Comptes Rendus
Algebra, Geometry and Topology
On Bousfield’s conjectures for the unstable Adams spectral sequence for SO and U
Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1789-1804.

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 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

Ext 𝒰 s H ˜ * P ,Σ t /2 n Ext 𝒰 s M n /M n-1 ,Σ t /2

where M 1 M 2 is the well known dyadic filtration of the 𝒜-module H ˜ * P ,/2𝔽 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.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.531
Classification: 55T15, 55Q52
Keywords: Injective resolution, Projective resolution, Unstable Adams spectral sequence, Unstable modules

Thế Cường Nguyễn 1

1 Department of Mathematics, Informatics and Mechanics, VNU University of Science, Vietnam National University, Hanoi
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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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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