Geometry & Topology

On invariants of Hirzebruch and Cheeger–Gromov

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 $τ(2):S(M)→ℝ$ 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
Accepted: 30 April 2003
First available in Project Euclid: 21 December 2017

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

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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