### 3-dimensional cohomology of the mod p Steenrod algebra.

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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...