On invariants of Hirzebruch and Cheeger–Gromov

We prove that, if $M$ is a compact oriented manifold of dimension $4 k + 3$ , where $k > 0$ , such that $π_{1} (M)$ 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 $τ_{(2)} : S (M) \to ℝ$ 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.

Keywords

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

Mathematical Subject Classification 2000

Primary: 57R67

Secondary: 46L80, 58G10

References

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Publication

Received: 28 March 2003

Accepted: 30 April 2003

Published: 17 May 2003

Proposed: Stephen Ferry

Seconded: Ronald Stern, Benson Farb

Authors

Stanley Chang
Department of Mathematics Wellesley College Wellesley Massachusetts 02481 USA

Shmuel Weinberger
Department of Mathematics University of Chicago Chicago Illinois 60637 USA

Volume 7, issue 1 (2003)

Stanley Chang and Shmuel Weinberger