Cyclic cohomology and characteristic classes for foliations

Abstract

This thesis deals with the cohomology theories and the theory of characteristic classes for leaf spaces of foliations, as well as with the interaction between the classical approach (of Grothendieck, Bott-Haefliger) and the non-commutative approach (of Connes-Moscovici) to these theories. Leaf spaces provide a large class of examples of "singular spaces" to which standard theories do not apply directly. Grothendieck, Bott and Haefliger overcome this problem by enlarging the category of spaces to those of etale groupoids, to which many of the classical constructions extend. Chapters 2 and 3 belong to this approach to leaf spaces. In Chapter 2 we introduce a homology theory which is in Poincare duality with Haefliger's cohomology, and we prove it has the expected properties (and these will be used in Chapter 4 when computing cyclic homology groups). In Chapter 3 we give a more geometrical (Cech-De Rham) model for Haefliger's cohomology, which allows us to geometrically construct characteristic classes for etale groupoids (hence leaf spaces) and to explain/extend Bott's formulas. These two chapters are joint work with Ieke Moerdijk. Leaf spaces also provide a large class of examples in non-commutative geometry. From this point of view, they are modelled by their associated convolution algebras. In Chapter 4 we compute the cyclic homology of convolution algebras of etale groupoids, which is the relevant theory from the non-commutative point of view. Here we find the connection with Grothendieck-Haefliger's approach (and this is based on our homology theory of Chapter 2). Our computations extend previous computations of Brylinski, Burghelea, Connes, Karoubi, Nistor. Motivated also by the connection with the longitudinal index theory, the last sections of this chapter concentrate on the case of holonomy proupoids of foliations, and associated Chern characters. In their approach to transversal index theorems for foliations, Connes and Moscovici have recently discovered a deep connection between teh (geometrical) characteristic classes for foliations and the non-commutative ones arising, via the Chern character, in the cyclic cohomology groups computed in Chapter 4. This connection is based on a cyclic cohomology theory of a particular Hopf algebra of the geometric operators. In Chapter 5 we study this cohomology theory. First of all, we show that it applies to general Hopf algebras as an extension of the classical Lie algebra homology. Secondly, we make the connection with Cuntz-Quillen's approach to cyclic cohomology in terms of X-complexes. Also, inspired by the (rather classical) construction of characteristic classes for foliations in terms of the truncated Weil complex (recalled in the prelimenaries), we describe a non-commutative version of the Weil complex. This turns out to be strongly related to Cuntz-Quillen's X-complex, and it is used to solve the problem of constructing characteristic maps associated to higher traces.