On the rational cohomology of spin hyperelliptic mapping class groups

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 $n-q\ge q$. 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 $H^{\ast }(P_n{)}^{\mathfrak{G}}$. Based on the study of Lehrer-Solomon, we prove that they are independent of $n$ and $q$ in degree $\ast \le q-1$. We also give a formula to calculate the dimension of $H^{\ast }(P_n{)}^{\mathfrak{G}}$ and calculate it in all degree for $q\le 3$.