isomorphic).

Proof

technique · direct

1.1F1F2F3

The leaf is a nice base. By [F1] the leaf $L$ is a connected smooth manifold with its own manifold topology and smooth structure, the inclusion $L ↪ M$ being a connected injective immersion. By [F2] $L$ is path-connected, locally path-connected and semilocally simply connected. Hence by [F3] there is a universal cover $q : \tilde{L} \to L$ and the deck group $Deck (q)$ is isomorphic to $π_{1} (L, x)$ via the assignment sending a loop class to the deck transformation moving a chosen point of the fibre over $x$ to the lifted endpoint. Fix $\tilde{x}$ over $x$ ; this fixes the isomorphism.

2.1F4step 1.1

The subgroup acts by a covering-space action. The deck group acts on the connected total space $\tilde{L}$ by a covering-space action by [F4]. Restricting the action to the subgroup $K$ (under the isomorphism of step 1.1) preserves the defining property: a neighbourhood $U$ with $γ U \cap U = \emptyset$ for all nonidentity $γ \in Deck (q)$ also satisfies it for all nonidentity elements of $K$ .

3.1F2F3F4step 2.1construct

The intermediate covering. Let $q_{K} : \tilde{L} \to \hat{L} : = \tilde{L} / K$ be the orbit covering supplied by [F4]. Since $q$ is constant on $K$ -orbits, it factors uniquely as $q = p \circ q_{K}$ through a continuous map $p : \hat{L} \to L$ . For a connected evenly covered coordinate neighborhood $W \subseteq L$ , the sheets of $q^{- 1} (W)$ are permuted by $K$ . Each $K$ -orbit of sheets projects under $q_{K}$ to one open set in $\hat{L}$ mapped homeomorphically by $p$ onto $W$ : choose one sheet to define its inverse, and the other sheets in its orbit give exactly the same quotient points. Distinct sheet orbits give disjoint sets. Thus $p$ is a covering. The space $\hat{L}$ is path connected as the continuous image of the path-connected universal cover.

4.1F3F4F5step 3.1

The image subgroup. Fix $\hat{x} = q_{K} (\tilde{x})$ . For a loop $a$ at $x$ , let $\tilde{a}$ be its lift through $q$ starting at $\tilde{x}$ . Its endpoint is $g \tilde{x}$ , where $g$ is the deck transformation corresponding to $[a]$ by [F3]. Then $q_{K} \circ \tilde{a}$ is its lift through $p$ , and this lift closes exactly when $q_{K} (g \tilde{x}) = q_{K} (\tilde{x})$ , equivalently $g \in K$ , since the deck action is free. If $[a] \in p_{*} π_{1} (\hat{L}, \hat{x})$ , an upstairs representing loop and uniqueness of lifts show this lift closes. Conversely, a closed lift is an upstairs loop projecting to $a$ . Hence $p_{*} π_{1} (\hat{L}, \hat{x}) = K$ ; injectivity of $p_{*}$ in [F5] also gives $π_{1} (\hat{L}, \hat{x}) ≅ K$ .

5.1F2F5step 3.1step 4.1∎

Uniqueness. Let $p^{'} : {\hat{L}}^{'} \to L$ be a connected covering with image subgroup $K$ at a point ${\hat{x}}^{'}$ over $x$ . Coverings of a locally path-connected manifold are locally path connected, so their connected total spaces are path connected. The lifting criterion [F5], applied to $p$ through $p^{'}$ and to $p^{'}$ through $p$ , gives based maps $u : \hat{L} \to {\hat{L}}^{'}$ and $v : {\hat{L}}^{'} \to \hat{L}$ over $L$ . The composites and the identities are based lifts of $p$ or $p^{'}$ through the same covering, so uniqueness in the lifting criterion gives $v u = i d$ and $u v = i d$ . Thus these coverings are isomorphic over $L$ . If a different point over $x$ was originally chosen, choose a point at which its image subgroup is $K$ , as required by the hypothesis.

DefinitionDefinition: Literature-sourcedProof: Not applicableOpen item page →

The holonomy cover of a leaf

Definition

Assume Countable Choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $F$ be a regular foliation of $M$ , let $L$ be a leaf with base point $x$ , let $T$ be a local transversal at $x$ , let $ρ_{x} : π_{1} (L, x) \to {Diff}_{x} (T)$ be the holonomy representation and let $K = \ker ρ_{x}$ (The holonomy representation and the holonomy group of a leaf). The holonomy cover of $L$ relative to $T$ is the connected covering $p : \hat{L} \to L$ with $p_{*} π_{1} (\hat{L}, \hat{x}) = K$ supplied by The covering of a leaf associated with the holonomy kernel exists, equipped with a base point $\hat{x}$ over $x$ . It exists and, by that lemma, is unique up to an isomorphism over $L$ : the construction takes the universal cover $\tilde{L} \to L$ , identifies $π_{1} (L, x)$ with its deck group and puts $\hat{L} = \tilde{L} / K$ , so the fibre of $p$ over $z \in L$ is the set of $K$ -orbits in the universal-cover fibre over $z$ .

By construction $π_{1} (\hat{L}, \hat{x}) ≅ K$ and the covering $p : \hat{L} \to L$ is the covering associated with the kernel of the holonomy representation; the covering class of $p$ is the leaf-level input for the finite-holonomy normal model of the Reeb stability pair, which consumes the holonomy cover rather than the universal cover. The kernel $K$ is independent of the choice of the local transversal $T$ , because replacing $T$ conjugates $ρ_{x}$ and conjugation preserves kernels, so the holonomy cover of $L$ does not depend on $T$ up to isomorphism over $L$ .

DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-6.1-sol)Open item page →

The monodromy groupoid of a foliation

Definition

Assume Countable Choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $F$ be a regular foliation of a smooth manifold $M$ with leaf-wise structure as in Leaves of a regular foliation, and let leafwise paths and leafwise homotopy relative to endpoints be as in Leafwise paths and leafwise homotopy relative to endpoints.

The monodromy groupoid $Mon (F)$ is the groupoid with object set $M$ whose arrows from $x$ to $y$ are the leafwise homotopy classes relative to endpoints of leafwise paths from $x$ to $y$ ; there is no arrow from $x$ to $y$ when $x$ and $y$ lie in different leaves. Composition is induced by concatenation of leafwise paths: if $a$ is a leafwise path from $x$ to $y$ and $b$ a leafwise path from $y$ to $z$ , the composite $[b] \circ [a]$ is the class of the concatenation of $b$ after $a$ . The identity at $x$ is the class of the constant leafwise path at $x$ , and the inverse of the class of $a$ is the class of the reversed path $a^{- 1}$ .

The groupoid laws have the following endpoint-fixed witnesses. If $A, B$ are homotopies of composable paths, their concatenation is $A (s, 2 t)$ for $t \leq 1 / 2$ and $B (s, 2 t - 1)$ for $t \geq 1 / 2$ ; the clauses agree at the common endpoint, so finite closed pasting makes this a leafwise homotopy. For any endpoint-fixing reparametrization $ϕ : I \to I$ , $a ((1 - s) t + s ϕ (t))$ is an endpoint-fixed leafwise homotopy from $a$ to $a \circ ϕ$ . This gives associativity and the two constant-path identities using the explicit reparametrizations in Loop classes form the group $π_{1} (X, x_{0})$ under concatenation, proof steps 2.1–2.2; those formulas work also when the path endpoints differ. For $a : x \to y$ , the path $a ((1 - s) \min (2 t, 2 - 2 t))$ contracts $a * a^{- 1}$ to the constant path at $x$ ; the same formula with $a^{- 1}$ contracts $a^{- 1} * a$ at $y$ . All these maps stay in the single leaf of their paths, and the formulas and finite pasting establish continuity in $M$ . Thus the displayed operations are well defined and satisfy all groupoid laws.

The groupoid is set-theoretic: no topology is imposed on the arrow set and no smooth structure on it is asserted here.

DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-6.1-sol)Open item page →

The holonomy groupoid of a foliation

Definition

Assume Countable Choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $F$ be a regular foliation of $M$ . Two leafwise paths with the same endpoints $x, y$ are holonomy-equivalent when their holonomy germs relative to some choice of local transversals at $x$ and $y$ agree (The holonomy germ is independent of the foliation chart chain, Local transversals to a regular foliation). By chain independence the answer does not depend on the choice of the transversals: by the concatenation law Holonomy respects path concatenation and reversal, passing from the pair of transversals $(T, T^{'})$ to another pair $(T_{1}, T_{1}^{'})$ replaces the germ $h_{c} (T^{'}, T)$ of any leafwise path $c$ from $x$ to $y$ by $β \circ h_{c} (T^{'}, T) \circ α^{- 1}$ , where $α$ and $β$ are the germs of constant-path transport $α : (T, x) \to (T_{1}, x)$ and $β : (T^{'}, y) \to (T_{1}^{'}, y)$ across the plaques at the endpoints; since the same two germs $α, β$ occur for every such path $c$ , the relation "the two germs agree" is the same for the two choices. So "one choice" and "every choice" give the same relation.

The holonomy groupoid $Hol (F)$ is the groupoid with object set $M$ whose arrows from $x$ to $y$ are the holonomy classes of leafwise paths from $x$ to $y$ ; there is no arrow between points in different leaves. Composition is induced by concatenation of leafwise paths, the identity at $x$ is the class of the constant path, and inverses are induced by reversal. That these operations are well defined on holonomy classes, and that $Hol (F)$ is a groupoid, is the content of Holonomy classes form a groupoid congruence ↗, which is recorded as the well-definedness certificate of this definition. By Holonomy depends only on leafwise homotopy relative to endpoints and Holonomy respects path concatenation and reversal leafwise homotopy relative to endpoints refines the holonomy relation, and multiplicativity of holonomy germs is what makes composition descend.

Thus $Hol (F)$ is the quotient of the monodromy groupoid $Mon (F)$ of The monodromy groupoid of a foliation by the relation that identifies arrows with equal holonomy germs: the projection $Mon (F) \to Hol (F)$ sends the leafwise homotopy class of a path to its holonomy class. Arrows whose endpoints are not composable have no composite. As for the monodromy groupoid, no topology on the arrow set is imposed and no smooth structure on it is asserted.

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

Holonomy classes form a groupoid congruence

Statement

Assume Countable Choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). On the set of leafwise paths of $F$ , the relation "same endpoints and equal holonomy germs" is an equivalence relation coarser than leafwise homotopy relative to endpoints, and it is a congruence for concatenation: if $a \sim a^{'}$ and $b \sim b^{'}$ , and $a * b$ and $a^{'} * b^{'}$ are defined, then $a * b \sim a^{'} * b^{'}$ . Consequently the quotient $Hol (F)$ is a groupoid and the projection $Mon (F) \to Hol (F)$ is a groupoid morphism (The holonomy groupoid of a foliation, The monodromy groupoid of a foliation).

Facts & Assumptions

Given: Leafwise paths of a regular foliation $F$ with chosen local transversals at their endpoints, and the relation of having equal holonomy germs.

[F1]

The holonomy germ of a leafwise path is well defined, depends only on the path and the endpoint transversals, and is invariant under leafwise homotopy relative to endpoints (The holonomy germ is independent of the foliation chart chain, Holonomy depends only on leafwise homotopy relative to endpoints, Local transversals to a regular foliation).

[F2]

Holonomy respects concatenation and reversal: $h_{a * b} = h_{b} \circ h_{a}$ for composable leafwise paths (with the appropriate endpoint transversals), and $h_{a^{- 1}} = (h_{a})^{- 1}$ (Holonomy respects path concatenation and reversal).

[F3]

For fixed pointed source and target manifolds, a germ is the equivalence class of a local diffeomorphism under agreement on a source neighborhood (Germs of local diffeomorphisms at a point). Smooth maps are continuous (Smooth maps are continuous).

[F4]

Leafwise homotopy relative to endpoints is an equivalence relation on leafwise paths with fixed endpoints, and concatenation of leafwise paths is the operation of the monodromy groupoid (Leafwise paths and leafwise homotopy relative to endpoints, The monodromy groupoid of a foliation).

Proof

technique · direct

1.1F1F3

Reflexivity, symmetry and transitivity. Two leafwise paths are related exactly when they have the same endpoints and their holonomy germs (computed with the chosen endpoint transversals) are equal. Equality of germs is reflexive, symmetric and transitive by [F3], and having the same endpoints is likewise; hence the relation is an equivalence relation on leafwise paths. Leafwise homotopy relative to endpoints refines it: homotopic relative-endpoint leafwise paths have equal holonomy germs by [F1].

1.2F2F3F4

Congruence for concatenation. Suppose $a \sim a^{'}$ and $b \sim b^{'}$ , with $a, a^{'}$ from $x$ to $y$ and $b, b^{'}$ from $y$ to $z$ , and fix local transversals $T, S, R$ at $x, y, z$ . By definition, $h_{a} (S, T) = h_{a^{'}} (S, T)$ and $h_{b} (R, S) = h_{b^{'}} (R, S)$ . By multiplicativity [F2], $h_{a * b} (R, T) = h_{b} (R, S) \circ h_{a} (S, T), h_{a^{'} * b^{'}} (R, T) = h_{b^{'}} (R, S) \circ h_{a^{'}} (S, T),$ and these composites are equal by the following representative argument, which applies between different transversals. Choose representatives $f, f^{'} : (T, x) \to (S, y)$ agreeing on an open neighborhood $A$ of $x$ , and $g, g^{'} : (S, y) \to (R, z)$ agreeing on an open neighborhood $B$ of $y$ . By continuity, $A \cap f^{- 1} (B)$ is an open neighborhood of $x$ ; on it $g \circ f = g^{'} \circ f^{'}$ , so the composite germs agree by [F3]. Inversion likewise respects germ equality: after restricting the equal representatives $f, f^{'}$ to a neighborhood on which they are diffeomorphisms, their inverses agree on its common open image about $y$ . Hence $a * b \sim a^{'} * b^{'}$ : the relation is a congruence.

2.1F2F4step 1.1step 1.2

The quotient is a groupoid. The composite of classes is well defined by step 1.2. Reversal is well defined by [F2] and the representative-inversion argument in step 1.2. The monodromy laws of [F4] supply endpoint-fixed leafwise homotopies from $c_{x} * a$ and $a * c_{y}$ to $a$ , from $a * a^{- 1}$ to $c_{x}$ , and from $a^{- 1} * a$ to $c_{y}$ , as well as between the two associative concatenations. By step 1.1 these homotopies imply equality of holonomy classes. Thus constant-path classes are identities, reversal gives inverses, and composition is associative, so $Hol (F)$ is a groupoid.

3.1F2step 1.1step 1.2step 2.1∎

The projection is a morphism. The projection sends the leafwise homotopy class of a path to its holonomy class; this is well defined by step 1.1 (a homotopy class is contained in a holonomy class), it preserves sources and targets, identities (constant paths), inverses (reversal) and composites (concatenation) by step 1.2 and [F2]. Hence it is a groupoid morphism.

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

The isotropy of the holonomy groupoid is the leaf holonomy group

Statement

Assume Countable Choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $F$ be a regular foliation, $x \in M$ , $L$ the leaf through $x$ , $T$ a local transversal at $x$ and $ρ_{x} : π_{1} (L, x) \to {Diff}_{x} (T)$ the holonomy representation (The holonomy representation and the holonomy group of a leaf). Then the isotropy group $Hol (F) (x, x)$ of the holonomy groupoid (The holonomy groupoid of a foliation) is canonically isomorphic to the holonomy group $Hol (L, x) = ρ_{x} (π_{1} (L, x))$ : the map sending the holonomy class of a leaf loop at $x$ to its holonomy germ is a group isomorphism. In particular the isotropy is trivial if and only if the holonomy representation is trivial.

Facts & Assumptions

Given: A regular foliation $F$ , a point $x \in M$ with leaf $L$ , a local transversal $T$ at $x$ , the holonomy representation $ρ_{x}$ , and the holonomy groupoid $Hol (F)$ .

