#### Volume 7, issue 1 (2003)

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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 ${\pi }_{1}\left(M\right)$ 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 ${\tau }_{\left(2\right)}:S\left(M\right)\to ℝ$ 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
##### Mathematical Subject Classification 2000
Primary: 57R67
Secondary: 46L80, 58G10