On the Heegaard Floer homology of a surface times a circle

TL;DR

This paper provides a detailed computation of Heegaard Floer homology for surface times circle, revealing new torsion phenomena and relations to the Jacobian torus, especially for genus greater than two.

Contribution

It completely determines HF^+ for the trivial first Chern class case and uncovers the first known torsion in integral Heegaard Floer homology for these manifolds.

Findings

HF^+ is explicitly computed for the trivial Chern class case.

Nontrivial 2-torsion in HF^+ appears for genus > 2.

HF^inity relates to the cohomology of a circle bundle over the Jacobian.

Abstract

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.