We prove that, if is a compact oriented manifold of dimension , where , such that is not torsion-free, then there are infinitely many manifolds that are homotopic equivalent to but not homeomorphic to it. To show the infinite size of the structure set of , we construct a secondary invariant that coincides with the –invariant of Cheeger–Gromov. In particular, our result shows that the –invariant is not a homotopy invariant for the manifolds in question.
"On invariants of Hirzebruch and Cheeger–Gromov." Geom. Topol. 7 (1) 311 - 319, 2003. https://doi.org/10.2140/gt.2003.7.311