Volume 7, issue 1 (2003)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
On invariants of Hirzebruch and Cheeger–Gromov

Stanley Chang and Shmuel Weinberger

Geometry & Topology 7 (2003) 311–319

arXiv: math.GT/0306247

Abstract

We prove that, if M is a compact oriented manifold of dimension 4k + 3, where k > 0, such that π1(M) is not torsion-free, then there are infinitely many manifolds that are homotopic equivalent to M but not homeomorphic to it. To show the infinite size of the structure set of M, we construct a secondary invariant τ(2): S(M) that coincides with the ρ–invariant of Cheeger–Gromov. In particular, our result shows that the ρ–invariant is not a homotopy invariant for the manifolds in question.

Keywords
signature, $L^2$–signature, structure set, $\rho$–invariant
Mathematical Subject Classification 2000
Primary: 57R67
Secondary: 46L80, 58G10
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Publication
Received: 28 March 2003
Accepted: 30 April 2003
Published: 17 May 2003
Proposed: Stephen Ferry
Seconded: Ronald Stern, Benson Farb
Authors
Stanley Chang
Department of Mathematics
Wellesley College
Wellesley
Massachusetts 02481
USA
Shmuel Weinberger
Department of Mathematics
University of Chicago
Chicago
Illinois 60637
USA