[F1]

Arrows $x \to x$ of $Hol (F)$ are the holonomy classes of leaf loops at $x$ , where two leaf loops are holonomy-equivalent exactly when their holonomy germs agree; the isotropy group consists of these arrows with composition induced by concatenation and identity the class of the constant loop (The holonomy groupoid of a foliation, Based loops and the fundamental group).

[F2]

The holonomy germ $h_{a} (T, T)$ of a leaf loop $a$ at $x$ is well defined and invariant under leafwise homotopy relative to endpoints; the holonomy representation is $ρ_{x} ([a]) = h_{a^{- 1}} (T, T)$ , it is a homomorphism, and $Hol (L, x) = ρ_{x} (π_{1} (L, x))$ (The holonomy representation and the holonomy group of a leaf, The holonomy germ is independent of the foliation chart chain).

[F3]

Holonomy germs satisfy $h_{a * b} = h_{b} \circ h_{a}$ ; the germs of local diffeomorphisms of $T$ at $x$ form a group under composition, so equal germs compose to equal germs (Holonomy respects path concatenation and reversal, Germs of local diffeomorphisms at a point form a group).

Proof

technique · direct

1.1F1F2

The map is well defined and injective. Define $Θ : Hol (F) (x, x) \to {Diff}_{x} (T)$ by $Θ (class of a) : = h_{a} (T, T)$ . If $a, a^{'}$ are holonomy-equivalent leaf loops, then by definition their holonomy germs agree, $h_{a} (T, T) = h_{a^{'}} (T, T)$ , so $Θ$ is well defined; conversely if the germs agree then the loops are holonomy-equivalent, so $Θ$ is injective.

1.2F1F3

The map is a homomorphism. The isotropy product is induced by concatenation, so for classes of leaf loops $a, b$ at $x$ , $Θ ([b] \cdot [a]) = Θ (class of a * b) = h_{a * b} (T, T) = h_{b} (T, T) \circ h_{a} (T, T) = Θ ([b]) \circ Θ ([a]),$ using multiplicativity of holonomy germs [F3]. The identity class is that of the constant loop, whose germ is the identity germ of $T$ , so $Θ$ preserves identities as well, and inverses are preserved because reversal inverts the germ.

2.1F2step 1.1step 1.2

The image is the holonomy group. Every holonomy class of a leaf loop at $x$ is represented by a leaf loop $a$ , and $Θ$ of its class is $h_{a} (T, T) = ρ_{x} ([a]^{- 1})$ by [F2]; hence the image of $Θ$ is exactly $ρ_{x} (π_{1} (L, x)) = Hol (L, x)$ . Since $Θ$ is an injective homomorphism onto this subgroup, it is a group isomorphism onto the holonomy group.

3.1step 2.1∎

Triviality criterion. The isotropy group is trivial exactly when its isomorphic image $Hol (L, x) = ρ_{x} (π_{1} (L, x))$ is trivial, that is, exactly when $ρ_{x}$ is the trivial homomorphism. This proves the proposition and the stated criterion.

DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-6.1-sol)Open item page →

Smooth maps transverse to a regular foliation

Definition

Assume Countable Choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )), the standing assumption of the smooth-distribution setting (Smooth distributions on a manifold). Let $F$ be a regular foliation of $M$ with tangent distribution $D = T F$ , the integrable smooth subbundle of $T M$ supplied by Regular foliations and integrable distributions correspond (Vector subbundles). Let $N$ be a smooth manifold and $f : N \to M$ a smooth map with differential $d f_{x} : T_{x} N \to T_{f (x)} M$ (The differential of a smooth map).

Then $f$ is transverse to $F$ at $x \in N$ when

$d f_{x} (T_{x} N) + D_{f (x)} = T_{f (x)} M,$

and transverse to $F$ , written $f ⋔ F$ , when this holds at every $x \in N$ . When $N \subseteq M$ and $f$ is the inclusion of an embedded submanifold, the condition reads $T_{x} N + D_{x} = T_{x} M$ at every $x \in N$ : this is the pointwise sum condition of Transverse smooth maps applied with the leaf distribution $D$ in place of the tangent space of a second submanifold, so the definition specialises the published transversality of smooth maps to the leaf distribution. The condition forces $\dim N \geq codim F$ at every point, since $\dim (d f_{x} (T_{x} N)) \leq \dim N$ and the sum with the $(n - q)$ -dimensional space $D_{f (x)}$ fills the $n$ -dimensional space $T_{f (x)} M$ . When $codim F = 0$ the condition is automatic, because then $D_{f (x)} = T_{f (x)} M$ .

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

The pullback foliation under a transverse map

Statement

Assume Countable Choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $F$ be a regular foliation of a smooth manifold $M$ of codimension $q$ , let $N$ be a smooth manifold and let $f : N \to M$ be a smooth map transverse to $F$ (Smooth maps transverse to a regular foliation). Put $D_{x}^{*} : = (d f_{x})^{- 1} (D_{f (x)})$ , where $D = T F$ . Then $D^{*}$ is a smooth rank- $(\dim N - q)$ distribution on $N$ , it is integrable, and the associated regular foliation $f^{*} F$ has as its leaves the connected components of the preimages $f^{- 1} (L)$ of the leaves $L$ of $F$ , with their intrinsic pullback manifold topology: identify $f^{- 1} (L)$ with $N \times_{M} L = {(z, ℓ) : f (z) = j_{L} (ℓ)}$ , where $j_{L} : L \to M$ uses the intrinsic leaf structure. The components here need not be the components in the subspace topology inherited from $N$ . Each leaf of $f^{*} F$ is mapped by $f$ into a leaf of $F$ .

Facts & Assumptions

Given: A regular foliation $F$ of $M$ of codimension $q$ with tangent distribution $D = T F$ , a smooth manifold $N$ , a smooth map $f : N \to M$ transverse to $F$ , and the family $D_{x}^{*} = (d f_{x})^{- 1} (D_{f (x)})$ .

[F1]

Transversality means $d f_{x} (T_{x} N) + D_{f (x)} = T_{f (x)} M$ for every $x \in N$ , and $D$ is a smooth rank- $(\dim M - q)$ subbundle of $T M$ (Smooth maps transverse to a regular foliation, Smooth distributions on a manifold, Regular foliations and integrable distributions correspond).

[F2]

A regular foliation has an atlas of foliation charts $φ = (x, y) : U \to R^{k} \times R^{q}$ with $D ∣_{U} = \ker d y$ ; the connected components of the level sets of $y$ are the plaques, and the leaves are the maximal connected integral manifolds of $D$ (Regular foliation atlases, Leaves of a regular foliation, Regular foliations and integrable distributions correspond).

[F3]

A smooth vector bundle map over the identity whose fibre rank is constant equal to $k$ has kernel and image that are smooth subbundles of rank $\dim E - k$ and $k$ (Constant-rank kernels and images of bundle maps over one base are subbundles).

[F4]

A rank- $k$ smooth distribution is integrable when through every point there passes an integral manifold of dimension $k$ , an integral manifold being a connected injectively immersed submanifold on which $d i$ identifies the tangent space with the distribution (Integrable distributions, Integral manifolds of a distribution).

[F5]

For an integrable distribution the $\sim$ -class $L_{p}$ of a point carries a unique smooth structure making the inclusion a connected injective immersion and an integral manifold; and any connected integral manifold through $p$ maps uniquely into $L_{p}$ (Existence and uniqueness of maximal connected integral manifolds).

[F6]

If $P$ is a connected manifold and $G : P \to M$ is smooth with $d G (T P) \subseteq D$ and $G (P)$ meeting a leaf $L$ of $F$ , then $G (P) \subseteq L$ ; the leaves are maximal connected integral manifolds (Every connected tangent map meeting a leaf factors uniquely through that leaf).

[F7]

A submersion is locally a coordinate projection (Local normal form for submersions). Transverse maps have a smooth embedded fibre product in the product of their domains (Transverse fibre products are embedded submanifolds).

