Integral homology groups of double coverings and rank one ℤ-local systems for a minimal CW complex

Abstract

Given a finite CW complex X X , a nonzero cohomology class ω ∈ H 1 ( X , Z 2 ) \omega \in H^1(X,\mathbb {Z}_2) determines a double covering X ω X^\omega and a rank one Z \mathbb {Z} -local system L ω \mathcal {L}_\omega . We investigate the relations between the homology groups H ∗ ( X ω , Z ) H_*(X^{\omega },\mathbb {Z}) and H ∗ ( X , L ω ) H_*(X,\mathcal {L}_\omega ) , when X X is homotopy equivalent to a minimal CW complex. In particular, this settles a conjecture recently proposed by Ishibashi, Sugawara and Yoshinaga [Betti numbers and torsions in homology groups of double coverings, arxiv.org/abs/2209.02236, 2022, Conjecture 3.3], for a hyperplane arrangement complement.