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 ${\pi }_1\left(M\right)$ 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.