Proof

technique · direct

1.1F1F2F3construct

$D^{*}$ is a smooth subbundle. Let $U$ be a foliation chart of $F$ with transverse coordinates $y : U \to R^{q}$ , so $D ∣_{U} = \ker d y$ by [F2], and put $V : = f^{- 1} (U)$ , an open subset of $N$ . Then $g : = y \circ f : V \to R^{q}$ is a submersion: for $x \in V$ , $d g_{x} = d y_{f (x)} \circ d f_{x}$ annihilates $\ker d f_{x}$ and maps $d f_{x} (T_{x} N)$ onto $d y_{f (x)} (T_{f (x)} M) = T_{y (f (x))} R^{q}$ , the last surjectivity because $d f_{x} (T_{x} N) + D_{f (x)} = T_{f (x)} M$ and $d y$ kills $D$ . Hence $\ker d g_{x} = (d f_{x})^{- 1} (\ker d y_{f (x)}) = D_{x}^{*}$ for every $x \in V$ . Thus $D^{*}$ is locally the kernel of a constant-rank bundle map $T N ∣_{V} \to V \times R^{q}$ , and by [F3] it is a smooth subbundle of rank $\dim N - q$ on $V$ . The local descriptions agree on overlaps, since all of them compute the same family of subspaces $D_{x}^{*}$ ; hence $D^{*}$ is a smooth distribution of rank $\dim N - q$ on all of $N$ .

2.1F4F7step 1.1

Local integral manifolds. In step 1.1, $g = y \circ f$ is a submersion. By [F7], locally it is a coordinate projection, so a small connected piece of $g^{- 1} (g (x))$ is an embedded submanifold of dimension $\dim N - q$ with tangent space $\ker d g = D^{*}$ . Thus $D^{*}$ has an integral manifold through every point.

3.1F1F2F4F5F7step 2.1construct

Intrinsic leaf preimages. Let $j_{L} : L \to M$ be the intrinsic integral immersion of a leaf. By [F1], $f$ and $j_{L}$ are transverse, so [F7] makes $P_{L} = N \times_{M} L$ an embedded manifold in $N \times L$ . Projection $P_{L} \to N$ is injective because $j_{L}$ is. In a plaque neighborhood of $ℓ \in L$ , its local image is a level set of $y \circ f$ , so step 2.1 shows that this projection is an immersion with tangent image $D^{*}$ . This gives the stated intrinsic topology on the set $f^{- 1} (L)$ . Each connected component of $P_{L}$ is therefore a connected integral manifold of $D^{*}$ . Other plaques of $L$ in the same ambient chart are different intrinsic neighborhoods; no single transverse value $c_{L}$ is assigned to all of $L \cap U$ .

3.2F4F5step 2.1

Global integrability. By [F4] and step 2.1, $D^{*}$ is integrable, and [F2] and Regular foliations and integrable distributions correspond associate to it a regular foliation $f^{*} F$ whose leaves are the maximal connected integral manifolds of $D^{*}$ ; by [F5] the leaf through a point is the $\sim_{D^{*}}$ -class of that point.

4.1F6step 3.2

Every leaf of $f^{*} F$ lies in a preimage of a leaf of $F$ . Let $L^{'}$ be a leaf of $f^{*} F$ , with inclusion $i : L^{'} \to N$ . For $u \in L^{'}$ , $d (f \circ i)_{u} (T_{u} L^{'}) = d f_{u} (D_{u}^{*}) \subseteq D_{f (u)}$ . Since $L^{'}$ is connected, [F6] applied to the smooth map $f \circ i$ shows that $f (L^{'})$ lies in a single leaf $L$ of $F$ . Hence $L^{'} \subseteq f^{- 1} (L)$ .

5.1F5F6step 3.1step 3.2step 4.1∎

The leaves are exactly the intrinsic components. Let $C$ be a connected component of $P_{L}$ and let $x$ be in its image in $N$ . By step 3.1 and [F5], this image lies in the pullback leaf $L^{'}$ through $x$ . Conversely, step 4.1 and the smooth factorization in [F6] give a smooth map $L^{'} \to L$ lifting $f ∣_{L^{'}}$ . Its graph defines a continuous map $L^{'} \to P_{L}$ ; its image is connected and meets $C$ , hence lies in $C$ . Thus $L^{'}$ is exactly the image of $C$ . Every point of $P_{L}$ belongs to such a component, proving the leaf description and the final mapping assertion.

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

The quotient foliation under a free and properly discontinuous foliated action

Statement

Assume Countable Choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $Γ$ be a group acting on a smooth manifold $M$ by diffeomorphisms, and suppose the action is free and properly discontinuous: $γ \cdot x = x$ implies $γ = e$ for all $x \in M$ , and for every compact subset $K \subseteq M$ the set ${γ \in Γ : γ K \cap K \neq \emptyset}$ is finite. Let $F$ be a regular foliation of $M$ preserved by $Γ$ , so every $γ$ maps leaves onto leaves, with tangent distribution $D = T F$ . Then:

$M / Γ$ carries a unique smooth structure for which the orbit map $π : M \to M / Γ$ is a local diffeomorphism, and with this structure $π$ is a covering map;
there is a unique regular foliation $F / Γ$ on $M / Γ$ whose leaves are the images $π (L)$ of the leaves of $F$ , and its codimension equals $codim F$ ;
the tangent distribution of $F / Γ$ is $d π (D)$ .

Facts & Assumptions

Given: A group $Γ$ acting freely and properly discontinuously by diffeomorphisms on a smooth manifold $M$ , a regular foliation $F$ of $M$ with tangent distribution $D = T F$ preserved by $Γ$ , the orbit map $π : M \to M / Γ$ , and the set $Γ x = {γ \cdot x : γ \in Γ}$ .

[F1]

A smooth manifold is a topological manifold: Hausdorff, second countable and locally Euclidean (Topological manifolds without boundary: Hausdorff, second-countable, and locally Euclidean spaces, Smooth manifolds and their smooth charts).

[F2]

Every point of a topological manifold has a neighbourhood basis of open sets with compact closures; in particular $M$ is locally compact and first countable (Topological manifolds are locally compact and locally path connected).

[F3]

For a surjection $q : X \to Y$ , the quotient topology makes $V \subseteq Y$ open exactly when $q^{- 1} (V)$ is open, and a set is open in the quotient exactly when it is the image of a saturated open set (The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection).

[F4]

An action by homeomorphisms is a covering-space action when every point has an open neighbourhood $U$ with $γ U \cap U = \emptyset$ for every $γ \neq e$ ; for a covering-space action the orbit map is a covering map (Covering-space actions by disjoint translates of neighbourhoods, The orbit map of a covering-space action is a covering, with the acting group equal to the deck group when the total space is path-connected).

[F5]

A covering map is a continuous surjection each of whose points has an evenly covered open neighbourhood $W$ , over which the preimage is a disjoint union of open sheets mapping homeomorphically onto $W$ (Covering maps, evenly covered neighbourhoods, fibres, sheets, and trivial coverings).

[F6]

A regular foliation atlas of codimension $n - k$ on an $n$ -manifold has charts whose overlaps preserve the transverse coordinates, and its leaves are the equivalence classes of the plaque-chain relation (Regular foliation atlases, Leaves of a regular foliation); regular foliations and integrable distributions determine each other, the leaves being the maximal connected integral manifolds (Regular foliations and integrable distributions correspond). Under the assumed Countable Choice, these leaves carry intrinsic second-countable smooth manifold structures; connected integral manifolds factor smoothly through them (Existence and uniqueness of maximal connected integral manifolds). In particular every plaque is intrinsically open: its factorization is a local diffeomorphism because its tangent image and the leaf tangent image both equal $D$ .

[F7]

If $G : M \to N$ is a local diffeomorphism and $U$ is open with $G ∣_{U}$ a diffeomorphism onto $G (U)$ , then the family $G_{*} D_{G (p)} = d G_{p} (D_{p})$ is a smooth distribution on $G (U)$ of the same rank, and images of integral manifolds are integral manifolds (Local diffeomorphisms carry distributions and integral manifolds).

[F8]

A smooth atlas is a family of pairwise smoothly compatible charts covering the space, and every smooth atlas is contained in exactly one maximal smooth atlas, which generates the same smooth structure (Smooth atlases, Each smooth atlas is contained in a unique maximal smooth atlas).

[F9]

A diffeomorphism is a bijective smooth map with smooth inverse, and a local diffeomorphism restricts near each point to a diffeomorphism onto an open set (Diffeomorphisms and local diffeomorphisms of manifolds).

