A \(K\)-theoretic relative index theorem and Callias-type Dirac operators
type: Item
instance of: scholarly article
title: A \(K\)-theoretic relative index theorem and Callias-type Dirac operators
zbMATH Open document ID: 0835.58035
DOI: 10.1007/BF01460989
author: Ulrich Bunke
published in: Mathematische Annalen
publication date: 1996-04-21T00:00:00Z
full work available at URL: https://eudml.org/doc/165367
review text: Motivated by the positive scalar curvature problem a relative index theorem for Dirac operators was first proved by Gromov/Lawson (1983). It expresses in topological terms the difference of the \(L^2\)-index of two Dirac operators which live on possibly different complete Riemannian manifolds. The essential assumptions are \(-\) that both operators are positive at infinity and \(-\) that the operators coincide outside of compact sets. In the paper under review the author extends a version of this theorem to operators \(D\) acting on bundles of modules over some \(C^*\)-algebra \(A\). The index \(\text{ind }D\) of such an operator is an element of the \(K\)- theory of \(A\). Let be given two Dirac operators \(D_1\), \(D_2\) on two complete Riemannian manifolds \(M_1\), \(M_2\) which are positive at infinity. Assume that there are compact hypersurfaces \(N_i \subset M_i\), \(i = 1,2\) cutting the \(M_i\) into two pieces and an isomorphism of all structures over a neighbourhood of \(N_1\) to those over a neighbourhood of \(N_2\). Then one can cut at those hypersurfaces and glue the pieces together interchanging the boundary components. One obtains new Dirac operators \(D_3\), \(D_4\). The relative index theorem proved in the paper states: \[ \text{ind } D_1 + \text{ind } D_2 = \text{ind } D_3 + \text{ind } D_4 \in K(A). \] This theorem is also valid for real operators. The above result is applied to show that a homomorphism \[ R_n (\pi) \to \text{KO}_n (C^*_r (\pi)) \] is well defined, where \(\pi\) is a finitely generated group, \(R_n (\pi)\) is the group of \(B \text{Spin} \times B \pi\)-bordisms with distinguished positive scalar curvature metric introduced by Stolz and Hajduk and \(\text{KO}_n (C^*_r (\pi))\) is the \(n\)-th real \(K\)-group of the reduced (real) group-\(C^*\)- algebra of \(\pi\). The above homomorphism is realized by the index of Dirac operators on manifolds with cylindrical ends twisted with flat bundles with fibre \(C^*_r (\pi)\). Another applic…
Mathematics Subject Classification ID: 58J22
Mathematics Subject Classification ID: 19K56
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 47A53
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 19K33
zbMATH DE Number: 840184
zbMATH Keywords: Callias type operators
zbMATH Keywords: relative index theorem
zbMATH Keywords: \(K\)-theory
zbMATH Keywords: Dirac operators
zbMATH Keywords: \(K\)-homology
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 840184