The Index of Operators on Foliated Bundles

✍ Scribed by Victor Nistor

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 588 KB
• Volume: 141
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1996.0135

✦ Synopsis

We compute the equivariant cohomology Connes Karoubi character of the index of elliptic operators along the leaves of the foliation of a flat bundle. The proof uses techniques similar to those developed in Algebraic Topology, in the study of noncommutative algebras.

1996 Academic Press, Inc.

1. Introduction

Let (V, ?), F Ä V w Ä ? B be a smooth fiber bundle with fiber F of dimension q. We assume that (V, ?) is endowed with a flat connection corresponding to an integrable subbundle F/TV, of dimension n=dim(B), transverse at any point to the fibers of ?. The pair (V, F) is a foliation.

The purpose of this paper to study invariants of differential operators along the leaves of the above foliation. The index of an elliptic operator along the leaves of the foliation F is an element in the K-Theory group K 0 (F)=K 0 (9 & (F)) where 9 & (F) is the algebra of regularizing operators along the leaves. In the case of a foliated bundle there exists a Connes Karoubi character Ch: K 0 (9 & (F)) Ä H* 1 &q (F, O)$ to the dual of equivariant cohomology with twisted coefficients, where 1 is the fundamental group of the base B acting on the fiber F via holonomy, and O is the orientation sheaf. Our main theorem computes the Connes Karoubi character of the index. This amounts to a proof of the ``higher index theorem for foliations'' in this special case. A very general higher index theorem for foliations can be found in [C3] and here we give a new proof of this theorem for flat bundles. Some very interesting results related to the results in this paper are contained in [CM2] where Diff-invariant structures are treated in detail. See also [C4]. In the even more special case of a family of elliptic operators our theorem recovers the computation of article no. 0135