TITLE: List of invariants that distinguish homotopy equivalent non-homeomorphic spaces
QUESTION [6 upvotes]: 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?

REPLY [4 votes]: Intersection (co)homology is a (co)homology theory, defined for stratified pseudomanifolds that is not homotopy invariant. 
Examples of applications to your question have been done by Dirk Schuetz in his papers:
"Intersection homology of linkage spaces" and "Intersection homology of linkage spaces in odd dimensional Euclidean space". 
Using intersection cohomology he is able to distinguish non-homeomorphic spaces.
Intersection cohomology can also be used to get the topological classification of weighted projective spaces:
"Intersection homology of weighted projective spaces and pseudo-lens spaces." Masato Kuwata