## 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.

## A Cup product input

For the study of symmetric groups in [10] we did not use the previously calculated cup coproduct in homology [6], which can be difficult to apply because of the need to account for Adem relations. Because Fox–Neuwirth cochains do not model cup product, contrary to what is sketched in [11], we use two such cup coproduct calculations for our study of alternating groups.

To set notation, we denote the product associated to the inclusion

(We prefer “lower index” notation.)
Given a sequence
*I*
*admissible*.
If such an *I* has no zeros, we call it *strongly admissible*. Let

Calculations of Nakaoka [19] imply that the homology of symmetric groups is a polynomial
algebra over the product

In [10] we define
*I* consists of
*m* blocks of

## Proposition A.1.

*The product
*

*m*blocks of three repeated 2’s.

*The product
*

*m*blocks of the form

## Proof.

We perform the arguments in parallel.
We start with
*x* is equal to that of

For pairings with
*u*, *v* and
*w* are all the equivalence class of

to

To pass to higher *m*, we perform an induction based on the detection
result of Madsen and Milgram [18], that the cohomology of symmetric groups is detected by coproduct, along with restriction to

The cup product representatives thus far have be obtained simply by adding Fox–Neuwirth entries of the factors.
In [11] we define a Hopf semi-ring structure on
Fox–Neuwirth cochains in this way, the homology of which agrees with the cohomology of symmetric groups. But, contrary to what is claimed and sketched in
[11], this abstract isomorphism is not induced by the map between then given by Alexander duality as in the proof of Theorem 4.9.
For example, while

## Acknowledgements

We thank Alejandro Adem and Paolo Salvatore for helpful comments throughout our time working on this project, and two referees for careful readings and useful comments.

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**Received:**2018-01-09

**Revised:**2020-01-09

**Published Online:**2020-08-05

**Published in Print:**2021-03-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston