It is written on wikipedia article (https://en.wikipedia.org/wiki/Analytic_torsion) that the Reidemeister torsion is the first invariant that could distinguish between spaces which are homotopy equivalent but not homeomorphic. Can any one give me examples of other invariants which do this?
[Math] List of invariants that distinguish homotopy equivalent non-homeomorphic spaces
at.algebraic-topology
Best Answer
Usual homotopy-type invariants of the configuration spaces associated to a given space may be able to distinguish between spaces which are homotopy equivalent but not homeomorphic. For instance, Riccardo Longoni and Paolo Salvatore [Configuration spaces are not homotopy invariant, Topology 44 (2005), no. 2, 375–380; MR2114713] distinguish between the lens spaces $L(7,1)$ and $L(7,2)$ by considering the universal covers of the (ordered) configuration spaces of $2$ points on these lens spaces, and showing that a certain Massey product vanishes for $\widetilde{F_2(L(7,1))}$ but not for $\widetilde{F_2(L(7,2))}$.