On the rational cohomology of spin hyperelliptic mapping class groups

Let [Formula: see text] be the subgroup [Formula: see text] of the [Formula: see text]th symmetric group [Formula: see text] for [Formula: see text]. In this paper, we study the [Formula: see text]-invariant part of the rational cohomology group of the pure braid group [Formula: see text]. The invariant part includes the rational cohomology of a spin hyperelliptic mapping class group of genus [Formula: see text] as a subalgebra when [Formula: see text], denoted by [Formula: see text]. Based on the study of [G. I. Lehrer and L. Solomon, On the action of the symmetric group on the cohomology of the complement of its reflecting hyperplanes, J. Algebra 104 (1986) 410–424], we prove that they are independent of [Formula: see text] and [Formula: see text] in degree [Formula: see text]. We also give a formula to calculate the dimension of [Formula: see text] and calculate it in all degrees for [Formula: see text].