On the deleted squares of lens spaces

The configuration space F2(M)F2(M) of ordered pairs of distinct points in a manifold M, also known as the deleted square of M, is not a homotopy invariant of M: Longoni and Salvatore produced examples of homotopy equivalent lens spaces M and N   of dimension three for which F2(M)F2(M) and F2(N)F2(N) are not homotopy equivalent. They also asked whether two arbitrary 3-dimensional lens spaces M and N   must be homeomorphic in order for F2(M)F2(M) and F2(N)F2(N) to be homotopy equivalent. We give a partial answer to this question using a novel approach with the Cheeger–Simons differential characters.