On Bousfield’s conjectures for the unstable Adams spectral sequence for $SO$ and $U$ - Nguyễn, Thế Cường (Comptes Rendus. Mathématique)
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 $.