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

Abstract

Let $X$ be a connected finite CW complex. A connected double covering of $X$ is classified by a non-zero cohomology class $\ddot{I}‰\in H^1(X,{\mathbb{Z}}_2)$. Denote the double covering space by $X^{\ddot{I}}‰$. There exists a corresponding non-trivial rank one $\mathbb{Z}$-local system ${\mathcal{L}}_{\ddot{I}}‰$ on $X$. What is the relation between the integral homology groups of $X^{\ddot{I}}‰$ and the homology groups of the local system ${\mathcal{L}}_{\ddot{I}}‰$? 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 ${\mathcal{L}}_{\ddot{I}}‰$ satisfies certain conditions, we show that $H_{\ast }(X^{\ddot{I}}‰,\mathbb{Z})$ is combinatorially determined.

Final version, to appear in Proc. Amer. Math. Soc