Kameko's homomorphism and the algebraic transfer

Tín, Nguyễn Khắc; Sum, Nguyễn

doi:10.1016/j.crma.2016.06.005

Topology

Kameko's homomorphism and the algebraic transfer
[Le morphisme de Kameko et le transfert algébrique]

Présenté par : Claire Voisin

Tín, Nguyễn Khắc ¹ ; Sum, Nguyễn ²

¹ Department of Foundation Sciences, University of Technical Education of Hồ Chí Minh City, 01 Võ Văn Ngân, Thủ Đức District, Hồ Chí Minh City, Viet Nam
² Department of Mathematics, Quy Nhơn University, 170 An Dương Vương, Quy Nhơn, Bình Định, Viet Nam

Comptes Rendus. Mathématique, Tome 354 (2016) no. 9, pp. 940-943.

Résumé
Abstract

Soit $P_{k} : = F_{2} [x_{1}, x_{2}, \dots, x_{k}]$ l'algèbre polynomiale graduée à k générateurs sur le corps à deux éléments $F_{2}$ , chaque générateur étant de degré 1. En tant que cohomologie mod-2 du classifant $B {(Z / 2)}^{k}$ , l'algèbre $P_{k}$ est dotée d'une structure naturelle de module sur l'algèbre de Steenrod $A$ . Dans cette Note, nous généralisons un résultat de Hưng pour le morphisme de Kameko ${\tilde{S q}}_{⁎}^{0} : F_{2} \otimes_{A} P_{k} ⟶ F_{2} \otimes_{A} P_{k}$ . En appliquant ce résultat, nous montrons que la conjecture de Singer pour le transfert algébrique est vraie pour $k = 5$ et le degré $7, 2^{s} - 5$ avec $s > 0$ .

Let $P_{k} : = F_{2} [x_{1}, x_{2}, \dots, x_{k}]$ be the graded polynomial algebra over the prime field of two elements $F_{2}$ , in k generators $x_{1}, x_{2}, \dots, x_{k}$ , each of degree 1. Being the mod-2 cohomology of the classifying space $B {(Z / 2)}^{k}$ , the algebra $P_{k}$ is a module over the mod-2 Steenrod algebra $A$ . In this Note, we extend a result of Hưng on Kameko's homomorphism ${\tilde{S q}}_{⁎}^{0} : F_{2} \otimes_{A} P_{k} ⟶ F_{2} \otimes_{A} P_{k}$ . Using this result, we show that Singer's conjecture for the algebraic transfer is true in the case $k = 5$ and the degree ${7.2}^{s} - 5$ with s an arbitrary positive integer.

Reçu le : 2016-02-15
Accepté le : 2016-06-27
Publié le : 2016-07-18

DOI : 10.1016/j.crma.2016.06.005

Affiliations des auteurs :

Tín, Nguyễn Khắc ¹ ; Sum, Nguyễn ²

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Tín, Nguyễn Khắc; Sum, Nguyễn. Kameko's homomorphism and the algebraic transfer. Comptes Rendus. Mathématique, Tome 354 (2016) no. 9, pp. 940-943. doi : 10.1016/j.crma.2016.06.005. http://www.numdam.org/articles/10.1016/j.crma.2016.06.005/

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