On the rational cohomology of spin hyperelliptic mapping class groups

TL;DR

This work studies the $\mathfrak{G}$-invariant part of the rational cohomology of the pure braid group, with $\mathfrak{G}=\mathfrak{S}_{n-q}\times\mathfrak{S}_q$, and connects it to the cohomology of spin hyperelliptic mapping class groups when $n=2g+2$. Building on the Lehrer–Solomon description, the authors introduce a partition-indexed framework and a combinatorial invariant $\chi(\delta)$ to determine which summands contribute to $H^*(P_n)^{\mathfrak{G}}$, and prove that these invariants yield independence from $n$ and $q$ in degrees $*\le q-1$. They derive an explicit dimension formula (Theorem 3.1) and compute it completely for $q\le 3$ (Theorems 3.4–3.6), with the counting reduced to $|\Pi(\lambda_i,d_i)|$ via Theorem 3.3. The results provide concrete, low-rank data on the unstable rational cohomology of spin hyperelliptic mapping class groups and link it to the combinatorics of partitions and centralizers in the symmetric group.

Abstract

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 $q\leq 3$.