By pawel Walczak, Lawrence Conlon, Remi Langevin

This quantity comprises surveys and learn articles concerning varied points of the speculation of foliation. the most elements main issue the topology of foliations of low-dimensional manifolds, the geometry of foliated Riemannian manifolds and the dynamical houses of foliations. one of the surveys are lecture notes dedicated to the research of a few operator algebras on foliated manifolds and the speculation of confoliations (objects outlined lately through W Thurston and Y Eliashberg, located among foliations and speak to structures). one of the learn articles you will find an in depth evidence of an unpublished theorem (due to Duminy) pertaining to ends of leaves in unheard of minimum units.

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**Extra resources for Foliations: Geometry and Dynamics**

**Example text**

A complete transversal T for T has finite length, so by definition of geometric entropy, for e > 0 small and i —> oo, there exists a sequence n* —> oo and collections of (e, rii)separated points {x\,... } C T where the sequence {p^ has exponential growth exp(ni/i(j F ))/pi —> 1. By a pigeon-hole principle, there must be subcollections of exponentially many points exponentially close which are DYNAMICS AND THE GODBILLON-VEY CLASS 47 (e,nj)-separated. We can assume the set transversal T is a subset of the line so well-ordered, and the sets are index with x\ < x\ for k < £.

We can define a relative tangential cohomology considering the complex n^M, u) = nrr(M) e n^iu) with differential djr(u),9) = (d^uj,u>\u —djrO). We have the exact sequence of the pair ( M , U) ••• - H^r\U) -* HrT(M, U) - Hrr(M) -+ H^(U) -^ ••• We define a product [5] in relative tangential cohomology and prove: T h e o r e m 5 . 4 c a t t ( M , J") > n i l ^ ( M ) . T h e tangential cohomology of a manifold is in general hard to calculate. Vogt [22] gave a cohomological bound for the tangential category using ordinary cohomology instead of foliated cohomology which allows explicit calculation of the tangential category in many cases.

3 Dummy's Theorem In a brilliant work growing out of the paper [28], G. Duminy [26] introduced the Godbillon measure gjr on the E-algebra Bo(^F) generated by the open saturated subsets of a foliation T of codimension one. Dummy's note was also highly original on two other points: T h e Godbillon measure is one half of the Godbillon-Vey invariant, constructed from the leaf cohomology class [77] 6 HX(M,T). T h e second half, the "Vey class" [drj\ € H2{M/T), was considered as a fixed invariant of T on which the Godbillon measure could be evaluated to give GV{T\U) for U 6 Bo{T).