Geometry & Topology, Vol. 7 (2003) Paper no. 9, pages 311--319.

On Invariants of Hirzebruch and Cheeger-Gromov

Stanley Chang, Shmuel Weinberger

Abstract. We prove that, if M is a compact oriented manifold of dimension 4k+3, where k>0, such that pi_1(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 tau_(2): S(M)-->R that coincides with the rho-invariant of Cheeger-Gromov. In particular, our result shows that the rho-invariant is not a homotopy invariant for the manifolds in question.

Keywords. Signature, L^2-signature, structure set, rho-invariant

AMS subject classification. Primary: 57R67. Secondary: 46L80, 58G10.

DOI: 10.2140/gt.2003.7.311

E-print: arXiv:math.GT/0306247

Submitted to GT on 28 March 2003. Paper accepted 30 April 2003. Paper published 17 May 2003.

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Stanley Chang, Shmuel Weinberger

Department of Mathematics, Wellesley College
Wellesley, MA 02481, USA
and
Department of Mathematics, University of Chicago
Chicago, IL 60637, USA

Email: shmuel@math.uchicago.edu, sschang@palmer.wellesley.edu

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