On the Heegaard Floer homology of a surface times a circle

We make a detailed study of the Heegaard Floer homology of the product of a closed surface Σg of genus g with S1. We determine HF+(Σg×S1,s;C) completely in the case c1(s)=0, which for g⩾3 was previously unknown. We show that in this case HF∞ is closely related to the cohomology of the total space of a certain circle bundle over the Jacobian torus of Σg, and furthermore that HF+(Σg×S1,s;Z) contains nontrivial 2-torsion whenever g⩾3 and c1(s)=0. This is the first example known to the authors of torsion in Z-coefficient Heegaard Floer homology. Our methods also give new information on the action of H1(Σg×S1) on HF+(Σg×S1,s) when c1(s) is nonzero.