Rank inequalities for the Heegaard Floer homology of branched covers

TL;DR

This paper establishes inequalities relating the Heegaard Floer homologies of 3-manifolds connected by branched covers, explores implications for the L-space conjecture, and extends results to sutured Floer homology.

Contribution

It introduces new inequalities for Heegaard Floer homology dimensions under branched covers and relates these to topological conjectures and sutured Floer homology.

Findings

Inequality between Floer homology dimensions for branched covers

Connections to the L-space conjecture

Analogous results for sutured Floer homology

Abstract

Given a double cover between 3-manifolds branched along a nullhomologous link, we establish an inequality between the dimensions of their Heegaard Floer homologies. We discuss the relationship with the L-space conjecture and give some other topological applications, as well as an analogous result for sutured Floer homology.