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

Kathryn Lesh

Subject Classification: 55T15, 55Q52, 55U99

Department of Mathematics
Union College
Schenectady, NY 12308
Telephone number: (518)388-6246

klesh@member.ams.org

     In this paper 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. In this paper,
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.