Rohlin's theorem about the signature of smooth, closed spin 4-manifolds leads to an interesting Z/2-valued invariant of 3-dimensional manifolds. I will describe some joint work with Nikolai Saveliev about a version of this invariant that can be applied to certain non-simply-connected 4-manifolds. We have given an interpretation of this invariant in terms of gauge theory, and I will discuss some applications of this interpretation.