[F11]

A nondegenerate real interval is uncountable (Every nondegenerate interval of $R$ is uncountable); a connected countable subset of $R$ must therefore be a singleton, since a missing intermediate value would separate it by open half-lines.

[F10]

A space is second countable when it has an at most countable basis, i.e. every open set is a union of members of that countable family (Second countability: an at most countable basis for the topology, Basis and subbasis for a topology, and the topology generated by a family of sets).

Proof

technique · direct

1.1F2givenchoose

The pointwise disjoint-translates condition. For each $x \in M$ there is an open neighbourhood $U$ of $x$ with $γ U \cap U = \emptyset$ for every nonidentity $γ \in Γ$ . Indeed, by [F2] choose a compact neighbourhood $K$ of $x$ . The set $F_{0} : = {γ \in Γ : γ K \cap K \neq \emptyset}$ is finite by proper discontinuity. For each $γ \in F_{0}$ with $γ \neq e$ freeness and Hausdorffness give disjoint open sets $V_{γ} ∋ x$ and $W_{γ} ∋ γ x$ ; then $U : = int (K) \cap ⋂_{γ \in F_{0} ∖ {e}} (V_{γ} \cap γ^{- 1} W_{γ})$ is an open neighbourhood of $x$ , and for $γ \in F_{0} ∖ {e}$ one has $γ U \subseteq W_{γ}$ and $U \subseteq V_{γ}$ , so $γ U \cap U = \emptyset$ , while for $γ \notin F_{0}$ one has $γ U \cap U \subseteq γ K \cap K = \emptyset$ .

1.2F1F3F10

$M / Γ$ is second countable. The orbit map is open: for open $B \subseteq M$ the saturation $π^{- 1} (π (B)) = Γ B$ is a union of translates of $B$ , hence open, so $π (B)$ is open by [F3]. Choose a countable basis $B$ of $M$ by [F1] and [F10]; then $π [B]$ is an at most countable family of open subsets of $M / Γ$ : given an open $V \subseteq M / Γ$ and $b \in V$ , choose $m \in π^{- 1} (b)$ and then $B \in B$ with $m \in B \subseteq π^{- 1} (V)$ ; then $b \in π (B) \subseteq V$ . So $π [B]$ is a basis, and $M / Γ$ is second countable.

2.1F4F9step 1.1given

The orbit map is a covering. By step 1.1 and [F4] the action on $M$ — which is by homeomorphisms, because $Γ$ acts by diffeomorphisms by [F9] — is a covering-space action, so the orbit map $π : M \to M / Γ$ is a covering map. Its fibres are exactly the orbits: $π (m) = π (m^{'})$ holds exactly when $m^{'} = γ m$ for some $γ \in Γ$ , by the definition of the orbit space.

3.1F5step 2.1choose

Local sections differ locally by group elements. Let $s_{1} : W_{1} \to M$ and $s_{2} : W_{2} \to M$ be continuous local sections of $π$ on open sets, so $π \circ s_{i} = {i d}_{W_{i}}$ , and let $w \in W_{1} \cap W_{2}$ . Then some open connected neighbourhood $W_{0} \subseteq W_{1} \cap W_{2}$ of $w$ and some $γ \in Γ$ satisfy $s_{2} ∣_{W_{0}} = γ \circ s_{1} ∣_{W_{0}}$ . Indeed, $s_{1} (w)$ and $s_{2} (w)$ lie in the same $π$ -fibre, which is an orbit by step 2.1, so $γ s_{1} (w) = s_{2} (w)$ for some $γ \in Γ$ . By [F5] choose an evenly covered open neighbourhood $W$ of $w$ ; shrinking inside $W \cap W_{1} \cap W_{2}$ (a smaller open set over an evenly covered one is again evenly covered) gives an open connected neighbourhood $W_{0}$ of $w$ on which both sections are defined and over which $π$ is evenly covered. The connected set $s_{2} (W_{0})$ lies in a single sheet $V$ of $π^{- 1} (W_{0})$ ; the connected set $γ s_{1} (W_{0})$ satisfies $π (γ s_{1} (W_{0})) = W_{0}$ and contains $γ s_{1} (w) = s_{2} (w)$ , so it likewise lies in the single sheet $V$ . Since $π ∣_{V}$ is injective and both $s_{2}$ and $γ \circ s_{1}$ are sections over $W_{0}$ , it follows that $s_{2} (t) = γ s_{1} (t)$ for every $t \in W_{0}$ .

3.2F1F3step 2.1choose

$M / Γ$ is Hausdorff. Let $x, y \in M$ with $π (x) \neq π (y)$ , so $y \notin Γ x$ . By [F2] choose compact neighbourhoods $K_{x}$ of $x$ and $K_{y}$ of $y$ with open interiors $U_{x}$ , $U_{y}$ . The set $F_{1} : = {γ \in Γ : γ K_{y} \cap K_{x} \neq \emptyset}$ is finite, because $γ K_{y} \cap K_{x} \neq \emptyset$ implies $γ (K_{x} \cup K_{y}) \cap (K_{x} \cup K_{y}) \neq \emptyset$ and $K_{x} \cup K_{y}$ is compact (the same argument as in step 1.1, applied with [F1]). For each $γ \in F_{1}$ one has $x \neq γ y$ , since $γ y = x$ would give $y \in Γ x$ ; by Hausdorffness there are disjoint open sets $V_{γ} ∋ x$ and $W_{γ} ∋ γ y$ . Put $V : = U_{x} \cap ⋂_{γ \in F_{1}} V_{γ}, W : = U_{y} \cap ⋂_{γ \in F_{1}} γ^{- 1} W_{γ} .$ Both are open neighbourhoods of $x$ and $y$ . If $γ \in F_{1}$ , then $V \subseteq V_{γ}$ and $γ W \subseteq W_{γ}$ , so $V \cap γ W = \emptyset$ ; if $γ \notin F_{1}$ , then $V \cap γ W \subseteq K_{x} \cap γ K_{y} = \emptyset$ . Hence $V \cap Γ W = \emptyset$ . The set $Γ V$ is open (a union of translates of an open set) and $Γ$ -invariant, and it is disjoint from the open $Γ$ -invariant set $Γ W$ ; by [F3] their images $π (Γ V) = π (V)$ and $π (Γ W) = π (W)$ are disjoint open sets in $M / Γ$ containing $π (x)$ and $π (y)$ . Hence $M / Γ$ is Hausdorff.

4.1F5F8F9step 3.1step 3.2step 1.2construct

A smooth atlas and the local diffeomorphism property. For every sheet $U$ over an evenly covered open set and every smooth chart $(U^{'}, φ)$ of $M$ with $U^{'} \subseteq U$ , define $ψ : π (U^{'}) \to R^{n}$ by $ψ (π (m)) : = φ (m)$ for $m \in U^{'}$ ; this is well defined because $π ∣_{U^{'}}$ is injective, and it is a homeomorphism onto the open set $φ (U^{'})$ because $π ∣_{U^{'}}$ is a homeomorphism onto the open set $π (U^{'})$ . Such pairs cover $M / Γ$ (every point has a neighbourhood contained in a sheet with a chart, by [F5]). Two of them, $(ψ_{1}, π (U_{1}^{'}))$ and $(ψ_{2}, π (U_{2}^{'}))$ , overlap in $π (U_{1}^{'}) \cap π (U_{2}^{'})$ ; writing $s_{i} : = (π ∣_{U_{i}^{'}})^{- 1}$ for the inverse sections, the transition on a point $π (m)$ of the overlap is $ψ_{2} \circ ψ_{1}^{- 1} (φ_{1} (m)) = φ_{2} (s_{2} (π (m))) = φ_{2} (γ s_{1} (π (m))) = φ_{2} (γ (m))$ for some $γ \in Γ$ and all $m$ in a neighbourhood of the given point, the middle equality by step 3.1. This is smooth, because $φ_{2} \circ γ \circ φ_{1}^{- 1}$ is a transition between charts of $M$ conjugated by the diffeomorphism $γ$ of $M$ ([F9]). Hence the $ψ$ form a smooth atlas $A$ on the topological manifold $M / Γ$ — Hausdorff by step 3.2, second countable by step 1.2, locally Euclidean by the $ψ$ — and $ψ \circ π \circ φ^{- 1} = i d$ on the appropriate domain shows that $π$ is a local diffeomorphism for the smooth structure on $M / Γ$ generated by $A$ .

