# nLab foliation

Contents

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## Idea

A foliation of a manifold $X$ is a decomposition into submanifolds. These submanifolds are called the leaves of the foliation and one says that $X$ is foliated by the leaves. In order to have a useful notion, leaves are required to behave sufficiently well locally. In particular, if all leaves have the same dimension, then one speaks of a regular foliation, which is the case discussed here. If the dimension of leaves is allowed to vary, one speaks instead a singular foliation; see there for more details.

For smooth manifolds, smooth foliations are decompositions into immersed submanifolds that locally can be expressed as fibers of a submersion (the projection to the space of leaves).

Historically (as they were introduced in Ehresmann and Reeb), one considered foliations for smooth manifolds $X$ as arising from subbundles of the tangent bundle $E \hookrightarrow T X$ that are integrable distributions in the sense that the Lie bracket of vector fields that are sections of $E$ is again a section of $E$: the leaves are the submanifolds whose tangent vectors are sections of $E$. If one thinks of $E$ as encoding a differential equation, then the leaves are the solution spaces to this equation.

Expressed in terms of higher Lie theory such an integrable distribution is a sub-Lie algebroid of the tangent Lie algebroid of $X$. Accordingly, under Lie integration of this structure foliations of $X$ are also equivalently encodes as Lie groupoids whose space of objects is $X$ and whose orbits are the leaves of the foliation.

Moreover, foliations are classified by Čech cohomology cocycles with coefficients in a topological groupoid/Lie groupoid called the Haefliger groupoid. These relations make foliation theory of sub-topic of Lie groupoid-theory. See also at motivation for higher differential geometry.

The Haefliger groupoids in fact classifies structures slightly more general than foliations: Haefliger structures.

## Definition

There are several equivalent definitions of foliations.

### Original definition

Let $M$ be an $n$-dimensional topological manifold. A decomposition of $M$ as a disjoint union of connected subsets $V_\alpha$, called leaves,

$M = \cup_\alpha V_\alpha$

is called a foliation if there is a cover of $M$ by a collection of “special” charts of the form $(U, \phi)$, $\phi = (\phi_1,\ldots,\phi_n) : U \to \mathbb{R}^n$ such that for each “special” chart and each $\alpha$ there is a number $p\leq n$, called the dimension of the foliation, such that the intersection of any given leaf $V_\alpha$ with $U$ is one of the level sets, i.e. the solution of the system $\phi_r(x) = const = const(r,U,\alpha)$ for all $r\gt p$.

If the manifold is a smooth manifold, the charts may be required to be smooth too, to obtain the notion of a smooth foliation or foliation in differential geometry. In this case, the $p$-dimensional foliations with underlying manifold $X$ are in 1-1 correspondence with integrable distributions of hyperplanes of dimension $p$ in the tangent bundle of $X$.

### Alternative definitions

The following equivalent definitions and their relation are discussed for instance in (IfLg, 1.2).

###### Definition

A foliation atlas of a manifold $X$ of dimension $n$ and leaf-codimension $q$ is an atlas $\{\phi_i^{-1}: R^n \to X\}_i$ such that the transition functions are globally of the form

$\phi_{i j} : (x,y) \mapsto (g_{i j}(x,y), h_{i j}(y))$

with respect to the canonical decomposition $\mathbb{R}^n = \mathbb{R}^{n-q} \times \mathbb{R}^q$.

$\array{ \mathbb{R}^n &\stackrel{\phi_{i j}}{\to}& \mathbb{R}^n && \\ \downarrow && \downarrow \\ \mathbb{R}^q &\stackrel{h_{i j}}{\to}& \mathbb{R}^q } \,.$
###### Definition

A foliation atlas of a manifold $X$ of dimension $n$ and leaf-codimension $q$ is an open cover $\{U_i \to X\}_i$ of $X$ equipped with submersions $\{ s_i \colon U_i \to \mathbb{R}^q \}$ such that there exists diffeomorphisms

$\gamma_{i j} \colon s_j(U_i \cap U_j) \to s_i(U_i \cap U_j)$

satisfying on each $U_i \cap U_j$ the condition

$s_i = \gamma_{i j} \circ s_j \,.$
###### Remark

Given a foliation atlas as in def. , the diffeomorphisms $\{\gamma_{i j}\}_{i,j}$ satisfy the Cech cocycle condition

$\gamma_{i j} \circ \gamma_{j k} = \gamma_{i k} \,.$

This is called the Haefliger cocycle of the foliation atlas.

