Variants of equivariant Seiberg-Witten Floer homology

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 [5] lead to a collection of variants HF^(SW,−)_(*,U(1))(Y,s), HF^(SW,∞)_(*,U(1)(Y,s) HF^(SW,+)_(*,U(1))(Y,s), HF^(SW)_*(Y,s) and HF^(SW)_(red,*)(Y,s) which are topological invariants. We establish a long exact sequence relating HF^(SW,±)_(*,U(1))(Y,s) and HF^(SW,∞) _(*,U(1))(Y,s). We show they satisfy a duality under orientation reversal, and we explain their relation to the equivariant Seiberg-Witten Floer (co)homologies introduced in [5]. We conjecture the equivalence of these versions of equivariant Seiberg-Witten Floer homology with the Heegaard Floer invariants introduced by Ozsváth and Szabó.

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