The equivalence of lattice and Heegaard Floer homology

TL;DR

This paper proves that for certain 3-manifolds, the lattice homology and Heegaard Floer homology are equivalent, establishing a significant connection between these two invariants in low-dimensional topology.

Contribution

It proves Némethi's conjecture relating lattice and Heegaard Floer homology for plumbed 3-manifolds, and proposes a conjectural description of the homology action when the first Betti number is positive.

Findings

Lattice homology coincides with Heegaard Floer homology for plumbed 3-manifolds.

Provides a conjectural description of the $H_1(Y)/ ext{Tors}$ action for $b_1(Y)>0$.

Establishes a fundamental equivalence between two important topological invariants.

Abstract

We prove N\'{e}methi's conjecture: if $Y$ is a 3-manifold which is the boundary of a plumbing of a tree of disk bundles over $S^2$, then the lattice homology of $Y$ coincides with the Heegaard Floer homology of $Y$. We also give a conjectural description of the $H_1(Y)/\mathrm{Tors}$ action when $b_1(Y)>0$.