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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 correspondingnon-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 thehomology groups of the local system $\mathcal{L}_\omega$? When $X$ is homotopyequivalent 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 anapplication, 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.

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