Mod-two cohomology rings of alternating groups

2017/05/02 by Giusti, Chad, Sinha, Dev
#Algebraic Topology (math.AT) #FOS: Mathematics #K-Theory and Homology (math.KT) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1705.01141

Abstract

We calculate the mod-two cohomology of all alternating groups together, with both cup and transfer product structures, which in particular determines the additive structure and ring structure of the cohomology of individual groups. We show that there are no nilpotent elements in the cohomology rings of individual alternating groups. We calculate the action of the Steenrod algebra and discuss individual component rings. A range of techniques is needed: an almost Hopf ring structure associated to the embeddings of products of alternating groups, the Gysin sequence relating the cohomology of alternating groups to that of symmetric groups, Fox-Neuwirth resolutions, and restriction to elementary abelian subgroups.

Related

• The mod 2 cohomology rings of the alternating subgroups of the Coxeter groups of Type B
• The cohomology of the Lyons group and double covers of alternating groups
• Invariants cohomologiques mod 2 et invariants de Witt des groupes alternés
• Cohomology rings of extended powers and free infinite loop spaces
• Stable cohomology of alternating groups