5.1F8F9step 4.1

Uniqueness of the smooth structure. In any smooth structure on $M / Γ$ for which $π$ is a local diffeomorphism, the charts $ψ$ of step 4.1 are smoothly compatible with every chart $θ$ of that structure. Indeed, $θ \circ ψ^{- 1} = θ \circ π \circ φ^{- 1}$ is smooth, and its inverse $ψ \circ θ^{- 1} = φ \circ (π ∣_{U^{'}})^{- 1} \circ θ^{- 1}$ is smooth because the local inverse of $π$ is smooth. By [F8] both atlases generate the same maximal atlas, proving uniqueness.

5.2F1F6F7F9F11givenstep 4.1

The descended distribution. Define, for $b \in M / Γ$ and any $m \in π^{- 1} (b)$ , the subspace $E_{b} : = d π_{m} (D_{m}) \subseteq T_{b} (M / Γ)$ . This does not depend on $m$ : if $m^{'} = γ m$ , then $π$ near $m^{'}$ equals $π$ near $m$ composed with $γ^{- 1}$ , so $d π_{m^{'}} (D_{m^{'}}) = d π_{m} (d γ_{γ m}^{- 1} (D_{γ m})) = d π_{m} (D_{m})$ , using $d γ (D) = D$ . To justify this implication from preservation of leaf sets, restrict $γ$ to a connected plaque neighborhood whose image lies in a target foliation chart. A leaf meets at most countably many target plaques, since these are disjoint open subsets of its intrinsic second-countable manifold ([F1], [F6]); the connected image has constant transverse coordinates, because a countable connected subset of $R$ is a singleton. Thus $γ$ maps this neighborhood smoothly into one target plaque and carries its tangent space into $D$ . Applying the same argument to $γ^{- 1}$ gives equality. The family $E$ is a smooth rank- $(\dim M - codim F)$ distribution: the charts $ψ$ of step 4.1 are local diffeomorphisms of $M / Γ$ obtained by pushing forward by $π$ along a sheet, so on each chart domain $E$ is the pushforward of the subbundle $D$ by a diffeomorphism, which is a smooth subbundle of the same rank by [F7].

6.1F6F7step 4.1step 5.2construct

Integrability and the quotient foliation. Around each $m \in M$ restrict a foliation chart to a sheet of $π$ . Its plaques push forward to integral manifolds of $E$ by [F7], and one passes through every point of the quotient. Thus $E$ is integrable. By [F6] it determines a regular foliation $F / Γ$ with maximal connected integral leaves, codimension $codim F$ , and tangent distribution $E = d π (D)$ . This uses existence of an atlas for an integrable distribution; it does not assert that all projected charts have a single transverse transition function on an entire overlap.

7.1F6F7step 2.1step 4.1step 5.2step 6.1given

The leaves are exactly the images of leaves of $F$ . Let $Q$ be the quotient leaf through $π (m)$ and $L$ the original leaf through $m$ . On each plaque patch of $L$ contained in a sheet, $π$ is an integral immersion for $E$ , so its image lies in one quotient leaf by [F6]. These patches cover the connected intrinsic manifold $L$ ; the inverse images of quotient leaves partition $L$ into open sets, so $π (L) \subseteq Q$ . Conversely, join $π (m)$ to any $z \in Q$ by a finite chain of quotient plaques. Subdivide each plaque path into finitely many pieces contained in sheets' images, using its compact parameter interval and the local plaque coordinates. Lift the first piece through $m$ , and each following piece through the preceding endpoint, using the inverse of $π$ on a sheet. Each lifted piece is an integral manifold patch of $D$ , since $d π$ identifies $D$ with $E$ , and therefore lies in one original leaf by [F6]. Consecutive pieces meet, so all lie in $L$ , and their final point maps to $z$ . Hence $Q = π (L)$ .

8.1F6F11step 2.1step 5.1step 5.2step 6.1step 7.1∎

Uniqueness of $F / Γ$ , and conclusion. If $F^{'}$ is a regular foliation of $M / Γ$ whose leaves are the sets $π (L)$ , then its tangent distribution $D^{'}$ satisfies $D_{b}^{'} = T_{b} (π (L))$ in its intrinsic leaf structure for $b = π (m)$ . More explicitly, apply the connected-plaque and countable-transverse-values argument of step 5.2 to a plaque of $F^{'}$ inside a chart of $F / Γ$ , and conversely to a plaque of $F / Γ$ inside a chart of $F^{'}$ . Since both foliations have the same leaf sets, these smooth inclusions give $D_{b}^{'} \subseteq E_{b}$ and $E_{b} \subseteq D_{b}^{'}$ . Hence $D_{b}^{'} = E_{b}$ ; since a regular foliation is determined by its tangent distribution and its leaves, $F^{'} = F / Γ$ . Thus step 2.1 gives claim 1 except uniqueness, step 5.1 gives that uniqueness, steps 6.1 and 7.1 give claim 2 with the codimension, and step 5.2 gives claim 3.

DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-6.1-sol)Open item page →

The suspension foliation of a representation of the fundamental group

Definition

Assume Countable Choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $B$ be a connected smooth manifold with base point $b_{0}$ (Smooth manifolds and their smooth charts), let $F$ be a smooth manifold, let $p : \tilde{B} \to B$ be a universal cover with a fixed point ${\tilde{b}}_{0}$ over $b_{0}$ (Universal covering spaces), and let $ρ : π_{1} (B, b_{0}) \to Diff (F)$ be a homomorphism into the diffeomorphism group of $F$ (Diffeomorphisms and local diffeomorphisms of manifolds). The universal cover has a canonical smooth manifold structure with $p$ a local diffeomorphism. To include the second-countability prerequisite, choose a countable cover of $B$ by coordinate balls: for each nonempty member of an enumerated basis that is contained in some coordinate ball, use ${A C}_{ω}$ to select one such ball. These selected balls cover $B$ because its coordinate balls form a neighborhood basis. Each is simply connected by its convex coordinates (Every nonempty convex subset of $R^{n}$ is contractible; a zero-dimensional connected base is a single point). Each pairwise intersection has at most countably many connected components, and they are path connected (Components of a topological manifold are open and at most countable). Using ${A C}_{ω}$ , choose a point in each nonempty overlap component and a path from it to a chosen center in each of the two balls. A subdivided loop can move every junction to the selected overlap point along a path in that component; paths with the resulting fixed endpoints inside a ball are homotopic by simple connectivity. Thus each loop class is represented by a finite word in countably many selected connecting paths. This proves that $π_{1} (B, b_{0})$ is at most countable (finite words over a countable set are countable under Countable unions of at most countable sets, assuming ${A C}_{ω}$ ). The sheets over each ball are indexed by this countable fibre. Lifting the countable coordinate cover gives a countable smooth atlas on $\tilde{B}$ : transitions are restrictions of base-chart transitions. The total space is Hausdorff, since points over different base points are separated downstairs, and points in one fibre lie in disjoint sheets. The lifted atlas also makes every deck transformation smooth with smooth inverse. The cover exists because $B$ is path connected, locally path connected and semilocally simply connected (Every nonempty path-connected locally path-connected semilocally simply connected space has a universal cover).

