Mod-two cohomology rings of alternating groups

Author:

Giusti Chad1ORCID,Sinha Dev2

Affiliation:

1. Department of Mathematical Sciences , University of Delaware , Newark , DE 19716 , USA

2. Mathematics Department , University of Oregon , Eugene , OR 97403-1222 , USA

Abstract

Abstract We calculate the direct sum of the mod-two cohomology of all alternating groups, 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 are developed, including an almost Hopf ring structure associated to the embeddings of products of alternating groups and Fox–Neuwirth resolutions, which are new techniques. We also extend understanding of the Gysin sequence relating the cohomology of alternating groups to that of symmetric groups and calculation of restriction to elementary abelian subgroups.

Publisher

Walter de Gruyter GmbH

Subject

Applied Mathematics,General Mathematics

Reference24 articles.

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2. A. Adem and R. J. Milgram, Cohomology of finite groups, 2nd ed., Grundlehren Math. Wiss. 309, Springer, Berlin 2004.

3. M. F. Atiyah, Power operations in K-theory, Quart. J. Math. Oxford Ser. (2) 17 (1966), 165–193.

4. R. R. Bruner, J. P. May, J. E. McClure and M. Steinberger, H ∞ H_{\infty} ring spectra and their applications, Lecture Notes in Math. 1176, Springer, Berlin 1986.

5. T. Church, Homological stability for configuration spaces of manifolds, Invent. Math. 188 (2012), no. 2, 465–504.

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