###### Definition

A smooth foliation of a smooth manifold $X$ is equivalently an integrable distribution (or an integrable subbundle) $E \hookrightarrow T X$.

### In terms of Lie algebroids and Lie groupoids

Definition above is immediately reformulated equivalently as the following statement in higher Lie theory.

###### Definition

For $X$ a smooth manifold, a foliation of $X$ is equivalently a Lie algebroid over $X$ such that the anchor map is an injection.

###### Remark

The Lie groupoids which under Lie differentiation give rise to Lie algebroids with injective anchors as in def. are precisely those which are Morita-equivalent to étale groupoids (hence are the foliation groupoids, see there for more details) (Crainic-Moerdijk 00, theorem 1).

One says:

###### Definition

A Lie groupoid integrates a given foliation, if it Lie integrates the coresponding Lie algebroid, according to def. .

###### Example

For a simple foliation $\mathcal{D}$ of a manifold $X$, example , hence one where there is a submersion

$p_{\mathcal{D}} \;\colon\; X \to X/\mathcal{D}$

to the leaf space, that map itself is the atlas of a Lie groupoid $\mathcal{G}$ which integrates the foliation, which is the Cech nerve

$\mathcal{G}_\bullet = \left( X \underset{X/\mathcal{D}}{\times} X \stackrel{\to}{\to} X/\mathcal{D} \right) \,.$
###### Example

Among all Lie groupoids that integrate a given foliation $\mathcal{F}$ of a manifold $X$, the two special extreme

1. holonomy groupoid$Hol(X,\mathcal{F})_\bullet$

2. monodromy groupoid$Monod(X,\mathcal{F})_\bullet$

###### Proposition

Let $\mathcal{G}_\bullet$ be a Lie groupoid with (for simplicity) connected source-fibers.

Then there are maps

$hol \;\colon\; Monod(X,\mathcal{F})_\bullet \stackrel{g_{\mathcal{G}}}{\to} \mathcal{G}_\bullet \stackrel{hol_G}{\to} Hol(X,\mathcal{F})$

which are surjective local diffeomorphisms and such that the composite is the holonomy morphism (…).

This is (Crainic-Moerdijk 00, prop. 1).

### Of higher smooth spaces

One can consider the generalization of the notion of foliation of manifolds to foliations of structures in higher differential geometry such as Lie groupoids and Lie algebroids. See at

### In terms of differential cohesive higher geometry

The following is a suggestion for an axiomatization of foliations in higher differential geometry in the formalization of differential cohesion, followed by some considerations showing how these axioms reproduce traditional theory.

Under construction.

###### Definition (Notation)

Let $\mathbf{H}$ be a cohesive (∞,1)-topos equipped with differential cohesion.

As usual, we write $(\int \dashv \flat \dashv \sharp)$ for the adjoint triple of modalities that defines the cohesion (shape modality $\dashv$ flat modality $\dashv$ sharp modality) and we write $(Red \dashv \int_{inf} \dashv \flat_{inf})$ for the adjoint triple of modalities that defines the differential cohesion (reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality).

Below we are going to axiomatize aspects of the traditional description of foliations by Lie groupoids/foliation groupoids as discussed above, so we start by briefly setting up some terminology on groupoid objects in differential cohesion.

###### Definition (Notation for groupoid objects and atlases)

By the Giraud-Rezk-Lurie axioms we may think of a 1-epimorphism $\mathcal{G}_0 \to \mathcal{G}$ in $\mathbf{H}$ as an atlas of the cohesive $\infty$-groupoid $\mathcal{G} \in \mathbf{H}$, exhibiting equivalently the corresponding groupoid object which we write

$\mathcal{G}_\bullet \coloneqq \mathcal{G}_0^{\times^{\bullet+1}_{\mathcal{G}}} \,.$

Hence we use notation where omitting the subscript decorationon a groupoid object $\mathcal{G}_\bullet \in \mathbf{H}^{\Delta^{op}}$ refers to its realization

$\mathcal{G} \coloneqq {\underset{\rightarrow}{\lim}}_n \mathcal{G}_{n} \;\;\; \in \mathbf{H} \,.$

We have the following “geometricity” constraints on groupoid objects.

