On the Heegaard Floer homology of a surface times a circle

Abstract

We make a detailed study of the Heegaard Floer homology of the product of a closed surface Σg of genus g with S 1 . We determine HF + (Σg × S 1 , s; C) completely in the case c 1 (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 × S 1 , s; Z) contains nontrivial 2-torsion whenever g ≥ 3 and c 1 (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 H 1 (Σg × S 1 ) on HF + (Σg × S 1 , s) when c 1 (s) is nonzero.

Product formulae for Ozsváth–Szabó 4–manifold invariants

“…† g I Z/-action has no correction terms. This was already observed by Ozsváth-Szabó [10,Theorem 9.3] in the case of Z coefficients (see also [3]). However, it is no longer the case that the action by H 1 .Y 0 / decreases degree by 1, or is even homogeneous.…”

“…The results from [3] show that F 0 ; F 1 W H fi 0g ! H fi 0 and j kg are given by With all these preliminaries out of the way and with our notation in place, we now turn to the actual calculations of the twisted Heegaard Floer groups of † g S 1 .…”

“…Indeed, our choice of Á is induced by the cobordism † g D 2 D 4 . The main input for this computation comes from [3] where we developed most of the technical tools required. We start this section by elucidating the new phenomena associated with working with twisted coefficients in surgery exact sequences.…”

Floer homology and covering spaces

“…In fact, there are no known examples of rational homology spheres with HF containing torsion. (For the first examples of Z-torsion in Heegaard Floer homology, see [JM08]. Those examples have b 1 > 0.)…”

Knot Floer homology and rational surgeries

“…The second square involves the identification between C fi 0g and C fj 0g, which sends the subset C fi j D tg to C fi j D tg, as explained in Proposition 5.2 of [27]. (Note that this is where we use most significantly the fact that we are using coefficients over F ; over coefficients in Z the involution has a more complicated form, described in [9].) Lemma 10. where here X.t/ is the homology of a chain complex satisfying the hypotheses of Lemma 10.5, for the function ı t W Z !…”