On the rational cohomology of spin hyperelliptic mapping class groups
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866909063390953472 |
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| author | Wang, Gefei |
| author_facet | Wang, Gefei |
| contents | Let $\mathfrak{G}$ be the subgroup $\mathfrak{S}_{n-q} \times \mathfrak{S}_{q}$ of the $n$-th symmetric group $\mathfrak{S}_{n}$ for $n-q \geq q$. In this paper, we study the $\mathfrak{G}$-invariant part of the rational cohomology group of the pure braid group $P_{n}$. The invariant part includes the rational cohomology of a spin hyperelliptic mapping class group of genus $g$ as a subalgebra when $n=2g+2$, denoted by $H^*(P_{n})^{\mathfrak{G}}$. Based on the study of Lehrer-Solomon, we prove that they are independent of $n$ and $q$ in degree $*\leq q-1$. We also give a formula to calculate the dimension of $H^*(P_{n})^{\mathfrak{G}}$ and calculate it in all degree for $q\leq 3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_03110 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the rational cohomology of spin hyperelliptic mapping class groups Wang, Gefei Geometric Topology 20F36 (Primary), 20J06, 57K20, 14H30 (Secondary) Let $\mathfrak{G}$ be the subgroup $\mathfrak{S}_{n-q} \times \mathfrak{S}_{q}$ of the $n$-th symmetric group $\mathfrak{S}_{n}$ for $n-q \geq q$. In this paper, we study the $\mathfrak{G}$-invariant part of the rational cohomology group of the pure braid group $P_{n}$. The invariant part includes the rational cohomology of a spin hyperelliptic mapping class group of genus $g$ as a subalgebra when $n=2g+2$, denoted by $H^*(P_{n})^{\mathfrak{G}}$. Based on the study of Lehrer-Solomon, we prove that they are independent of $n$ and $q$ in degree $*\leq q-1$. We also give a formula to calculate the dimension of $H^*(P_{n})^{\mathfrak{G}}$ and calculate it in all degree for $q\leq 3$. |
| title | On the rational cohomology of spin hyperelliptic mapping class groups |
| topic | Geometric Topology 20F36 (Primary), 20J06, 57K20, 14H30 (Secondary) |
| url | https://arxiv.org/abs/2401.03110 |