  This article surveys our ongoing project about the relationship between
invariants extending the classical Rohlin invariant of homology spheres and
those coming from 4-dimensional (Yang-Mills) gauge theory. The main conjecture
towards which this project is directed is that the Rohlin invariant and the
gauge theoretic invariant coincide for a homology $S^1\times S^3$. We discuss
the implications of this conjecture for some classical problems in
low-dimensional topology, and progress we have made towards proving the
conjecture.
