A conjecture on the unstable Adams spectral sequences for $SO$ and $U$
Volume 174 / 2002
Abstract
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~$U$.
