On the $\mathfrak{S}_{q} \overleftrightarrow{\times} \mathfrak{S}_q$-invariant rational cohomology of the pure braid group $P_{2q}$

Yan Fu; Gefei Wang

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arXiv · 2503.23787

On the $\mathfrak{S}_{q} \overleftrightarrow{\times} \mathfrak{S}_q$-invariant rational cohomology of the pure braid group $P_{2q}$

Yan Fu·Gefei Wang

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Abstract

Let $\mathfrak{S}_{q} \overleftrightarrow{\times} \mathfrak{S}_q$ be the $\mathbb{Z}/2$-extension of the product of two symmetric groups $\mathfrak{S}_{q} \times \mathfrak{S}_q$. In this paper, we compute the $\mathfrak{S}_{q} \overleftrightarrow{\times} \mathfrak{S}_q$-invariant part of the rational cohomology of the pure braid group $P_{2q}$. This includes the rational cohomology of a spin hyperelliptic mapping class group of genus $g$, where $g+1=q$.

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Yan Fu, Gefei Wang. 2025-03-31. On the $\mathfrak{S}_{q} \overleftrightarrow{\times} \mathfrak{S}_q$-invariant rational cohomology of the pure braid group $P_{2q}$. https://arxiv.org/abs/2503.23787

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