In an old answer to an old question of mine, Peter Teichner commented that it is an open problem to determine which homotopy 2-types arise from 4-manifolds. In some instances we know that a 4-manifold *is* a 2-type with one extra piece of data (the intersection form), so it's a very interesting question as to when we can go the other way.

I'd like to know what is the latest on this problem, in the form of references or results.