$\mathrm{HF} = \mathrm{HM}$, V: Seiberg–Witten Floer homology and handle additions

Abstract

This is the last of five papers that construct an isomorphism between the Seiberg–Witten Floer homology and the Heegaard Floer homology of a given compact, oriented $3$–manifold. See Theorem 1.4 for a precise statement. As outlined in paper I (Geom. Topol. 24 (2020) 2829–2854), this isomorphism is given as a composition of three isomorphisms. In this article, we establish the third isomorphism, which relates the Seiberg–Witten Floer homology on the auxiliary manifold with the appropriate version of Seiberg–Witten Floer homology on the original manifold. This constitutes Theorem 4.1 in paper I, restated in a more refined form as Theorem 1.1 below. The tool used in the proof is a filtered variant of the connected sum formula for Seiberg–Witten Floer homology, in special cases where one of the summand manifolds is $S^1\times S^2$ (referred to as “handle-addition” in all five articles in this series). Nevertheless, the arguments leading to the aforementioned connected sum formula are general enough to establish a connected sum formula in the wider context of Seiberg–Witten Floer homology with nonbalanced perturbations. This is stated as Proposition 6.7 here. Although what is asserted in this proposition has been known to experts for some time, a detailed proof has not appeared in the literature, and therefore of some independent interest.