Models for the Leaf Space of a Foliation

Abstract

The aim of this talk is to explain and compare some approaches to the leaf space (or " transverse structure ") of a foliation. A foliation is a certain partition F of a manifold M into immersed submanifolds, the leaves of the foliation. Identifying each of the leaves to a single point yields a very uninformative, " coarse " quotient space, and the problem is to define a more refined quotient M/F, which captures aspects of that part of the geometric structure of the foliation which is constant and/or trivial along the leaves. It is possible to distinguish (at least) three approaches to this problem. One is in the spirit of non-commutative geometry [4], and uses the duality between the manifold M and the ring C ∞ c (M) of compactly supported smooth functions on M . The quotient M/F is then modelled, dually, by an extension of this ring C ∞ c (M), the so-called convolution algebra of the foliation. Completion of such convolution algebras leads one into C * -algebras. Important invariants are the cyclic type (i.e. Hochschild, cyclic, periodic cyclic) homologies and the K-theory of these convolution and C * -algebras. A second approach, which predates non-commutative geometry, is to con-struct a quotient " up to homotopy " . Like all such homotopy colimits in algebraic topology, this construction takes the form of a classifying space. This approach goes back to Haefliger, who constructed a classifying space BΓ q for foliations of codimension q, as the leaf space of the " universal " foliation [9, 2]. Important in-variants are the cohomology groups of these classifying spaces, in particular the universal or characteristic classes coming from the cohomology of the universal leaf space BΓ q . A third approach, even older, is due to Grothendieck. Not surprisingly, Gro-thendieck uses the 'duality' between the space M and the collection of all its sheaves, which form a topos Sh(M). The quotient M/F can then be constructed as a suitable topos " Sh(M/F) " , consisting of sheaves on M which are invariant along the leaves in a suitable sense. One can then apply the whole machinery of [17], and study the Grothendieck fundamental group of Sh(M/F), its sheaf cohomology groups, etc. etc.. Central to all these approaches is the construction [18] of a smooth groupoid out of the foliated manifold (M, F), called the holonomy groupoid and denoted Hol(M, F). The three approaches above then become special instances of the gen-eral procedure of associating to a smooth (or " Lie ") groupoid G a convolution algebra C ∞