The mod 2 cohomology of the infinite families of Coxeter groups of type B and D as almost Hopf rings

Abstract

We describe a Hopf ring structure on the direct sum of the cohomology groups ${⨁}_{n\ge 0}{H}^{\ast }\left(W_{B_n};{\mathbb{F}}_2\right)$ of the Coxeter groups of type $B_n$, and an almost-Hopf ring structure on the direct sum of the cohomology groups ${⨁}_{n\ge 0}{H}^{\ast }\left(W_{D_n};{\mathbb{F}}_2\right)$ of the Coxeter groups of type $D_n$, with coefficient in the field with two elements ${\mathbb{F}}_2$. We give presentations with generators and relations, determine additive bases and compute the Steenrod algebra action. The generators are described both in terms of a geometric construction by De Concini and Salvetti and in terms of their restriction to elementary abelian 2-subgroups.

66 pages, 5 figures. Revised on June, 11, 2021: the contents have been reorganized and some parts expanded, some arguments have been modified because of mathematical error. Revised on June, 30, 2021: the contents are unchanged from the previous version, but the results have been renumbered and there are minor typographical changes to improve clarity