Triple Products and Cohomological Invariants for Closed Three-manifolds

Motivated by conjectures in Heegaard Floer homology, we introduce an invariant HC∞ ∗ (Y ) of the cohomology ring of a closed 3manifold Y whose behavior mimics that of the Heegaard Floer homology HF∞(Y, s) for s a torsion spin structure. We derive from this a numerical invariant h(Y ) ∈ Z, 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. Heegaard Floer homology groups are a powerful tool for the study of lowdimensional topology introduced and studied by Ozsváth and Szabó ([8], [9], [10], etc.), and they have generated much interest among topologists ([1], [5], [6], etc.). The groups are associated to a closed oriented 3-manifold Y together with a choice of spinc structure s, and comprise a number of variations: HF+, HF−, ĤF , HF∞. Of these, HF∞ is considered to be the least interesting as an invariant, due to the apparent fact (formulated as a conjecture by Ozsváth and Szabó [8]) that it is determined by the cohomology ring of Y . However, while all evidence supports Ozsváth’s and Szabó’s conjecture, the structure of HF∞ can be rather more complicated than a cursory inspection of the cohomology ring of Y might suggest (c.f. [2]). Furthermore, in various situations it can be useful for other purposes to understand the behavior of HF∞—for example, it plays a key role in Ozsváth’s and Szabó’s proof of Donaldson’s diagonalizability theorem for definite 4-manifolds and generalizations [7]. With these ideas in mind, we introduce here an invariant HC∞ ∗ (Y ) of the cohomology ring of Y that we call the “cup cohomology;” it is closely related to HF∞(Y, s) for any torsion spinc structure s, granted the conjecture mentioned above (more precisely, in this case HC∞ ∗ (Y ) is the E∞ term of a spectral sequence converging to HF∞(Y, s), possibly after a grading shift). The cup cohomology satisfies various properties: (1) It is the homology of a free complex C∞ ∗ (Y ) over Z whose underlying group is Λ∗H1(Y ;Z) ⊗ Z[U,U−1] (where U is a formal variable of degree −2), and whose differential is defined in terms of triple products 〈a ∪ b ∪ c, [Y ]〉 of elements a, b, c ∈ H1(Y ). 1