# Variants of equivariant Seiberg-Witten Floer homology

Marcolli, M; Wang, B-L (2005). Variants of equivariant Seiberg-Witten Floer homology. In: Booss-Bavnbek, B; Grubb, G; Wojciechowski, K P. Spectral geometry of manifolds with boundary and decomposition of manifolds. Providence, RI: American Mathematical Society, 225-238.

## Abstract

For a rational homology 3-sphere Y with a Spin c structure s, we show that simple algebraic manipulations of our construction of equivariant Seiberg-Witten Floer homology in lead to a collection of variants, which are all topological invariants. We establish a long exact sequence relating them and we show that they satisfy a duality under orientation reversal. We explain their relation to the equivariant Seiberg-Witten Floer (co)homologies introduced in [loc. cit.]. We conjecture the equivalence of these versions of equivariant Seiberg-Witten Floer homology with the Heegaard-Floer invariants introduced by Ozsváth and Szabó.

## Abstract

For a rational homology 3-sphere Y with a Spin c structure s, we show that simple algebraic manipulations of our construction of equivariant Seiberg-Witten Floer homology in lead to a collection of variants, which are all topological invariants. We establish a long exact sequence relating them and we show that they satisfy a duality under orientation reversal. We explain their relation to the equivariant Seiberg-Witten Floer (co)homologies introduced in [loc. cit.]. We conjecture the equivalence of these versions of equivariant Seiberg-Witten Floer homology with the Heegaard-Floer invariants introduced by Ozsváth and Szabó.