#### 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.

Notes on file formats
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

GT home page

## Archival Version

**These pages are not updated anymore.
They reflect the state of
.
For the current production of this journal, please refer to
http://msp.warwick.ac.uk/.
**