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 pi(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 tau((2)) : S( M) --> 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.