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The Twisted Homology of Simplicial Set

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Abstract

In this article, we give a generalization of δ-twisted homology introduced by Jingyan Li, Vladimir Vershinin and Jie Wu, called Δ-twisted homology, which enriches the theory of δ-(co)homology introduced by Alexander Grigor’yan, Yuri Muranov and Shing-Tung Yau. We show that the Mayer—Vietoris sequence theorem holds for Δ-twisted homology. Applying the Δ-twisted ideas to Cartesian products, we introduce the notion of Δ-twisted Cartesian product on simplicial sets, which generalizes the classical work of Barratt, Gugenheim and Moore on twisted Cartesian products of simplicial sets. Under certain hypothesis, we show that the coordinate projection of Δ-twisted Cartesian product admits a fibre bundle structure.

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Acknowledgements

The authors are very grateful to the referees for their time and comments.

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Authors and Affiliations

  1. School of Mathematical Sciences, Hebei Normal University, Shijiazhuang, 050024, P. R. China

    Meng Meng Zhang & Jie Wu

  2. Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, Beijing, 101408, P. R. China

    Meng Meng Zhang, Jing Yan Li & Jie Wu

Authors
  1. Meng Meng Zhang
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  2. Jing Yan Li
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  3. Jie Wu
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Corresponding author

Correspondence to Jie Wu.

Additional information

Dedicated to Professor Banghe Li on His 80th Birthday

Supported by NSFC (Grant No. 11971144), High-level Scientific Research Foundation of Hebei Province and the start-up research fund from BIMSA. The first author is also supported by Postgraduate Innovation Funding Project of Hebei Province (Grant No. CXZZBS2022073)

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Zhang, M.M., Li, J.Y. & Wu, J. The Twisted Homology of Simplicial Set. Acta. Math. Sin.-English Ser. 38, 1781–1802 (2022). https://doi.org/10.1007/s10114-022-2190-3

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  • DOI: https://doi.org/10.1007/s10114-022-2190-3

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