The Heegaard Floer d-invariant for more rational homology spheres

TL;DR

This paper proves Némethi's conjecture that a lattice homology formula for the Heegaard Floer d-invariant applies to all negative-definite plumbed rational homology spheres, using Zemke's isomorphism.

Contribution

It establishes the validity of Némethi's formula for the d-invariant across a broad class of rational homology spheres, confirming a longstanding conjecture.

Findings

Proves Némethi's conjecture for all negative-definite plumbed rational homology spheres.

Uses Zemke's isomorphism to connect lattice and Heegaard Floer homology.

Confirms the universality of the lattice homology formula for the d-invariant.

Abstract

The Heegaard Floer d-invariant for a rational homology sphere Y and spin$^c$-structure $\mathfrak{s}$ is defined as the minimal absolute grading of a generator of $HF^+(Y; \mathfrak{s})$. In 2005, N\'emethi used lattice homology to compute the d-invariant for a particular class of negative-definite plumbed rational homology spheres, and conjectured that his formula should hold for all negative-definite plumbed rational homology spheres. In this paper, we use Zemke's isomorphism between lattice and Heegaard Floer homology to prove N\'emethi's conjecture.