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