Using the isomorphism $π_{1} (B, b_{0}) ≅ Deck (\tilde{B} \to B)$ (For a path-connected locally path-connected semilocally simply connected base, the deck group of a universal cover is isomorphic to the fundamental group), take the deck identification defined by lifted endpoints starting at ${\tilde{b}}_{0}$ , and let $π_{1} (B, b_{0})$ act on the product $\tilde{B} \times F$ (Products of smooth manifolds have a canonical product smooth structure) diagonally by

$γ \cdot (x, y) : = (γ \cdot x, ρ (γ) (y)) .$

This is a free and properly discontinuous action by diffeomorphisms, so that the quotient-foliation proposition applies in the form recorded below. It is free: $γ \cdot (x, y) = (x, y)$ forces $γ x = x$ on the connected total space $\tilde{B}$ , and a deck transformation fixing a point is the identity (On a connected covering space, a deck transformation is determined by one point and the deck action is free). It is properly discontinuous: projections of compact sets are compact (A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism), and if a compact $K \subseteq \tilde{B} \times F$ meets $γ K$ then the compact projection $C \subseteq \tilde{B}$ of $K$ meets $γ C$ , so it suffices to show that ${γ : γ C \cap C \neq \emptyset}$ is finite for a compact $C \subseteq \tilde{B}$ . Suppose $γ_{1}, γ_{2}, \dots$ are distinct with points $x_{n}, γ_{n} x_{n} \in C$ for every $n$ . The manifold $\tilde{B}$ is locally Euclidean, hence first countable, and a sequence in a compact first-countable space has a convergent subsequence: the closed tails have the finite intersection property, so they have a common point, and a nested neighbourhood basis at that point produces the subsequence. Passing to subsequences twice we may therefore assume $x_{n} \to x$ and $γ_{n} x_{n} \to y$ with $x, y \in C$ (Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right). Let $W$ be a connected evenly covered coordinate neighbourhood of $p (x)$ and let $U$ be the sheet of $p^{- 1} (W)$ containing $x$ (Covering maps, evenly covered neighbourhoods, fibres, sheets, and trivial coverings); choose a connected open neighbourhood $Ω$ of $y$ with $p (Ω) \subseteq W$ on which $p$ is injective, which exists because $p$ is a local homeomorphism. For all large $n$ one has $x_{n} \in U$ and $γ_{n} x_{n} \in Ω$ . Now $γ_{n} (U)$ is connected with $p (γ_{n} (U)) = p (U) = W$ , so it is a sheet over $W$ ; and $Ω$ is connected with $p (Ω) \subseteq W$ , so $Ω$ lies in a single sheet over $W$ , which must be $γ_{n} (U)$ because $γ_{n} x_{n}$ lies in both. Hence $γ_{n} (U)$ is the same sheet $S$ over $W$ for all large $n$ , and $γ_{n} ∣_{U} = (p ∣_{S})^{- 1} \circ (p ∣_{U})$ for all large $n$ . Two deck transformations of the connected cover $\tilde{B}$ agreeing on the nonempty open set $U$ are equal (On a connected covering space, a deck transformation is determined by one point and the deck action is free), so $γ_{m} = γ_{n}$ for all large $m, n$ , contradicting distinctness. Hence only finitely many $γ$ meet $C$ , and therefore only finitely many meet $K$ . The action is in particular a covering-space action (Covering-space actions by disjoint translates of neighbourhoods, The deck group of a connected covering acts by a covering-space action).

The product foliation of $\tilde{B} \times F$ by the leaves $\tilde{B} \times {y}$ ( $y \in F$ ) is a regular foliation of codimension $\dim F$ , and it is invariant under the action, since $γ \cdot (\tilde{B} \times {y}) = \tilde{B} \times {ρ (γ) y}$ . By The quotient foliation under a free and properly discontinuous foliated action the quotient $M_{ρ} : = (\tilde{B} \times F) / π_{1} (B, b_{0})$ is therefore a smooth manifold, the orbit map $π : \tilde{B} \times F \to M_{ρ}$ is a covering map, and the product foliation descends to a regular foliation $F_{ρ}$ of $M_{ρ}$ of codimension $\dim F$ . This foliation is the suspension foliation of the representation $ρ$ .

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

Suspension holonomy is the germ of the represented monodromy action

Statement

Assume Countable Choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). In the suspension $M_{ρ}$ of a representation $ρ : π_{1} (B, b_{0}) \to Diff (F)$ as in The suspension foliation of a representation of the fundamental group let $γ : [0, 1] \to B$ be a loop at $b_{0}$ with class $[γ] \in π_{1} (B, b_{0})$ , let $\tilde{x} = {\tilde{b}}_{0}$ be the fixed point of $\tilde{B}$ used for the deck identification in that definition, let ${\tilde{x}}_{γ}$ be the lift of $γ$ with ${\tilde{x}}_{γ} (0) = \tilde{x}$ , so that ${\tilde{x}}_{γ} (1) = [γ] \cdot \tilde{x}$ Based loops and the fundamental group, and let $y \in F$ . Then $a (t) : = π ({\tilde{x}}_{γ} (t), y)$ is a leafwise path of $F_{ρ}$ , and its holonomy germ between the local transversal $T = π ({\tilde{x}} \times F)$ at the start and the corresponding slice at the end is the germ at $y$ of $ρ ([γ])^{- 1}$ : $h_{a} = {germ}_{y} (ρ ([γ])^{- 1}) .$ Moreover the leaf $L_{y} = π (\tilde{B} \times {y})$ is diffeomorphic to $\tilde{B} / K_{y}$ with $K_{y} = {γ : ρ (γ) y = y}$ the stabiliser of $y$ , and under $π_{1} (L_{y}) ≅ K_{y}$ the holonomy representation of the leaf (as a homomorphism with traversal-order loop multiplication) is $γ \mapsto {germ}_{y} (ρ (γ))$ . The forward-path holonomy is the inverse germ. Both maps have the same image and kernel.

Facts & Assumptions

Given: A representation $ρ$ of $π_{1} (B, b_{0})$ in the diffeomorphism group of $F$ , the diagonal action on $\tilde{B} \times F$ , the quotient $M_{ρ} = (\tilde{B} \times F) / π_{1} (B, b_{0})$ with orbit map $π$ and suspension foliation $F_{ρ}$ , a loop $γ$ at $b_{0}$ , a point $\tilde{x} \in \tilde{B}$ over $b_{0}$ , its lift ${\tilde{x}}_{γ}$ with ${\tilde{x}}_{γ} (1) = [γ] \cdot \tilde{x}$ , and a point $y \in F$ .

[F1]

In the suspension the quotient $M_{ρ}$ is a smooth manifold, $π$ is a covering map and a local diffeomorphism, and the product foliation of $\tilde{B} \times F$ with leaves $\tilde{B} \times {y^{'}}$ descends to the regular foliation $F_{ρ}$ whose leaves are the images of those product leaves (The suspension foliation of a representation of the fundamental group, The quotient foliation under a free and properly discontinuous foliated action).

[F2]

The slice ${\tilde{x}} \times F$ is a local transversal to the product foliation at each of its points: the product foliation has tangent distribution $T \tilde{B} \times {0}$ and the slice has tangent space ${0} \times T F$ , a complementary direct summand (Local transversals to a regular foliation, Plaques of a flat chart).

[F3]

In a product chart of the product foliation the plaques keep the $F$ -coordinate fixed, so the transport between two slices of the form ${{\tilde{x}}_{0}} \times F$ and ${{\tilde{x}}_{1}} \times F$ along a leafwise path in a leaf $\tilde{B} \times {y}$ keeps the second coordinate: it sends $({\tilde{x}}_{0}, y_{0})$ to $({\tilde{x}}_{1}, y_{0})$ (The holonomy germ is independent of the foliation chart chain, Plaques of a flat chart).

[F4]

The $π_{1} (B, b_{0})$ -action on $\tilde{B} \times F$ is diagonal, $γ_{0} \cdot ({\tilde{x}}_{0}, y_{0}) = (γ_{0} {\tilde{x}}_{0}, ρ (γ_{0}) y_{0})$ , so $(γ_{0} {\tilde{x}}_{0}, y_{0})$ and $({\tilde{x}}_{0}, ρ (γ_{0})^{- 1} y_{0})$ lie in the same orbit, and $π$ identifies them; moreover $π ({\tilde{x}}_{0}, y_{0}) = π ({\tilde{x}}_{1}, y_{1})$ holds exactly when $({\tilde{x}}_{1}, y_{1}) = γ_{0} \cdot ({\tilde{x}}_{0}, y_{0})$ for some $γ_{0}$ (The suspension foliation of a representation of the fundamental group, The quotient foliation under a free and properly discontinuous foliated action).

[F5]

Holonomy germs are well defined, invariant under leafwise homotopy relative to endpoints, and multiplicative under concatenation (The holonomy germ is independent of the foliation chart chain, Holonomy depends only on leafwise homotopy relative to endpoints, Holonomy respects path concatenation and reversal).

