Set-theoretic Yang-Baxter cohomology of cyclic biquandles
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866913543667843072 |
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| author | Liang, Minyi Wang, Xiao Yang, Seung Yeop |
| author_facet | Liang, Minyi Wang, Xiao Yang, Seung Yeop |
| contents | We completely determine the free parts of the set-theoretic Yang-Baxter (co)homology groups of finite cyclic biquandles, along with fully computing the torsion subgroups of their 1st and 2nd homology groups. Furthermore, we provide upper bounds for the orders of torsions in the 3rd and higher dimensional homology groups. This work partially solves the conjecture that the normalized set-theoretic Yang-Baxter homology of cyclic biquandles satisfy $H_{n}^{NYB}(C_{m}) = \mathbb{Z}^{(m-1)^{n-1}} \oplus \mathbb{Z}_{m}$ when $n$ is odd and $H_{n}^{NYB}(C_{m}) = \mathbb{Z}^{(m-1)^{n-1}}$ when $n$ is even. In addition, we obtain cocycle representatives of a basis for the rational cohomology group of a cyclic biquandle and introduce several non-trivial torsion homology classes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2108_03019 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Set-theoretic Yang-Baxter cohomology of cyclic biquandles Liang, Minyi Wang, Xiao Yang, Seung Yeop Geometric Topology Algebraic Topology Primary: 20G10, 55N35, 57K18, Secondary: 58H10, 57K12, 55S20 We completely determine the free parts of the set-theoretic Yang-Baxter (co)homology groups of finite cyclic biquandles, along with fully computing the torsion subgroups of their 1st and 2nd homology groups. Furthermore, we provide upper bounds for the orders of torsions in the 3rd and higher dimensional homology groups. This work partially solves the conjecture that the normalized set-theoretic Yang-Baxter homology of cyclic biquandles satisfy $H_{n}^{NYB}(C_{m}) = \mathbb{Z}^{(m-1)^{n-1}} \oplus \mathbb{Z}_{m}$ when $n$ is odd and $H_{n}^{NYB}(C_{m}) = \mathbb{Z}^{(m-1)^{n-1}}$ when $n$ is even. In addition, we obtain cocycle representatives of a basis for the rational cohomology group of a cyclic biquandle and introduce several non-trivial torsion homology classes. |
| title | Set-theoretic Yang-Baxter cohomology of cyclic biquandles |
| topic | Geometric Topology Algebraic Topology Primary: 20G10, 55N35, 57K18, Secondary: 58H10, 57K12, 55S20 |
| url | https://arxiv.org/abs/2108.03019 |