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Comptes Rendus Mathématique
Volume 357, n° 2
pages 111-114 (février 2019)
Doi : 10.1016/j.crma.2019.01.001
Received : 9 July 2018 ;  accepted : 2 January 2019
The algebraic transfer for the real projective space
Transfert algébrique pour l'espace réel projectif
 

Nguyễn H.V. Hưng , Lưu X. Trường
 Department of Mathematics, Vietnam National University, Hanoi, 334 Nguyễn Trãi Street, Hanoi, Viet Nam 

Abstract

A chain-level representation of the Singer transfer for any left  -module is constructed. We prove that the image of the Singer transfer   for the infinite real projective space is a module over the image of the transfer   for the sphere. Further, the algebraic Kahn–Priddy homomorphism is an epimorphism from   onto   in positive stems. The indecomposable elements   for   and  ,  ,  ,  ,   for   are detected, whereas the ones   for   and  ,   for   are not detected by the Singer transfer  . 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   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.

The full text of this article is available in PDF format.
Résumé

Une description au niveau des chaînes du transfert de Singer pour tout  -module à gauche est construite. Nous démontrons que l'image du transfert de Singer   pour l'espace projectif réel infini est un module sur l'image du transfert   pour la sphère. De plus, l'homomorphisme algébrique de Kahn–Priddy est un épimorphisme de   sur   en degré positif. Les éléments indécomposables   pour   et  ,  ,  ,  ,   pour   sont détectés, alors que les   pour   et  ,   pour   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   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.

The full text of this article is available in PDF format.

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


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