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Mod-two cohomology rings of alternating groups

Chad Giusti ORCID logo and Dev Sinha

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 𝒮 n × 𝒮 m 𝒮 n + m by * , which is thus dual to our coproduct in cohomology. There are “wreath product” operations q i : H k ( B 𝒮 n ) H 2 k + i ( B 𝒮 2 n ) which satisfy Adem relations:

For  m > n , q m q n = i ( i - n - 1 2 i - m - n ) q m + 2 n - 2 i q i .

(We prefer “lower index” notation.) Given a sequence I = i 1 , , i k of non-negative integers, let q I = q i 1 q i k . Using the Adem relations, these relations are spanned by q I whose entries are non-decreasing. We call such an I admissible. If such an I has no zeros, we call it strongly admissible. Let ι H 0 ( B 𝒮 1 ) be the non-zero class, and by abuse let q I denote q I ( ι )

Calculations of Nakaoka [19] imply that the homology of symmetric groups is a polynomial algebra over the product * generated by strongly admissible q I . Cohen, Lada and May [6] developed the theory much further, including showing that the cup coproduct is given by

Δ q I = J + K = I q J q K .

In [10] we define γ , m to be the linear dual of q I * m , where I consists of ones, in the Nakaoka basis. In [10, Theorem 4.9] we essentially show that γ , m is represented by the Fox–Neuwirth cocycle with m blocks of 2 - 1 repeated ones. To verify cup product relations, we establish Fox–Neuwirth representatives of two types of products of these.

Proposition A.1.

The product γ 2 , m 2 is represented by the Fox–Neuwirth cochain α 2 , m ( 2 ) with m blocks of three repeated 2’s.

The product γ , m γ 1 , m 2 - 1 is represented by the Fox–Neuwirth cochain α , m ( 1.5 ) , which has m blocks of the form [ 2 , 1 , 2 , , 1 , 2 ] , each of length 2 - 1 .

Proof.

We perform the arguments in parallel. We start with m = 1 , showing that both the product γ 2 , 1 2 and [ 2 , 2 , 2 ] are linear dual in the Dyer–Lashof basis of q 2 , 2 (respectively, both γ , 1 γ 1 , 2 - 1 and [ 2 , 1 , 2 , , 1 , 2 ] are the linear dual of q 1 , , 1 , 2 ). We calculate pairings, starting with γ 2 , 1 2 , whose value on some x is equal to that of γ 2 , 1 γ 2 , 1 (respectively, γ , 1 γ 1 , 2 - 1 ) on Δ x . For x = q 2 , 2 (respectively, q 1 , , 1 , 2 ) we see that the only term in Δ q 2 , 2 in the correct pair of degrees is q 1 , 1 q 1 , 1 (respectively, q 1 , , 1 q 0 , , 0 , 1 ), which pairs to one by definition. Any *-decomposable will have decomposable coproduct, and these will evaluate to zero on a tensor factor of γ , 1 by definition.

For pairings with [ 2 , 2 , 2 ] (respectively, [ 2 , 1 , 2 , , 1 , 2 ] ), * -decomposable classes also evaluate to zero, by the analogue of Theorem 4.16. We then apply the symmetric groups version of Proposition 4.15 to evaluate [ 2 , 2 , 2 ] on q 2 , 2 , which by definition is represented by a map from S 2 × 𝒮 2 ( P 2 × P 2 ) to Conf ¯ 4 ( ) sending ( u , v , w ) / to the configuration with points at u ± ε v and - u ± ε w . We get a single transversal intersection when u, v and w are all the equivalence class of ( 0 , 0 , 1 ) . To evaluate [ 2 , 1 , 2 , , 1 , 2 ] on q 1 , , 1 , 2 , we represent the latter by a standard map of the iterated wreath product S 1 P 2 , where

S 1 X = S 1 × 𝒮 2 ( X × X ) ,

to Conf ¯ 2 ( ) . Once again, we will get a single transversal intersection when each coordinate entry of S 1 is ( 0 , ± 1 ) and each of P 2 is the equivalence class of ( 0 , 0 , 1 ) .

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 V n for 𝒮 2 n . The formulas for coproducts of γ 2 , m 2 and α 2 , m ( 2 ) (respectively, γ , m γ 1 , m 2 - 1 and α , m ( 1.5 ) ), the former given by Theorem 3.1 and Hopf semi-ring distributivity and the latter given by Theorem 4.16, will be equal by inductive assumption, and application of detection establishes the induction step when m 2 n . When m = 2 n , we note that by having the same coproducts they must agree up to the kernel of restriction to V n , which is generated by γ n , 1 . But there are no non-zero multiples of γ n , 1 in the relevant degrees. ∎

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 γ 1 , 2 is represented by [ 1 , 0 , 1 ] , its cube γ 1 , 2 3 cannot be represented by [ 3 , 0 , 3 ] . Its cube evaluates non-trivially with the class q 2 , 2 in homology, whose coproduct includes q 2 , 0 q 0 , 2 = q 0 , 1 q 0 , 2 by the first Adem relation. But the image of q 2 , 2 can be embedded in configurations in 3 and so cannot pair with [ 3 , 0 , 3 ] . Indeed, γ 1 , 2 3 must be represented by [ 3 , 0 , 3 ] + [ 2 , 2 , 2 ] . There are variants of Fox–Neuwirth cochains which should result in cup product models as well. Thankfully, our need for cup product input at the cochain level was limited in the present work.

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

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