The unstable Adams spectral sequence for two-stage towers

Let KP denote a generalized mod 2 Eilenberg-MacLane space and let Y be the fiber of a map X --> KP to which the Massey-Peterson theorem applies. We study the relationship of the mod2 unstable Adams spectral sequence (UASS) for X and for Y. Given conditions on X, we split the E-2-term for Y, and we use a primary level calculation to compute d(2) for Y up to an error term. If the UASS for X collapses at E-2 (for example, if X is an Eilenberg-MacLane space), the UASS for Y collapses at E-3, and we have the entire UASS for Y. We also give examples and address a conjecture of Bousfield on the UASS for the Lie group SO. (C) 2000 Published by Elsevier Science B.V. All rights reserved.