Algèbre, Géométrie et Topologie

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

Nguyễn, Thế Cường¹

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

Comptes Rendus. Mathématique, Tome 361 (2023) no. G11, pp. 1789-1804

Résumé

The unstable Adams spectral sequence is a spectral sequence that starts from algebraic information about the mod $2$ cohomology $H^{*}\left(X\right)$ 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 $SO$ or $U$. Bousfield and Davis created algebraic spectral sequences and conjectured that they agreed with the unstable Adams spectral sequences for $SO$ and $U$. To this end the following algebraic decomposition must hold

$${\mathrm{Ext}}_{𝒰}^s\left({\tilde{H}}^{*}\left(ℝP^{\infty },{\Sigma }^t\mathbb{Z}/2\right)\right)\cong \underset{n}{⨁}{\mathrm{Ext}}_{𝒰}^s\left(M_n/M_{n-1},{\Sigma }^t\mathbb{Z}/2\right)$$

where $M_1\subset M_2\subset \cdots$ is the well known dyadic filtration of the $𝒜$-module ${\tilde{H}}^{*}\left(ℝP^{\infty },\mathbb{Z}/2\right)\cong {𝔽}_2\left[u\right]$ 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 $t$.

Détail

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
Keywords: Injective resolution, Projective resolution, Unstable Adams spectral sequence, Unstable modules

Licence : CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits