The algebraic transfer for the real projective space

Hưng, Nguyễn H.V.; Trường, Lưu X.

doi:10.1016/j.crma.2019.01.001

Homological algebra/Topology

The algebraic transfer for the real projective space
[Transfert algébrique pour l'espace réel projectif]

Présenté par : the Editorial Board

Hưng, Nguyễn H.V.¹ ; Trường, Lưu X.¹

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

Comptes Rendus. Mathématique, Tome 357 (2019) no. 2, pp. 111-114.

Résumé
Abstract

Une description au niveau des chaînes du transfert de Singer pour tout $A$ -module à gauche est construite. Nous démontrons que l'image du transfert de Singer ${Tr}_{⁎}^{R P^{\infty}}$ pour l'espace projectif réel infini est un module sur l'image du transfert ${Tr}_{⁎}$ pour la sphère. De plus, l'homomorphisme algébrique de Kahn–Priddy est un épimorphisme de $Im {Tr}_{⁎}^{R P^{\infty}}$ sur $Im {Tr}_{⁎}$ en degré positif. Les éléments indécomposables ${\hat{h}}_{i}$ pour $i \geq 1$ et ${\hat{c}}_{i}$ , ${\hat{d}}_{i}$ , ${\hat{e}}_{i}$ , ${\hat{f}}_{i}$ , ${\hat{p}}_{i}$ pour $i \geq 0$ sont détectés, alors que les ${\hat{g}}_{i}$ pour $i \geq 1$ et ${\hat{D}}_{3} (i)$ , ${\hat{p}}_{i}^{'}$ pour $i \geq 0$ ne le sont pas. Ce transfert n'est pas injectif en chaque degré homologique positif. Le transfert est aussi étudié au voisinage des « éléments critiques ». Nous montrons que le morphisme de Kameko sur le domaine de ${Tr}_{⁎}^{R P^{\infty}}$ est un isomorphisme sur son image après un nombre suffisant d'itérations. Ce phénomène mène à la « stabilité » du transfert pour l'espace projectif réel infini sous l'action du morphisme de Kameko et sous l'action de l'élévation au carré itérée.

A chain-level representation of the Singer transfer for any left $A$ -module is constructed. We prove that the image of the Singer transfer ${Tr}_{⁎}^{R P^{\infty}}$ for the infinite real projective space is a module over the image of the transfer ${Tr}_{⁎}$ for the sphere. Further, the algebraic Kahn–Priddy homomorphism is an epimorphism from $Im {Tr}_{⁎}^{R P^{\infty}}$ onto $Im {Tr}_{⁎}$ in positive stems. The indecomposable elements ${\hat{h}}_{i}$ for $i \geq 1$ and ${\hat{c}}_{i}$ , ${\hat{d}}_{i}$ , ${\hat{e}}_{i}$ , ${\hat{f}}_{i}$ , ${\hat{p}}_{i}$ for $i \geq 0$ are detected, whereas the ones ${\hat{g}}_{i}$ for $i \geq 1$ and ${\hat{D}}_{3} (i)$ , ${\hat{p}}_{i}^{'}$ for $i \geq 0$ are not detected by the Singer transfer ${Tr}_{⁎}^{R P^{\infty}}$ . This transfer is shown to be not monomorphic in every positive homological degree. The transfer behavior is also investigated near “critical elements”. We prove that Kameko's squaring operation on the domain of ${Tr}_{⁎}^{R P^{\infty}}$ is eventually isomorphic. This phenomenon leads to the so-called “stability” of the Singer transfer for the infinite real projective space under the iterated squaring operation.

Reçu le : 2018-07-09
Accepté le : 2019-01-02
Publié le : 2019-01-21

DOI : 10.1016/j.crma.2019.01.001

Affiliations des auteurs :

Hưng, Nguyễn H.V. ¹ ; Trường, Lưu X. ¹

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

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Hưng, Nguyễn H.V.; Trường, Lưu X. The algebraic transfer for the real projective space. Comptes Rendus. Mathématique, Tome 357 (2019) no. 2, pp. 111-114. doi : 10.1016/j.crma.2019.01.001. http://www.numdam.org/articles/10.1016/j.crma.2019.01.001/

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^☆ This research is funded by the National Foundation for Science and Technology Development (NAFOSTED) of Vietnam under grant number 101.04-2014.19.