On Invariants of Hirzebruch and Cheeger–Gromov

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. AMS Classification numbers Primary: 57R67 Secondary: 46L80, 58G10