Volume 22, issue 6 (2018)

 Download this article For screen For printing
 Recent Issues
 The Journal About the Journal Editorial Board Subscriptions Editorial Interests Editorial Procedure Submission Guidelines Submission Page Ethics Statement ISSN (electronic): 1364-0380 ISSN (print): 1465-3060 Author Index To Appear Other MSP Journals
$C^*$–algebraic higher signatures and an invariance theorem in codimension two

Nigel Higson, Thomas Schick and Zhizhang Xie

Geometry & Topology 22 (2018) 3671–3699
Abstract

We revisit the construction of signature classes in ${C}^{\ast }$–algebra $K$–theory, and develop a variation that allows us to prove equality of signature classes in some situations involving homotopy equivalences of noncompact manifolds that are only defined outside a compact set. As an application, we prove a counterpart for signature classes of a codimension-two vanishing theorem for the index of the Dirac operator on spin manifolds (the latter is due to Hanke, Pape and Schick, and is a development of well-known work of Gromov and Lawson).

Keywords
$K$–theory, $C^*$–algebraic signature, partitioned manifold theorem, eventual homotopy equivalence
Mathematical Subject Classification 2010
Primary: 19K56, 57R19
Publication
Received: 12 October 2017
Accepted: 2 April 2018
Published: 23 September 2018
Proposed: Benson Farb
Seconded: Tobias H Colding, Martin Bridson
Authors
 Nigel Higson Department of Mathematics Pennsylvania State University University Park, PA United States Thomas Schick Mathematisches Institut Georg-August-Universität Göttingen Göttingen Germany Zhizhang Xie Department of Mathematics Texas A&M University College Station, TX United States