Integral homology groups of double coverings and rank one $\mathbb{Z}$-local system for minimal CW complex

TL;DR

This paper explores the relationship between the homology groups of double coverings of CW complexes and local systems, providing a complete answer for minimal CW complexes and confirming a conjecture for hyperplane arrangements.

Contribution

It establishes a precise connection between the homology of double coverings and local systems for minimal CW complexes, settling a recent conjecture for hyperplane arrangements.

Findings

Complete characterization for minimal CW complexes.

Confirmation of the conjecture for hyperplane arrangement complements.

Homology groups are combinatorially determined under certain conditions.

Abstract

Let $X$ be a connected finite CW complex. A connected double covering of $X$ is classified by a non-zero cohomology class $\omega \in H^1(X,\mathbb{Z}_2)$. Denote the double covering space by $X^\omega$. There exists a corresponding non-trivial rank one $\mathbb{Z}$-local system $\mathcal{L}_\omega$ on $X$. What is the relation between the integral homology groups of $X^\omega$ and the homology groups of the local system $\mathcal{L}_\omega$? When $X$ is homotopy equivalent to a minimal CW complex, we give a complete answer to this question. In particular, this settles a conjecture recently proposed by Ishibashi, Sugawara and Yoshinaga for hyperplane arrangement complement. As an application, when $X$ is a hyperplane arrangement complement and $\mathcal{L}_\omega$ satisfies certain conditions, we show that $H_*(X^\omega,\mathbb{Z})$ is combinatorially determined.