Covering spaces and Q-gradings on Heegaard Floer homology

TL;DR

This paper explores an alternative method to define the relative Q-grading in Heegaard Floer homology by analyzing the homology of covering spaces, providing new insights into the grading structure.

Contribution

It introduces a novel approach to construct the relative Q-grading in Heegaard Floer homology using covering space techniques.

Findings

Provides an explicit construction of the relative Q-grading via covering spaces

Establishes connections between covering space homology and Floer gradings

Enhances understanding of grading structures in Heegaard Floer homology

Abstract

Heegaard Floer homology, first introduced by P. Ozsvath and Z. Szabo, associates to a 3-manifold Y a family of relatively graded Abelian groups HF(Y,t), indexed by Spin^c structures t on Y. In the case that Y is a rational homology sphere, Ozsvath and Szabo lift the relative Z-grading to an absolute Q-grading. This induces a relative Q-grading on \bigoplus_{t\in Spin^c(Y)} HF(Y,t). In this paper we describe an alternate construction of this relative Q-grading by studying the Heegaard Floer homology of covering spaces.