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.References
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Bibliographic Information
- Ye Liu
- Affiliation: Department of Pure Mathematics, Xi’an Jiaotong-Liverpool University, 111 Ren’ai Road, Suzhou, Jiangsu 215123, People’s Republic of China
- ORCID: 0000-0002-3972-5880
- Email: yeliumath@gmail.com
- Yongqiang Liu
- Affiliation: The Institute of Geometry and Physics, University of Science and Technology of China, 96 Jinzhai Road, Hefei, Anhui 230026, People’s Republic of China
- MR Author ID: 1135046
- Email: liuyq@ustc.edu.cn
- Received by editor(s): October 8, 2022
- Received by editor(s) in revised form: March 3, 2023, March 27, 2023, and April 21, 2023
- Published electronically: August 4, 2023
- Additional Notes: The first author was supported by NSFC grant No. 11901467.
The second author was supported by National Key Research and Development Project SQ2020YFA070080, the starting grant from University of Science and Technology of China, NSFC grant No. 12001511, the Project of Stable Support for Youth Team in Basic Research Field, CAS (YSBR-001), the project “Analysis and Geometry on Bundles” of Ministry of Science and Technology of the People’s Republic of China and Fundamental Research Funds for the Central Universities. - Communicated by: Julie Bergner
- © Copyright 2023 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 151 (2023), 5007-5012
- MSC (2020): Primary 55N25, 52C35
- DOI: https://doi.org/10.1090/proc/16528
- MathSciNet review: 4634902