There are many conjecturally-equivalent three-manifold [Floer homologies](https://en.wikipedia.org/wiki/Floer_homology), of which my understanding is the most-computable is [Heegaard Floer homology](http://math.mit.edu/~petero/Introduction.pdf). 



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



What is the Heegaard Floer Homology of a connect sum of $S^2 \times S^1$s?
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](http://annals.math.princeton.edu/wp-content/uploads/annals-v159-n3-p03.pdf), 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](http://www.ihes.fr/~gromov/PDF/8[27].pdf). 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, $\widetilde{HM}$.


