The algebraic transfer for the real projective space

Nguyễn H.V. Hưng; Lưu X. Trường

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

Nguyễn H.V. Hưng ¹ ; Lưu X. Trường ¹

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

Comptes Rendus. Mathématique, Volume 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 :

Nguyễn H.V. Hưng ¹ ; Lưu X. Trường ¹

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

@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},
}

TY  - JOUR
AU  - Nguyễn H.V. Hưng
AU  - Lưu X. Trường
TI  - The algebraic transfer for the real projective space
JO  - Comptes Rendus. Mathématique
PY  - 2019
SP  - 111
EP  - 114
VL  - 357
IS  - 2
PB  - Elsevier
DO  - 10.1016/j.crma.2019.01.001
LA  - en
ID  - CRMATH_2019__357_2_111_0
ER  -

%0 Journal Article
%A Nguyễn H.V. Hưng
%A Lưu X. Trường
%T The algebraic transfer for the real projective space
%J Comptes Rendus. Mathématique
%D 2019
%P 111-114
%V 357
%N 2
%I Elsevier
%R 10.1016/j.crma.2019.01.001
%G en
%F CRMATH_2019__357_2_111_0

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/

Bibliographie
Cité par

[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 ${Ext}_{A}^{5, ⁎} (Z / 2, Z / 2)$ , 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 ${Ext}_{A}^{4, ⁎} (Z / 2, Z / 2)$ and ${Ext}_{A}^{5, ⁎} (Z / 2, Z / 2)$ , 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

Cité par Sources :

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

Commentaires - Politique