On Bousfield’s conjectures for the unstable Adams spectral sequence for $SO$ and $U$
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 $\mathcal{A}$, and converges, in good cases, to the $2 $-localized homotopy gr...
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Académie des sciences
2023-12-01
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| Series: | Comptes Rendus. Mathématique |
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| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.531/ |
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| author | Nguyễn, Thế Cường |
| author_facet | Nguyễn, Thế Cường |
| author_sort | Nguyễn, Thế Cường |
| collection | DOAJ |
| description | 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 $\mathcal{A}$, 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
\[ \mathrm{Ext} _{ \mathcal{U} } ^{ s } \left(\tilde{ H } ^{ * } \left(\mathbb{R} P ^{ \infty }, \Sigma ^{ t } \mathbb{Z} /2 \right) \right) \cong \bigoplus _{ n } \mathrm{Ext}_{ \mathcal{U} } ^{ s } \left(M _{ n } / M _{ n - 1 },\Sigma ^{ t } \mathbb{Z} /2 \right) \]
where $M _{ 1 } \subset M _{ 2 } \subset \cdots $ is the well known dyadic filtration of the $\mathcal{A}$-module $\tilde{ H } ^{ * } \left(\mathbb{R} P ^{ \infty }, \mathbb{Z} /2 \right) \cong \mathbb{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 $. |
| format | Article |
| id | doaj-art-1b069446a0f34a1cba6e42ac27efe0b6 |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2023-12-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-1b069446a0f34a1cba6e42ac27efe0b62025-02-07T11:12:14ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692023-12-01361G111789180410.5802/crmath.53110.5802/crmath.531On Bousfield’s conjectures for the unstable Adams spectral sequence for $SO$ and $U$Nguyễn, Thế Cường0Department of Mathematics, Informatics and Mechanics, VNU University of Science, Vietnam National University, HanoiThe 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 $\mathcal{A}$, 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 \[ \mathrm{Ext} _{ \mathcal{U} } ^{ s } \left(\tilde{ H } ^{ * } \left(\mathbb{R} P ^{ \infty }, \Sigma ^{ t } \mathbb{Z} /2 \right) \right) \cong \bigoplus _{ n } \mathrm{Ext}_{ \mathcal{U} } ^{ s } \left(M _{ n } / M _{ n - 1 },\Sigma ^{ t } \mathbb{Z} /2 \right) \] where $M _{ 1 } \subset M _{ 2 } \subset \cdots $ is the well known dyadic filtration of the $\mathcal{A}$-module $\tilde{ H } ^{ * } \left(\mathbb{R} P ^{ \infty }, \mathbb{Z} /2 \right) \cong \mathbb{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 $.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.531/Injective resolutionProjective resolutionUnstable Adams spectral sequenceUnstable modules |
| spellingShingle | Nguyễn, Thế Cường On Bousfield’s conjectures for the unstable Adams spectral sequence for $SO$ and $U$ Comptes Rendus. Mathématique Injective resolution Projective resolution Unstable Adams spectral sequence Unstable modules |
| title | On Bousfield’s conjectures for the unstable Adams spectral sequence for $SO$ and $U$ |
| title_full | On Bousfield’s conjectures for the unstable Adams spectral sequence for $SO$ and $U$ |
| title_fullStr | On Bousfield’s conjectures for the unstable Adams spectral sequence for $SO$ and $U$ |
| title_full_unstemmed | On Bousfield’s conjectures for the unstable Adams spectral sequence for $SO$ and $U$ |
| title_short | On Bousfield’s conjectures for the unstable Adams spectral sequence for $SO$ and $U$ |
| title_sort | on bousfield s conjectures for the unstable adams spectral sequence for so and u |
| topic | Injective resolution Projective resolution Unstable Adams spectral sequence Unstable modules |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.531/ |
| work_keys_str_mv | AT nguyenthecuong onbousfieldsconjecturesfortheunstableadamsspectralsequenceforsoandu |