A general families index theorem
type: Item
instance of: scholarly article
title: A general families index theorem
zbMATH Open document ID: 0938.58021
DOI: 10.1023/A:1007886112487
author: James L. Heitsch
author: Connor Lazarov
published in: \(K\)-Theory
publication date: 2000-06-14T00:00:00Z
review text: The authors prove a families index theorem for a family of leafwise generalized Dirac operators associated with a foliation, having Hausdorff graph, of a compact Riemannian manifold. The proof is based on the definition of an appropriate Bismut superconnection and on the asymptotics associated with rescaling the manifold's metric by a factor of \(1/t\). A previous paper by \textit{J. L. Heitsch} [K-Theory 9, No. 6, 507-528 (1995; Zbl 0843.58117)] used the heat operator of the superconnection's curvature to define a Haefliger cohomology class that, in the \(t\rightarrow 0\) limit, gave the analogue of the classical cohomological families index formula. In the current paper, the authors identify regularity conditions (notably the condition that the Dirac operators' strong Novikov-Shubin invariants exceed three times the foliation's codimension) and show that these regularity conditions imply that the heat operator's \(t\rightarrow \infty\) limit gives the Chern character of the family's index bundle. A number of index theorems are special cases of the authors' general families index theorem. These include the Atiyah-Singer families index theorem and the Atiyah \(L^{2}\) covering index theorem. Indeed, the general families index theorem implies a common generalization of these two theorems. The order zero term of the general families index recovers A. Connes's index for measured foliations. (The regularity assumption is unnecessary here.) The relationship between the general families index theorem and the Connes-Skandalis longitudinal index theorem has not yet been determined.
reviewed by: Peter Haskell
Mathematics Subject Classification ID: 58J22
Mathematics Subject Classification ID: 19K56
Mathematics Subject Classification ID: 58J35
Mathematics Subject Classification ID: 58J20
Mathematics Subject Classification ID: 19L10
zbMATH DE Number: 1384367
zbMATH Keywords: families index theorem
zbMATH Keywords: Dirac operator
zbMATH Keywords: Bismut superconnection
zbMATH Keywords: foliation
zbMATH Keywords: heat operator asymptotics
MaRDI profile type: Publication
full work available at URL: a:1007886112487
OpenAlex ID: W2090284889
description: scientific article; zbMATH DE number 1384367