Longoni and Salvatore used Massey products, computed through explicit intersection theory, to provide the first example of two homotopy equivalent manifolds (of the same dimension) with inequivalent configuration spaces. In particular, they show that the orbit configuration space of two points for the Lens space L(7,1) is formal, but the orbit configuration space for L(7,2) admits a nontrivial Massey product.
In work in progress, I determine the rational homotopy DGA (differential graded algebra) models for orbit configurations of L(p,q). In this generality, the algebra becomes much more complicated, and there are problems with transverality (which I am in the process of solving). From these DGA models, I define a numerical homeomorphism invariant for Lens spaces. This invariant is defined in terms of the Lie cobracket structure on a cobar construction of these DGA's, as recently introduced by Sinha and Walter.