  Motivated by conjectures in Heegaard Floer homology, we introduce an
invariant HC(Y) of the cohomology ring of a closed 3-manifold Y whose behavior
mimics that of the Heegaard Floer homology HF^\infty(Y,s) for s a torsion
spin-c structure. We derive from this a numerical invariant h(Y), and obtain
upper and lower bounds on h(Y). We describe the behavior of h(Y) under
connected sum, and deduce some topological consequences. Examples show that the
structure of HC(Y) can be surprisingly complicated, even for 3-manifolds with
comparatively simple cohomology rings.
