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

Nguyễn, Thế Cường

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

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 $t$.

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

[1] Bousfield, A. K.; Curtis, Edward B. A spectral sequence for the homotopy of nice spaces, Trans. Am. Math. Soc., Volume 151 (1970), pp. 457-479 | DOI | MR | Zbl

[2] Bousfield, A. K.; Davis, Donald M. On the unstable Adams spectral sequence for $SO$ and $U$, and splittings of unstable $\mathrm{Ext}$ groups, Bol. Soc. Mat. Mex., II. Ser., Volume 37 (1992) no. 1-2, pp. 41-53 papers in honor of José Adem (Spanish) | MR | Zbl

[3] Bousfield, A. K.; Kan, Daniel M. Homotopy limits, completions and localizations, Lecture Notes in Mathematics, 304, Springer, 1972, v+348 pages | MR | DOI

[4] Lannes, Jean; Zarati, Saïd Sur les $𝒰$-injectifs, Ann. Sci. Éc. Norm. Supér., Volume 19 (1986) no. 2, pp. 303-333 | MR | DOI | Zbl

[5] Lesh, Kathryn A conjecture on the unstable Adams spectral sequences for $SO$ and $U$, Fundam. Math., Volume 174 (2002) no. 1, pp. 49-78 | DOI | MR | Zbl

[6] Massey, William S.; Peterson, Franklin P. The $\mathrm{mod}2$ cohomology structure of certain fibre spaces, Memoirs of the American Mathematical Society, 74, American Mathematical Society, 1967, 97 pages | MR

[7] Nguyễn, The Cuong The Pseudo-hyperresolution and Applications (2016) | arXiv

[8] Nguyễn, The Cuong Algebraic EHP sequences revisited, Proc. Edinb. Math. Soc., II. Ser., Volume 62 (2019) no. 3, pp. 625-640 | DOI | MR | Zbl

[9] Schwartz, Lionel Unstable modules over the Steenrod algebra and Sullivan’s fixed point set conjecture, Chicago Lectures in Mathematics, University of Chicago Press, 1994, x+229 pages | MR

[10] Steenrod, Norman E. Cohomology operations, Annals of Mathematics Studies, 50, Princeton University Press, 1962, vii+139 pages (lectures by N. E. Steenrod written and revised by D. B. A. Epstein) | MR

[11] Wood, Reginald M. W. A note on bases and relations in the Steenrod algebra, Bull. Lond. Math. Soc., Volume 27 (1995) no. 4, pp. 380-386 | DOI | MR | Zbl