at.algebraic topology - How can you tell if a space is homotopy equivalent to a manifold?

Jacob Lurie gave a talk last week at Peter May's birthday conference on noncommutative Poincare duality. The idea is to take an n-manifold M and a (n-1)-connected space X. Then, he showed that the compact mapping space Map_c(M,X) is isomorphic to a certain homotopy colimit over a certain category of open subsets of M. This is equivalent to the usual commutative Poincare duality. However, it is not clear (to me) what the natural generalization of the statement is to non-manifolds. So, I am not sure how to use it as a test. However, if you could use it as a test of being a manifold, it seems feasible that if the noncommutative statement held for your test space M and all (n-1)-connected spaces X for some X, then it would seem reasonable ask whether your test space is the homotopy type of an n-manifold.

The category of open sets over which Lurie takes the colimit is the category of disjoint balls (homeomorphic to R^n) in M. Thus, a guess might be something like: if M is a space, and if U is a category of open sets of M that cover M, and if Map_c(M,X) is equivalent to the homotopy colimit of Map_c(U_i,X) for all U_i in U for all (n-1)-connected X for some n, then M has the homotopy type of an n-manifold.

I have no idea if this is true, and even if it is true, it is not clear if it would be useful.