HF D HM, V: Seiberg-witten floer homology and handle additions

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 S¹ × S² (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.