Seiberg-Witten-Floer homology of a surface times a circle for non-torsion spin-c structures

TL;DR

This paper computes the Seiberg-Witten-Floer homology for surface times circle manifolds with non-torsion spin-c structures, revealing new algebraic structures and applications to four-manifold invariants.

Contribution

It explicitly determines the Floer homology groups and ring structure for these manifolds with non-trivial spin-c structures, extending previous results.

Findings

Computed Seiberg-Witten-Floer homology groups for surface x circle

Established ring structure of the homology groups

Applied results to four-manifold invariants and inequalities

Abstract

We determine the Seiberg-Witten-Floer homology groups of the three-manifold which is the product of a surface of genus $g \geq 1$ times the circle, together with its ring structure, for spin-c structures which are non-trivial on the three-manifold. We give applications to computing Seiberg-Witten invariants of four-manifolds which are connected sums along surfaces and also we reprove the higher type adjunction inequalities previously obtained by Oszv\'ath and Szab\'o.