###### Definition

For $f \colon X \to Y$ any morphism in $\mathbf{H}$, write

$X \stackrel{L(f)}{\to} Y \underset{\int_{inf} Y}{\times} \int_{inf} X$

for the canonical morphism induced by the naturality of the $\int_{inf}$-unit. We say that

1. $f$ is a formally smooth morphism (or submersion) if $L(f)$ is a 1-epimorphism;

2. $f$ is a formally étale morphism (or local diffeomorphism) if $L(f)$ is an equivalence.

Now if $\pi \colon \mathcal{G}_0 \to \mathcal{G}$ is a 1-epimorphism, hence an atlas for the cohesive $\infty$-groupoid $\mathcal{G}$, then we say about the corresponding groupoid object as in def. , that

1. $\mathcal{G}_\bullet$ is an geometric ∞-groupoid if its atlas $\pi$ is a formally smooth morphism/submersion.

2. $\mathcal{G}_\bullet$ is an étale ∞-groupoid if its atlas $\pi$ is a formally étale morphism/local diffeomorphism.

###### Definition

For $X \in \mathbf{H}$, a foliation of $X$ is a morphism $\mathcal{D} \colon X \to X//\mathcal{D}$ in $\mathbf{H}$ which is

Equivalently a foliation of $X$ is a map that exhibits $X$ as an atlas for a geometric ∞-groupoid, def. .

Given a foliation $\mathcal{D}$ on $X$ we say that the leaf decomposition of $X$ induced by the foliation is the (∞,1)-pullback

$LeafDec(\mathcal{D}) \coloneqq \flat(X//\mathcal{D}) \underset{X//\mathcal{D}}{\times} X$

in

$\array{ LeafDec(\mathcal{D}) &\stackrel{\iota_{\mathcal{D}}}{\to}& X \\ \downarrow && \downarrow^{\mathrlap{\mathcal{D}}} \\ \flat (X//\mathcal{D}) &\to& X//\mathcal{D} } \,,$

where the bottom map is the counit of the flat modality.

Now let $\mathbb{G} \in Grp_2(\mathbf{H})$ a braided ∞-group. Write

$\Omega^2_{cl} := \Omega^2_{flat}(-,\mathbb{G})$

for the corresponding coefficient object for curvature forms of $\mathbb{G}$-principal ∞-connections (as discussed there).

###### Definition

Given a closed 2-form

$\omega \;\colon\; X \to \Omega^2_{cl}$

a foliation of $X$ by $\omega$-isotropic subspaces is a foliation $\mathcal{D} \colon X \to X//\mathcal{D}$ as in def. such that the restriction of $\omega$ to the leaf decomposition is equivalent to the 0-form

$\iota_{\mathcal{D}}^* \omega \simeq 0 \,,$

hence such that the top composite morphism in the diagram

$\array{ (\flat \mathcal{E}) \underset{\mathcal{E}}{\times} X &\to& X &\stackrel{\omega}{\to}& \Omega^2_{cl} \\ \downarrow && \downarrow \\ \flat (X//\mathcal{D}) &\to& X//\mathcal{D} }$

factors through the point.

We now discuss how low-degree examples of this axiomatics interpreted in $\mathbf{H} \coloneqq$ SynthDiff∞Grpd reproduces the traditional notions of foliations and isotropic submanifolds of pre-symplectic manifolds.

In the following we regard smooth manifolds canonically under the embedding

SmoothMfd$\hookrightarrow$ Smooth∞Grpd $\stackrel{i_!}{\hookrightarrow}$ SynthDiff∞Grpd $= \mathbf{H}$

###### Example

A smooth function $f \colon X \to Y$ between smooth manifolds is

1. a local diffeomorphism in the traditional sense precisely if it is a formally étale morphism in the sense of def. ;

2. a submersion in the traditional sense precisely if it is a formally smooth morphism in the sense of def. .

