Local index theory over foliation groupoids

Local Index Theory over Foliation Groupoids

We give a local proof of an index theorem for a Dirac-type operator that is invariant with respect to the action of a foliation groupoid G. If M denotes the space of units of G then the input is a G-equivariant fiber bundle P → M along with a G-invariant fiberwise Dirac-type operator D on P . The index theorem is a formula for the pairing of the index of D, as an element of a certain K-theory g...

Local index theory over étale groupoids

We give a superconnection proof of Connes’ index theorem for proper cocompact actions of étale groupoids. This includes Connes’ general foliation index theorem for foliations with Hausdor¤ holonomy groupoid.

Extension Theory for Local Groupoids

We intend to provide an algebraic framework in which some of the algebraic theory of connections become identical to some of the (Eilenberg Mac Lane) theory of extensions of non-abelian groups. In particular, the Bianchi identity for the curvature of a connection is related to one of the crucial equations in extension theory. The Bianchi identity as a purely combinatorial fact was dealt with in...

0 Foliation groupoids and their cyclic homology ∗

Foliation groupoids and their cyclic homology

The purpose of this paper is to prove two theorems which concern the position of étale groupoids among general smooth (or ”Lie”) groupoids. Our motivation comes from the non-commutative geometry and algebraic topology concerning leaf spaces of foliations. Here, one is concerned with invariants of the holonomy groupoid of a foliation [4, 34], such as the cohomology of its classifying space [14],...