Topologie algébrique
The mod 2 Margolis homology of the Dickson algebra
Comptes Rendus. Mathématique, Tome 358 (2020) no. 4, pp. 505-510.

Dans cette note on calcule entièrement l’homologie de Margolis modulo 2 de l’algèbre de Dickson D n , i.e. l’homologie de D n en choisissant pour différentielles les opérations de Milnor Q j , pour tous n et j. La motivation pour cette étude est le rôle clé joué par cette homologie dans l’étude de la K-théorie de Morava K(j) * (BS m ) du groupe symétrique S m en m lettres.

Nous montrons que la conjecture de Pengelley–Sinha sur H * (D n ;Q j ) pour nj est vraie si et seulement si n=1,2. Pour 3nj notre résultat montre que la conjecture est fausse à cause de l’occurence d’éléments « critiques » h s 1 ,,s k de degré (2 j+1 -2 n )+ i=1 k (2 n -2 s i ) dans cette homologie pour 0<s 1 <<s k <n et k>1.

We completely compute the mod 2 Margolis homology of the Dickson algebra D n , i.e. the homology of D n with the differential to be the Milnor operation Q j , for every n and j. The motivation for this problem is that, the Margolis homology of the Dickson algebra plays a key role in study of the Morava K-theory K(j) * (BS m ) of the symmetric group on m letters S m .

We show that Pengelley–Sinha’s conjecture on H * (D n ;Q j ) for nj is true if and only if n=1 or 2. For 3nj, our result proves that this conjecture turns out to be false since the occurrence of some “critical elements” h s 1 ,,s k ’s of degree (2 j+1 -2 n )+ i=1 k (2 n -2 s i ) in this homology for 0<s 1 <<s k <n and k>1.

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DOI : 10.5802/crmath.68
Classification : 55S05, 55S10, 55N99
Hưng, Nguyễn H. V. 1

1 Department of Mathematics, HUS, Vietnam National University, Hanoi, 334 Nguyễn Trãi Street, Hanoi, Vietnam
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Hưng, Nguyễn H. V. The mod 2 Margolis homology of the Dickson algebra. Comptes Rendus. Mathématique, Tome 358 (2020) no. 4, pp. 505-510. doi : 10.5802/crmath.68. http://www.numdam.org/articles/10.5802/crmath.68/

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