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.
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.
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Nguyễn H.V. Hưng 1; Lưu X. Trường 1
@article{CRMATH_2019__357_2_111_0, author = {Nguyễn H.V. Hưng and Lưu X. Trường}, title = {The algebraic transfer for the real projective space}, journal = {Comptes Rendus. Math\'ematique}, pages = {111--114}, publisher = {Elsevier}, volume = {357}, number = {2}, year = {2019}, doi = {10.1016/j.crma.2019.01.001}, language = {en}, }
Nguyễn H.V. Hưng; Lưu X. Trường. The algebraic transfer for the real projective space. Comptes Rendus. Mathématique, Volume 357 (2019) no. 2, pp. 111-114. doi : 10.1016/j.crma.2019.01.001. https://comptes-rendus.academie-sciences.fr/mathematique/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.
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