On the vanishing of the Lannes–Zarati homomorphism

Nguyễn H.V. Hưng; Võ T.N. Quỳnh; Ngô A. Tuấn

doi:10.1016/j.crma.2014.01.013

Topology

On the vanishing of the Lannes–Zarati homomorphism
[Sur l'annulation de l'homomorphisme de Lannes–Zarati]

Présenté par : the Editorial Board

Nguyễn H.V. Hưng ¹ ; Võ T.N. Quỳnh ¹ ; Ngô A. Tuấn ¹

¹ Department of Mathematics, Vietnam National University, Hanoi, 334 Nguyễn Trãi Street, Hanoi, Viet Nam

Comptes Rendus. Mathématique, Volume 352 (2014) no. 3, pp. 251-254.

Abstract (VO)
Résumé

The conjecture on spherical classes states that the Hopf invariant one and the Kervaire invariant one classes are the only elements in $H_{⁎} (Q_{0} S^{0})$ belonging to the image of the Hurewicz homomorphism. The Lannes–Zarati homomorphism is a map that corresponds to an associated graded (with a certain filtration) of the Hurewicz map. The algebraic version of the conjecture predicts that the s-th Lannes–Zarati homomorphism vanishes in any positive stems for $s > 2$ . In the article, we prove the conjecture for the fifth Lannes–Zarati homomorphism.

La conjecture sur les classes sphériques affirme que les classes détectées par l'invariant de Hopf et l'invariant de Kervaire sont les seules dans $H_{⁎} (Q_{0} S^{0})$ dans l'image de l'homomorphisme de Hurewicz. L'homomorphisme de Lannes–Zarati est l'application correspondant au gradué (pour une certaine filtration) de l'homomorphisme de Hurewicz. La version algébrique de la conjecture prédit que le s-ième homomorphisme de Lannes–Zarati s'annule en tout degré positif pour $s > 2$ . Dans cette note, nous démontrons la conjecture pour le cinquième homomorphisme de Lannes–Zarati.

Reçu le : 2013-10-15
Accepté le : 2014-01-17
Publié le : 2014-02-06

DOI : 10.1016/j.crma.2014.01.013

Affiliations des auteurs :

Nguyễn H.V. Hưng ¹ ; Võ T.N. Quỳnh ¹ ; Ngô A. Tuấn ¹

¹ Department of Mathematics, Vietnam National University, Hanoi, 334 Nguyễn Trãi Street, Hanoi, Viet Nam

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     title = {On the vanishing of the {Lannes{\textendash}Zarati} homomorphism},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {251--254},
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Nguyễn H.V. Hưng; Võ T.N. Quỳnh; Ngô A. Tuấn. On the vanishing of the Lannes–Zarati homomorphism. Comptes Rendus. Mathématique, Volume 352 (2014) no. 3, pp. 251-254. doi : 10.1016/j.crma.2014.01.013. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.01.013/

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Cité par Sources :

^☆ The work was supported in part by a Grant of the NAFOSTED.

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