nLab
foliation

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

$$M={\cup }_{\alpha }{V}_{\alpha }$$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 $(U,\varphi )$, $\varphi =({\varphi }_1,\dots ,{\varphi }_n):U\to {ℝ}^n$ such that for each “special” chart and each $\alpha$ there is a number $p=p\le n$, 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 ${\varphi }_r(x)=\mathrm{const}=\mathrm{const}(r,U,\alpha )$ for all $r>p$.

If the manifold is smooth, the charts are required to be smooth too. In smooth case, the $p$-dimensional foliations with underlying manifold $M$ are in 1-1 correspondence with integrable distributions of hyperplanes of dimension $p$ in tangent bundle of $N$.

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

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

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

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

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

• Ieke Moerdijk, Janez Mrčun, Introduction to foliations and Lie groupoids, Cambridge Studies in Advanced Mathematics 91, 2003. x+173 pp. ISBN: 0-521-83197-0

• wikipedia, Springer Online Enc. of Math.: foliation

• 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 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

• 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; Rankin-Cohen brackets and the Hopf algebra of transverse geometry, Mosc. Math. J., 4:1 (2004), 111–130

• 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)

• Yu. A. Kordyukov, Index theory and non-commutative geometry on foliated manifolds, Russian Math. Surveys, 64:2 (2009), 273–391 (original: УМН, 64:2(386) (2009), 73–202)