This is discussed at SynthDiff∞Grpd. The idea of the proof is to use the ∞-cohesive site of definition CartSp${}_{synthdiff}$ and evaluate the homotopy pullback in def. first on all representables of the form $U \times D_1$ where $U$ ranges over Cartesian spaces and where $D_1$ is the first order ininfitesimal neighbourhood of the origin on $\mathbb{R}^1$ (whose smooth algebra of fucntions is the ring of dual numbers). Then the homotopy pullback is represented as an ordinary pullback of sheaves over Cartesian spaces and the naturality diagram in question is the diagram of tangent bundles

$\array{ T X &\stackrel{d f}{\to}& T Y \\ \downarrow && \downarrow \\ X &\stackrel{f}{\to}& Y } \,.$

With this now the claim is reduced to the traditional characterization of submersions and local diffeomorphisms.

###### Example

Let $\mathcal{G}$ be a smooth groupoid which has a presentation by a simplicial presheaf $\mathcal{G}_\bullet$ with values in 2-coskeletak Kan complexes where objects and morphisms are represented by a smooth manifold each, and consider it equipped with the induced atlas $\mathcal{G}_0 \to \mathcal{G}$. Then

• if the presentation $\mathcal{G}_\bullet$ is a Lie groupoid then $\mathcal{G}_0 \to \mathcal{G}$ is a geometric ∞-groupoid

• if the presentation $\mathcal{G}_\bullet$ is an étale groupoid then $\mathcal{G}_0 \to \mathcal{G}$ is an étale ∞-groupoid

in the sense of def. .

This follows by the corresponding discussion at SynthDiff∞Grpd. The idea of the proof is that one presents the atlas in the projective model structure on simplicial presheaves by the décalage fibration resolution, schematically

$\array{ && g \\ & \swarrow && \searrow & &&& &&& \mathcal{G}_0 \\ g_1 &&\to&& g_2 \\ \\ &&&& &&&&&& \downarrow \\ \\ g_1 &&\to&& g_2 &&&&&& \mathcal{G} } \,.$

Then the homotopy pullback $\mathcal{G} \underset{\int_{inf}\mathcal{G}} {\times}\int_{inf} X$ is presented by an ordinary pullback and so example applies degreewise. In degree 0 the above resolution is the target map in the groupoid $\mathcal{G}$ and so by example this is a submersion or local diffeomorphism, respectively, as claimed.

###### Example

Let $X$ be a smooth manifold and let $\mathcal{D}$ be a traditional foliation on $X$ which is a simple foliation, example , in that the leaf space $X/\mathcal{D}$ exists as a smooth manifold and the projection map $X \to X/\mathcal{D}$ is a submersion.

Then by the discussion at synthetic differential ∞-groupoid, this projection map is also a formally smooth morphism in $\mathbf{H}$ according to def. . Moreover, being a quotient projection it is a 1-epimorphism and hence exhibits the corresponding foliation groupoid

$\left( X \underset{X/\mathcal{D}}{\times} X \stackrel{\to}{\to} X/\mathcal{D} \right)$

as a geometric ∞-groupoid in the sense of def. .

Now $\flat ( X// \mathcal{D})$ is the underlying set of points of the leaf space, regarded as a discrete ∞-groupoid. So we have the pasting diagram of pullbacks

$\array{ L_{l} &\to& \coprod_{l \in X/\mathcal{D}} L_l &\to& X \\ \downarrow && \downarrow && \downarrow^{\mathrlap{\mathcal{D}}} \\ {*} &\stackrel{\vdash l}{\to}& \coprod_{l \in D/\mathcal{X}}{*} &\to& X/\mathcal{D} }$

for every leaf $L_l$ labeled by the point $l \in X/\mathcal{D}$ in leaf space, which exhibits the leaf decomposition of $X$ under $\mathcal{D}$ according to def. as the disjoint union of the leaves of $(X,\mathcal{D})$ in the traditional sense, injected into $X$ in the canonical way.

###### Example

Consider now $\mathcal{G}_\bullet$ any Lie groupoid, hence in particular a smooth groupoid $\mathcal{G} \in \mathbf{H}$ equipped with an atlas $\mathcal{G}_0 \to \mathcal{G}$, which hence by example exhibits a geometric ∞-groupoid in the sense of def. , hence a foliation $\mathcal{D} \;\colon\; \mathcal{G}_0 \to \mathcal{G}$ in the sense of def. .

