On Bousfield’s conjectures for the unstable Adams spectral sequence for $SO$ and $U$

Thế Cường Nguyễn

doi:10.5802/crmath.531

Algèbre, Géométrie et Topologie

On Bousfield’s conjectures for the unstable Adams spectral sequence for

S O

and

U

Thế Cường Nguyễn ¹

¹ Department of Mathematics, Informatics and Mechanics, VNU University of Science, Vietnam National University, Hanoi

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1789-1804.

Résumé

The unstable Adams spectral sequence is a spectral sequence that starts from algebraic information about the mod $2$ cohomology $H^{*} (X)$ of a space $X$ as an unstable algebra over the Steenrod algebra $𝒜$ , and converges, in good cases, to the $2$ -localized homotopy groups of $X$ . Bousfield and Don Davis looked at the case when $X$ was either of the infinite matrix groups $S O$ or $U$ . Bousfield and Davis created algebraic spectral sequences and conjectured that they agreed with the unstable Adams spectral sequences for $S O$ and $U$ . To this end the following algebraic decomposition must hold

{Ext}_{𝒰}^{s} ({\tilde{H}}^{*} (ℝ P^{\infty}, Σ^{t} ℤ / 2)) ≅ ⨁_{n} {Ext}_{𝒰}^{s} (M_{n} / M_{n - 1}, Σ^{t} ℤ / 2)

where $M_{1} \subset M_{2} \subset \dots$ is the well known dyadic filtration of the $𝒜$ -module ${\tilde{H}}^{*} (ℝ P^{\infty}, ℤ / 2) ≅ 𝔽_{2} [u]$ given by the dyadic expansion of the powers of $u$ . This paper aims at showing that this decomposition holds for numerous values of $s$ and $t$ .

Reçu le : 2022-01-20
Révisé le : 2023-03-02
Accepté le : 2023-07-01
Publié le : 2023-12-21

DOI : 10.5802/crmath.531

Classification : 55T15, 55Q52
Mots clés : Injective resolution, Projective resolution, Unstable Adams spectral sequence, Unstable modules

Affiliations des auteurs :

Thế Cường Nguyễn ¹

¹ Department of Mathematics, Informatics and Mechanics, VNU University of Science, Vietnam National University, Hanoi

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Thế Cường Nguyễn},
     title = {On {Bousfield{\textquoteright}s} conjectures for the unstable {Adams} spectral sequence for $SO$ and $U$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1789--1804},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     year = {2023},
     doi = {10.5802/crmath.531},
     language = {en},
}

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Thế Cường Nguyễn. On Bousfield’s conjectures for the unstable Adams spectral sequence for $SO$ and $U$. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1789-1804. doi : 10.5802/crmath.531. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.531/

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