Fukaya-Floer homology of Σ×S1 and applications.

Abstract

We determine the Fukaya-Floer (co)homology groups of the three-manifold y = S x S 1 , where S is a Riemann surface of genus g > 1. These are of two kinds. For the 1-cycle S1 C Y, we compute the Fukaya-Floer cohomology HFF*(Y, S1) and its ring structure, which is a sort of deformation of the Floer cohomology HF*(Y). On the other hand, for 1-cycles ö C 'S CY, we determine the Fukaya-Floer homology HFF*(Y,S) and its i?-F*(Y)-module structure. We give the following applications: We show that every four-manifold with 6+ > 1 is of finite type. Four-manifolds which arise as connected sums along surfaces of fourmanifolds with 6i = 0 are of simple type and we give constraints on their basic classes. We find the invariants of the product of two Riemann surfaces both of genus greater than or equal to one.