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There are many conjecturally-equivalent three-manifold Floer homologies, of which my understanding is the most-computable is Heegaard Floer homology.

What is the (Heegaard) Floer homology of a connect sum of $k$ copies of $S^2 \times S^1$? Is it trivial?

I am an outsider, and so please forgive me if my question is not quite well formed. E.g. I gather that HF depends on a $\mathrm{Spin}^c$ structure, so at best I would understand the answer for all choices; the value of Heegaard Floer homology is really an exact triangle, not a vector space, so I guess I'm asking for that triangle; perhaps there are other technicalities I am unaware of.

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The explicit calculation of this Heegaard-Floer homology (at least, $HF^-$, and of the unique $\text{Spin}^c$ structure for which $c_1(\mathfrak s) = 0$) was carried out in the paper in which it was introduced, and indeed is an important part of the proof that Heegaard-Floer homology is well-defined. See Lemma 9.1. One also has the isomorphisms $HF^+ \cong \Bbb Z[U,U^{-1}]/U\Bbb Z[U] \otimes H_*(T^g;\Bbb Z)$ and $HF^\infty \cong \Bbb Z[U^{-1},U] \otimes H_*(T^g;\Bbb Z)$. The exact sequence relating the triple ($HF^+, HF^-, HF^\infty)$ is exactly what you think it is. When the $\text{Spin}^c$ structure is nonzero, all the Floer groups are zero. (These four groups are all computable by hand, as done in the original paper.)

In this case, the Seiberg-Witten Floer homology is computable, because $S^2 \times S^1$ has positive scalar curvature, as do all connect-sums with itself by Gromov-Lawson. When $c_1(\mathfrak s)$ is not torsion (in this case, not zero), all the Floer groups are trivial, and in the case that $c_1(\mathfrak s) = 0$ all the Floer groups are explicitly calculable, as is the exact sequence (one of the connecting maps is zero) - for the precise result, see Chapter 36 of Kronheimer-Mrowka, "Monopoles and 3-manifolds".

To compare these results, one should note the relationship (now proved, thanks to Colin-Ghiggini-Honda and Kuthulan-Lee-Taubes) $\overline{HM} \cong HF^\infty$, $\check{HM} \cong HF^-$, $\widehat{HM} \cong HF^+$. $\widehat{HF}$ corresponds to a reduced Seiberg-Witten Floer homology not described in that chapter of Kronheimer-Mrowka, $\widetilde{HM}$.

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  • $\begingroup$ If anyone knows how to get the wide-check that really should be over HM-to, please feel free to edit it in... I don't know how to do that in MathJaX. $\endgroup$
    – mme
    Commented Apr 20, 2016 at 6:57
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    $\begingroup$ If someone's interested in the absolute grading and the action by $H_2$ of the 3-manifold: the grading on $HF^+$ needs to be shifted by $-g/2$ (so that the bottom-most homogeneous element sits in degree $-g/2$); and the action is so-called "standard" in this case. I think that the reference for both statements is the Absolutely graded paper by Ozsváth and Szabó. $\endgroup$ Commented Apr 20, 2016 at 8:08

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