Vol. 1, No. 2, 2016

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The surgery exact sequence, K-theory and the signature operator

Paolo Piazza and Thomas Schick

Vol. 1 (2016), No. 2, 109–154
Abstract

The main result of this paper is a new and direct proof of the natural transformation from the surgery exact sequence in topology to the analytic K-theory sequence of Higson and Roe.

Our approach makes crucial use of analytic properties and new index theorems for the signature operator on Galois coverings with boundary. These are of independent interest and form the second main theme of the paper. The main technical novelty is the use of large-scale index theory for Dirac-type operators that are perturbed by lower-order operators.

Keywords
signature operator, K-theory, exact surgery sequence, index classes, rho-classes
Mathematical Subject Classification 2010
Primary: 46L80, 46L87, 58J22
Milestones
Received: 9 January 2015
Accepted: 7 April 2015
Published: 20 October 2015
Authors
Paolo Piazza
Dipartimento di Matematica
Università degli Studi di Roma "La Sapienza"
P.le Aldo Moro 5
I-00185 Roma
Italy
Thomas Schick
Mathematisches Institut
Georg-August-Universität Göttingen
Bunsenstr. 3
D-37073 Göttingen
Germany