Computation of the homotopy pullback

$\array{ LeafDec_{\mathcal{D}}(\mathcal{G}_0) &\to& \mathcal{G}_0 \\ \downarrow && \downarrow^{\mathrlap{\mathcal{D}}} \\ \flat( \mathcal{G} ) &\to& \mathcal{G} & = \mathcal{G}_0//\mathcal{D} }$

by the method as in example shows that $LeafDec_{\mathcal{D}}(\mathcal{G}_0)$ is the smooth groupoid presented by the presheaf of groupoids whose

• smoothyl $U$-parameterized collection of objects are smoothly $U$-parameterized collections of morphisms $\{g_0 \to g(u)\}_{u \in U}$ in $\mathcal{G}_\bullet$ with $g_0$ held fixed;

• morphisms are given by precomposing these collections with a fixed (not varying with $U$) morphism in $\mathcal{G}_\bullet$.

This means that if $\mathcal{G}_\bullet$ is an étale groupoid to start with, then $LeafDec_{\mathcal{D}}(\mathcal{G}_0)$ is the disjoint union of all its orbit leaves (as smooth manifolds), hence that the abstractly defined $LeafDec_{\mathcal{D}}(\mathcal{G}_0)$ reproduces the decomposition of $\mathcal{G}_0$ by the foliation encoded by the foliation groupoid $\mathcal{G}_\bullet$ as in traditional theory.

###### Remark

We may suggestively summarize example in words as:

“In cohesive higher geometry, every foliation is a simple foliation.”

Because the quotient map to the leaf space of a general foliation is always a submersion/formally smooth morphism, just not always onto a manifold, but onto a higher space.

###### Remark

If the $\mathcal{G}_\bullet$ in example is not an étale groupoid to start with but a more general Lie groupoid, then $LeafDec_{\mathcal{D}}(\mathcal{G}_0)$ in general retains information of non-discrete isotropy groups of $\mathcal{G}_\bullet$.

We might decide to rule out this possibility by adding to the axioms in def. the clause that $X//\mathcal{G}$ (here $\mathcal{G}_0//\mathcal{D}$) be étale.

However, we might also keep that case and regard it as the first instance of what is certainly a natural phenomenon as we pass to higher geometry, namely that leaves of a foliation no longer need to manifolds but will be (higher) groupoids themselves.

Finally, given the above it is clear how isotropic appear in the cohesive axiomatics.

###### Example

For $\mathbb{G} = U(1)$ the smooth circle group, $\Omega^2_{cl}$ is the ordinary sheaf of closed differential 2-forms under the canonical embedding

$Sh(CartSp) \hookrightarrow Sh_\infty(CartSp) \simeq Smooth\infty Grpd \stackrel{i_!}{\hookrightarrow} SynthDiff\infty\mathrm{Grpd} \,.$

Then for $X$ a smooth manifold a morphism $\omega \;\colon\; X \to \Omega^2_{cl}$ is equivalently a differential 2-form.

Then for $\mathcal{D} \;\colon\; X \to X//\mathcal{D}$ a traditional foliation of $X$ regarded as a foliation in $SynthDiff\infty Grpd$ by example , it follows with the discussion there that $\iota_{\mathcal{D}}^* \omega$ is precisely the collection of restriction of $\omega$ to each of the leaves of the foliation. Therefore this is a foliation by isotropic submanifolds in the traditional sense precisely if it is an $\omega$-isotropc foliation in the sense of def. .

## Examples

###### Example

For $X \to Y$ a submersion of smooth manifolds, the connected fibers of the submersion constitute a foliation of $X$ whose codimension is the dimension of $Y$. Foliations of this form are called simple foliations.

###### Example

Every Lie groupoid gives a folitation on its space of objects: the leaves are the orbits. Conversely, every regular foliation gives rise to its holonomy groupoid. This is a (not necessarily Hausdorff) Lie groupoid whose orbits are the leaves of the original foliation, and which in some sense is minimal with this condition.

###### Example

Every Poisson manifold has a canonical structure of a foliation whose leaves are its maximal symplectic submanifolds, called symplectic leaves.

## Properties

### Leaf space

The set of components of a foliation is typically non-Hausdorff, which is one of the motivations of the Connes-style noncommutative geometry.

### Classification

Folitation are classified by the Haefliger groupoid. See at Haefliger theorem.

