Question

We know that Monopole Floer homology of a 3-manifold $M$ depends on a spin-c structure. My question is that if $M$ is $F\times S^1$ ($F$ is a surface of genus larger than 1) then how can we compute the Floer homology for it?

For the spin-c structures satisfying $\langle c_1(L),F\rangle>2g-2$ ($L$ is the determinant bundle of the spin-c structure,$g$ is the genus of $F$), Kronheimer and Mrowka prove that the Floer homology vanishes. They also proved that if $\langle c_1(L),F\rangle=2g-2$ then the Floer homology is $\mathbb{Z}$. But what about the other spin-c structure (when $\langle c_1(L),F\rangle<2g-2$)?

Also, what is the answer for this question if we consider Heegaard Floer Homology instead of Monopole Floer Homology?

Answer 1

I would assume you are interested in $HM$-to as opposed to $HM$-bar ($HM$-bar is mostly computed in the book Monopoles and 3-manifolds by Kronheimer and Mrowka). For the case of $HM$-to, you should use (as answered above) that monopole is isomorphic to Heegaard Floer (Kutluhan-Lee-Taubes or Taubes + Colin-Ghiggini-Honda).

If you want the trivial torsion Spin$^c$ structure, this is computed by Jabuka and Mark: http://arxiv.org/pdf/math/0502328v4.pdf

This paper also has the references to the earlier computations for the other Spin$^c$ structures, done by Ozsvath and Szabo.