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 E₂-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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm174-1-3,
author = {Kathryn Lesh},
title = {A conjecture on the unstable Adams spectral sequences for SO and U},
journal = {Fundamenta Mathematicae},
volume = {173},
year = {2002},
pages = {49-78},
zbl = {1007.55015},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm174-1-3}
}
Kathryn Lesh. A conjecture on the unstable Adams spectral sequences for SO and U. Fundamenta Mathematicae, Tome 173 (2002) pp. 49-78. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm174-1-3/