  We make a detailed study of the Heegaard Floer homology of the product of a
closed surface Sigma_g of genus g with S^1. We determine HF^+ for this
3-manifold completely for the spin^c structure having trivial first Chern
class, which for g>2 was previously unknown. We show that in this case
HF^\infty is closely related to the cohomology of the total space of a certain
circle bundle over the Jacobian torus of Sigma_g, and furthermore that HF^+ of
a surface times a circle with integral coefficients contains nontrivial
2-torsion whenever g>2. This is the first example known to the authors of
torsion in Heegaard Floer homology with integral coefficients. Our methods also
give new information on the action of H_1 of a surface times the circle on HF^+
of the same with spin^c-structures with nonzero first Chern class.