[F7]

Intrinsic leaves are integral immersions with plaque charts (Existence and uniqueness of maximal connected integral manifolds). With traversal-order loop multiplication, the holonomy representation uses the inverse of forward-path holonomy (The holonomy representation and the holonomy group of a leaf).

[F6]

For a covering-space action of a group $G$ on a path-connected space $E$ the orbit map $E \to E / G$ is a covering whose deck group consists exactly of the transformations supplied by $G$ ; for a universal cover the deck group is isomorphic to the fundamental group of the base, the isomorphism moving a chosen fibre point to the lifted endpoint of the corresponding loop (The orbit map of a covering-space action is a covering, with the acting group equal to the deck group when the total space is path-connected, For a path-connected locally path-connected semilocally simply connected base, the deck group of a universal cover is isomorphic to the fundamental group, Universal covering spaces).

Proof

technique · direct

1.1F1F2construct

The lifted path is leafwise. The path $\tilde{t} \mapsto ({\tilde{x}}_{γ} (t), y)$ lies in the product leaf $\tilde{B} \times {y}$ ; applying the local diffeomorphism $π$ by [F1] gives a leafwise path $a (t) = π ({\tilde{x}}_{γ} (t), y)$ of $F_{ρ}$ . Both $T = π ({\tilde{x}} \times F)$ and the end slice $π ({{\tilde{x}}_{γ} (1)} \times F) = π ({[γ] \tilde{x}} \times F)$ are local transversals to $F_{ρ}$ at the endpoints, being local diffeomorphic images of the transversals of [F2].

1.2F3

Transport upstairs. In the product foliation the transport along the path $t \mapsto ({\tilde{x}}_{γ} (t), y)$ from the slice ${\tilde{x}} \times F$ to the slice ${{\tilde{x}}_{γ} (1)} \times F$ keeps the $F$ -coordinate by [F3]: a point $(\tilde{x}, y_{0})$ is sent to $({\tilde{x}}_{γ} (1), y_{0})$ .

1.3F1F4F6F7construct

The leaf through $y$ . The restriction $\tilde{B} \times {y} \to M_{ρ}$ is tangent to the descended distribution. In the quotient's local product charts it maps into plaque neighborhoods of the intrinsic leaf $L_{y}$ , so it factors smoothly as $Φ : \tilde{B} \to L_{y}$ and is a local diffeomorphism between manifolds of dimension $\dim B$ by [F7]. By [F4], its fibres are exactly the $K_{y}$ -orbits. The group $K_{y}$ acts freely and properly discontinuously on $\tilde{B}$ by the deck action; the quotient proposition supplies its smooth quotient and orbit local diffeomorphism. Hence $Φ$ descends to a bijective local diffeomorphism $\tilde{B} / K_{y} \to L_{y}$ , and is a diffeomorphism. In particular $\tilde{B} \to L_{y}$ is the corresponding orbit covering.

2.1F1F3F4step 1.1step 1.2

Descending the transport. In $M_{ρ}$ the point $({\tilde{x}}_{γ} (1), y_{0}) = ([γ] \tilde{x}, y_{0})$ is identified by [F4] with $(\tilde{x}, ρ ([γ])^{- 1} y_{0})$ . Since $π$ is a local diffeomorphism and the holonomy germ of $a$ is computed by transporting along the descended local product structure, which is the corresponding chart-wise transport, the holonomy germ $h_{a}$ satisfies, after identifying both the start and the end transversal with $F$ through the maps $y_{0} \mapsto π (\tilde{x}, y_{0})$ , $h_{a} = {germ}_{y} (y_{0} \mapsto ρ ([γ])^{- 1} y_{0}) .$

2.2F6step 1.3

The fundamental group of the leaf. The group $K_{y}$ acts on $\tilde{B}$ by a covering-space action (the restriction of the deck action, which consists of homeomorphisms over $B$ ) and $\tilde{B}$ is path-connected and simply connected, being a universal cover; the covering $\tilde{B} \to \tilde{B} / K_{y} ≅ L_{y}$ is then a universal cover of $L_{y}$ whose deck group consists exactly of the transformations from $K_{y}$ by [F6]. Hence $π_{1} (L_{y}) ≅ K_{y}$ .

3.1F5F7step 1.3step 2.1step 2.2

The holonomy representation of the leaf. For $γ \in K_{y}$ , the projected path associated to a based loop representing $γ$ closes because $ρ (γ) y = y$ , and corresponds to $γ$ under step 2.2. Its forward holonomy is ${germ}_{y} (ρ (γ)^{- 1})$ by step 2.1. The representation in [F7] uses the reversed loop, and thus takes the inverse germ, namely ${germ}_{y} (ρ (γ))$ . This is a homomorphism on $K_{y}$ ; the unreversed transport is an antihomomorphism.

4.1step 1.1step 2.1step 1.3step 2.2step 3.1∎

Conclusion. Steps 1.1 and 2.1 give the stated holonomy germ $h_{a} = {germ}_{y} (ρ ([γ])^{- 1})$ of a base loop, and steps 1.3, 2.2 and 3.1 give the description of the leaf $L_{y}$ as $\tilde{B} / K_{y}$ together with its holonomy representation.

RemarkRemark: Literature-sourcedProof: Not applicableOpen item page →

Holonomy is a germ, not a globally defined return map

Remark

Assume Countable Choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). The holonomy of a leafwise path is a germ of a local diffeomorphism between local transversals, not a globally defined return map: a representative is defined only on some open neighbourhood of the base point in the transversal, and different representatives of the same germ may differ arbitrarily far from that point. Indeed, the construction of A leafwise path determines a germ of a transverse diffeomorphism composes finitely many chart transports, each of which is produced by the inverse function theorem on a neighbourhood of one point of the path; the resulting map is defined only on a neighbourhood of the base point, and The holonomy germ is independent of the foliation chart chain compares only germs, so nothing in the construction defines values away from the base point, and no global continuation is asserted.

The Poincaré return map of a periodic orbit of a flow is the special case in which the first-return construction defines a map on a fixed section; that is additional structure, not part of the general definition. Statements about "the" return map along a leaf loop must therefore be read as statements about the holonomy germ, as in The holonomy representation and the holonomy group of a leaf, where only the germ $ρ_{x} ([a]) = h_{a^{- 1}} (T, T)$ is used.

5 · Examples, counterexamples and false statements

None yet.

Foliation Holonomy and the Holonomy Groupoid

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

Local transversals to a regular foliation

Definition

Leafwise paths and leafwise homotopy relative to endpoints

Definition

Leaf topology under Countable Choice

Germs of local diffeomorphisms at a point

Definition

Germs of local diffeomorphisms at a point form a group

Statement

Facts & Assumptions

Proof

A leafwise path determines a germ of a transverse diffeomorphism

Statement

Facts & Assumptions

Proof

The holonomy germ is independent of the foliation chart chain

Statement

Facts & Assumptions

Proof

Holonomy depends only on leafwise homotopy relative to endpoints

Statement

Facts & Assumptions

Proof

Holonomy respects path concatenation and reversal

Statement

Facts & Assumptions

Proof

The holonomy representation and the holonomy group of a leaf

Definition

The deck group of a connected covering acts by a covering-space action

Statement

Facts & Assumptions

Proof

The covering of a leaf associated with the holonomy kernel exists

Statement

Facts & Assumptions

Proof

The holonomy cover of a leaf

Definition

The monodromy groupoid of a foliation

Definition

The holonomy groupoid of a foliation

Definition

Holonomy classes form a groupoid congruence

Statement

Facts & Assumptions

Proof

The isotropy of the holonomy groupoid is the leaf holonomy group

Statement

Facts & Assumptions

Proof

Smooth maps transverse to a regular foliation

Definition

The pullback foliation under a transverse map

Statement

Facts & Assumptions

Proof

The quotient foliation under a free and properly discontinuous foliated action

Statement

Facts & Assumptions

Proof

The suspension foliation of a representation of the fundamental group

Definition

Suspension holonomy is the germ of the represented monodromy action

Statement

Facts & Assumptions

Proof

Holonomy is a germ, not a globally defined return map

Remark

5 · Examples, counterexamples and false statements

Sources