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

Guerra, Lorenzo

doi:10.2140/agt.2023.23.3221

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

Lorenzo Guerra

Algebraic & Geometric Topology 23 (2023) 3221–3292

DOI: 10.2140/agt.2023.23.3221

Abstract

We describe a Hopf ring structure on the direct sum of the cohomology groups $\oplus_{n \geq 0} H^{*} (W_{B_{n}}; 𝔽_{2})$ of the Coxeter groups of type $W_{B_{n}}$ , and an almost-Hopf ring structure on the direct sum of the cohomology groups $\oplus_{n \geq 0} H^{*} (W_{D_{n}}; 𝔽_{2})$ of the Coxeter groups of type $W_{D n}$ , with coefficients in the field with two elements $𝔽_{2}$ . 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 their restriction to elementary abelian $2$ –subgroups.

Keywords

Hopf ring, Coxeter groups, group cohomology, configuration spaces

Mathematical Subject Classification

Primary: 20F55, 20J06

Secondary: 20J05

References

Publication

Received: 7 September 2021

Revised: 30 April 2022

Accepted: 18 May 2022

Published: 26 September 2023

Authors

Lorenzo Guerra
Dipartimento di Matematica Università di Roma Tor Vergata Rome Italy

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Volume 23, issue 7 (2023)