Connected Heegaard Floer homology of sums of Seifert fibrations

Abstract

We compute the connected Heegaard Floer homology (defined by Hendricks, Hom, and Lidman) for a large class of $3$–manifolds, including all linear combinations of Seifert fibered homology spheres. We show that for such manifolds, the connected Floer homology completely determines the local equivalence class of the associated $\iota$–complex. Some identities relating the rank of the connected Floer homology to the Rokhlin invariant and the Neumann–Siebenmann invariant are also derived. Our computations are based on combinatorial models inspired by the work of Némethi on lattice homology.