Comptes Rendus
Homological algebra/Topology
The algebraic transfer for the real projective space
Comptes Rendus. Mathématique, Volume 357 (2019) no. 2, pp. 111-114.

A chain-level representation of the Singer transfer for any left A-module is constructed. We prove that the image of the Singer transfer TrRP 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 ImTrRP onto ImTr in positive stems. The indecomposable elements hˆi for i1 and cˆi, dˆi, eˆi, fˆi, pˆi for i0 are detected, whereas the ones gˆi for i1 and Dˆ3(i), pˆi for i0 are not detected by the Singer transfer TrRP. 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 TrRP 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 A-module à gauche est construite. Nous démontrons que l'image du transfert de Singer TrRP 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 ImTrRP sur ImTr en degré positif. Les éléments indécomposables hˆi pour i1 et cˆi, dˆi, eˆi, fˆi, pˆi pour i0 sont détectés, alors que les gˆi pour i1 et Dˆ3(i), pˆi pour i0 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 TrRP 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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2019.01.001

Nguyễn H.V. Hưng 1; Lưu X. Trường 1

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

[1] J.F. Adams On the non-existence of elements of Hopf invariant one, Ann. of Math. (2), Volume 72 (1960), pp. 20-104

[2] J.M. Boardman Modular representations on the homology of powers of real projective space (M.C. Tangora, ed.), Algebraic Topology: Oaxtepec 1991, Contemp. Math., vol. 146, 1993, pp. 49-70

[3] R.R. Bruner; L.M. Hà; N.H.V. Hưng On behavior of the algebraic transfer, Trans. Amer. Math. Soc., Volume 357 (2005), pp. 473-487

[4] T.W. Chen Determination of ExtA5,(Z/2,Z/2), Topol. Appl., Volume 158 (2011), pp. 660-689

[5] L.M. Hà Sub-Hopf algebras of the Steenrod algebra and the Singer transfer (J. Hubbuck; N.H.V. Hưng; L. Schwartz, eds.), Proc. Hanoi 2004 School and Conf. in Alg. Topology, Geom. Topol. Monogr., vol. 11, 2007, pp. 101-124

[6] N.H.V. Hưng The weak conjecture on spherical classes, Math. Z., Volume 231 (1999), pp. 727-743

[7] N.H.V. Hưng The cohomology of the Steenrod algebra and representations of the general linear groups, Trans. Amer. Math. Soc., Volume 357 (2005), pp. 4065-4089

[8] N.H.V. Hưng; V.T.N. Quỳnh The image of Singer's fourth transfer, C. R. Acad. Sci. Paris, Ser. I, Volume 347 (2009), pp. 1415-1418

[9] M. Kameko Products of Projective Spaces as Steenrod Modules, Johns Hopkins University, Baltimore, MD, USA, 1990 (Thesis)

[10] W.H. Lin ExtA4,(Z/2,Z/2) and ExtA5,(Z/2,Z/2), Topol. Appl., Volume 155 (2008), pp. 459-496

[11] J.P. May (Lect. Notes Math.), Volume vol. 168, Springer-Verlag (1970), pp. 153-231

[12] H. Mùi Modular invariant theory and the cohomology algebras of symmetric group, J. Frac. Sci. Univ. Tokyo Sect. IA Math., Volume 22 (1975), pp. 319-369

[13] W.M. Singer Invariant theory and the Lambda algebra, Trans. Amer. Math. Soc., Volume 280 (1983), pp. 673-693

[14] W.M. Singer The transfer in homological algebra, Math. Z., Volume 202 (1989), pp. 493-523

[15] M.C. Tangora On the cohomology of the Steenrod algebra, Math. Z., Volume 116 (1970), pp. 18-64

[16] J.S.P. Wang On the cohomology of the mod 2 Steenrod algebra and the non-existence of elements of Hopf invariant one, Ill. J. Math., Volume 11 (1967), pp. 480-490

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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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