Geometry & Topology

On invariants of Hirzebruch and Cheeger–Gromov

Stanley Chang and Shmuel Weinberger

Full-text: Open access

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.

Article information

Source
Geom. Topol., Volume 7, Number 1 (2003), 311-319.

Dates
Received: 28 March 2003
Accepted: 30 April 2003
First available in Project Euclid: 21 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513883100

Digital Object Identifier
doi:10.2140/gt.2003.7.311

Mathematical Reviews number (MathSciNet)
MR1988288

Zentralblatt MATH identifier
1037.57028

Subjects
Primary: 57R67: Surgery obstructions, Wall groups [See also 19J25]
Secondary: 46L80: $K$-theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22] 58G10

Keywords
signature $L^2$–signature structure set $\rho$–invariant

Citation

Chang, Stanley; Weinberger, Shmuel. On invariants of Hirzebruch and Cheeger–Gromov. Geom. Topol. 7 (2003), no. 1, 311--319. doi:10.2140/gt.2003.7.311. https://projecteuclid.org/euclid.gt/1513883100


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