Connected Heegaard Floer homology of sums of Seifert fibrations

TL;DR

This paper computes the connected Heegaard Floer homology for a broad class of 3-manifolds, especially linear combinations of Seifert fibered homology spheres, revealing its role in classifying local equivalence classes and relating to classical invariants.

Contribution

It provides explicit computations of connected Floer homology for many 3-manifolds and establishes its connection to classical invariants like the Rokhlin and Neumann-Siebenmann invariants.

Findings

Connected Floer homology determines the local equivalence class of the $omplex.

Identities relate the rank of connected Floer homology to classical invariants.

Computations are based on combinatorial models inspired by lattice homology.

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 N\'emethi on lattice homology.