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

Liu, Ye; Liu, Yongqiang

doi:10.1090/proc/16528

Integral homology groups of double coverings and rank one $\mathbb {Z}$-local systems for a minimal CW complex
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by Ye Liu and Yongqiang Liu;

Proc. Amer. Math. Soc. 151 (2023), 5007-5012

DOI: https://doi.org/10.1090/proc/16528

Published electronically: August 4, 2023

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Abstract:

Given a finite CW complex $X$, a nonzero cohomology class $\omega \in H^1(X,\mathbb {Z}_2)$ determines a double covering $X^\omega$ and a rank one $\mathbb {Z}$-local system $\mathcal {L}_\omega$. We investigate the relations between the homology groups $H_*(X^{\omega },\mathbb {Z})$ and $H_*(X,\mathcal {L}_\omega )$, when $X$ 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.

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