A conjecture on the unstable Adams spectral sequences for $SO$ and $U$

Kathryn Lesh¹

¹ Department of Mathematics Union College Schenectady, NY 12308, U.S.A.

Fundamenta Mathematicae, Tome 174 (2002) no. 1, pp. 49-78

Résumé

We give a systematic account of a conjecture suggested by Mark Mahowald on the unstable Adams spectral sequences for the groups $SO$ and $U$. The conjecture is related to a conjecture of Bousfield on a splitting of the $E_{2}$-term and to an algebraic spectral sequence constructed by Bousfield and Davis. We construct and realize topologically a chain complex which is conjectured to contain in its differential the structure of the unstable Adams spectral sequence for $SO$. A filtration of this chain complex gives rise to a spectral sequence that is conjectured to be the unstable Adams spectral sequence for $SO$. If the conjecture is correct, then it means that the entire unstable Adams spectral sequence for $SO$ is available from a primary level calculation. We predict the unstable Adams filtration of the homotopy elements of $SO$ based on the conjecture, and we give an example of how the chain complex predicts the differentials of the unstable Adams spectral sequence. Our results are also applicable to the analogous situation for the group~$U$.

Détail

DOI : 10.4064/fm174-1-3

Keywords: systematic account conjecture suggested mark mahowald unstable adams spectral sequences groups conjecture related conjecture bousfield splitting term algebraic spectral sequence constructed bousfield davis construct realize topologically chain complex which conjectured contain its differential structure unstable adams spectral sequence filtration chain complex gives rise spectral sequence conjectured unstable adams spectral sequence conjecture correct means entire unstable adams spectral sequence available primary level calculation predict unstable adams filtration homotopy elements based conjecture example chain complex predicts differentials unstable adams spectral sequence results applicable analogous situation group