### Characteristic classes

There is a theory of characteristic classes for foliations. A most well known example is the Godbillon-Vey characteristic class? for codimension one foliations.

The notion of foliated manifolds was introduced in the 1950s, motivated from partial differential equation theory, in

• Reeb, …

• Eli Cartan, Sur l’intégration des équations différentiels completement intégrable, Oeuvres Complètes, Pt. II, Vol. I, 555-561.

A discussion in differential geometry is in

• Robert Hermann, On the differential geometry of foliations, Annals of Mathematics, Second Series, Vol. 72, No. 3 pp. 445-457 (jstor)

A textbook account with a view to the modern formulation in Lie groupoid theory is

Foliations in Lie groupoid theory are discussed in more detail in

The corresponding groupoid algebras are discussed in chapter 2, section 8 of

A survey by Fuks in Russian Itogi:

• Д. Б. Фукс, Слоения, Итоги науки и техн. Сер. Алгебра. Топол. Геом., 1981, том 18, стр. 151–-213, pdf

Cohomology of formal vector fields and characteristic classes of foliations were originally studied in the papers

• D. B. Fuks, Cohomology of infinite-dimensional Lie algebras and characteristic classes of foliations (book, Rus. and Eng. versions)

• I. M. Gelʹfand, B. L. Feĭgin, D. B. Fuks, Cohomology of the Lie algebra of formal vector fields with coefficients in its dual space and variations of characteristic classes of foliations, Funkcional. Anal. i Priložen. 8 (1974), no. 2, 13–29 (Russian original mathnet.ru, pdf)

• Claude Godbillon, Cohomologies d’algèbres de Lie de champs de vecteurs formels, Séminaire Bourbaki, 25ème année (1972/1973), Exp. No. 421, pp. 69–87. Lecture Notes in Math. 383, Springer 1974.

• И. М. Гельфанд, Д. Б. Фукс, Когомологии алгебры Ли формальных векторных полей, Изв. АН СССР. Сер. матем., 1970, 34, в. 2, стр. 322–-337, pdf

In a series of works of Connes and Moscovici, the local index formulas in the context of transverse geometry of foliations has been studied in connection to a new cyclic homology of a Hopf algebra arising in this context:

• A. Connes, H. Moscovici, Modular Hecke algebras and their Hopf symmetry, Mosc. Math. J., 4:1 (2004), 67–109; math.QA/0301089, ams; Hopf algebras, cyclic cohomology and the transverse index theory, math.DG/9806109, Comm. Math. Phys. 198, n.1, 1998 MR99m:58186 doi; Rankin-Cohen brackets and the Hopf algebra of transverse geometry, Mosc. Math. J., 4:1 (2004), 111–130; Differentiable cyclic cohomology and Hopf algebraic structures in transverse geometry, in: Essays on geometry and related topics, Vol. 1, 2, Monogr. Enseign. Math. 38, p. 217–255. Enseignement Math., Geneva, 2001 MR2003k:58042
• A. Connes, Cyclic cohomology and the transverse fundamental class of a foliation, in: Geom. methods in operator algebras (Kyoto, 1983), Pitman Res. Notes in Math. 123, p. 52–144, 1986 MR88k:58149

More general issues of index theory in noncommutative geometry applied to foliations is in

• Yu. A. Kordyukov, Noncommutative geometry of foliations, J. K-Theory, 2:2, Special issue in memory of Y. P. Solovyev, Part 1 (2008), 219–327 MR2009m:58018; Index theory and non-commutative geometry on foliated manifolds, Russian Math. Surveys, 64:2 (2009), 273–391 (original: Ю. А. Кордюков, УМН, 64:2(386) (2009), 73–202); Формула следов для трансверсально-эллиптических операторов на римановых слоениях, Алгебра и анализ, 12:3 (2000), 81–105 pdf

• W. P. Thurston, Existence of codimension-one foliations, Ann. of Math. (2) 104 (1976), no. 2, 249–268 (doi); Foliations and groups of diffeomorphisms, Bull. Amer. Math. Soc. 80 (1974), 304–307 (pdf); The theory of foliations of codimension greater than one, Comment. Math. Helv. 49 (1974), 214–231 (link)

The first of a series on foliations in derived geometry