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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), 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?

2 added 90 characters in body; edited tags; edited title

# How to compute the Monopole Floer Homology for SufaceproductS1Surface$\timesS^1$ ?

We know that Monopole Floer homology of a 3-mfd M 3-manifold $M$ depends on spin-C a spin-c structureon M. My question is that if M $M$ is F*S1(F $F\times S^1$ ($F$ is a surface of genus larger than 1), 1) then how can we compute the Floer homology for it?

For the spinC spin-c structures satisfying (C1(L),F)>2g-2 $\langle c_1(L),F\rangle>2g-2$ (L $L$ is the determine determinant bundle of the spin-c structure), Kronheimer and Mrowka proves prove that the Floer homology vanishes. They also proved that if (C1(L),F)=2g-2, $\langle c_1(L),F\rangle=2g-2$ then the Floer homology is $Z$. \mathbb{Z}$. But what about the other spin-C structure?(when spin-c structure (C1(L),F)<2g-2).when$\langle c_1(L),F\rangle<2g-2$)? Also, what is the answer for this question if we consider Heegaard Floer Homology instead of Monopole Floer Homology? 1 # How to compute the Monopole Floer Homology for Suface product S1? We know that Monopole Floer homology of a 3-mfd M depends on spin-C structure on M. My question is that if M is F*S1(F is a surface of genus larger than 1), how can we compute the Floer homology for it? For the spinC structures satisfying (C1(L),F)>2g-2 (L is the determine bundle of spin-c structure), Kronheimer and Mrowka proves that the Floer homology vanishes. They also proved that if (C1(L),F)=2g-2, the Floer homology is$Z\$. But what about the other spin-C structure?(when (C1(L),F)<2g-2).

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