A Spectral Sequence for Classifying Liftings in Fiber Spaces

where pg « ƒ and p is a fibration with fiber F. Suppose that X is a CW-complex of dimension ^2conn(/ ) and conn (F) è l (conn = connectivity). Let [X, Y]B be the set of homotopy classes of pointed maps over f(H : X XI~*Y is a homotopy over ƒ if pHt—f for each / £ / ) . Becker proved in [2], [3] that under these hypotheses [X, Y]B can be given an abelian group structure with [g] as zero element. The purpose of this note is to describe a spectral sequence of the Adams type which converges to [Xf Y]B. The differentials of the spectral sequence are the twisted operations described in [ö], [7]. The sequence has the same relation to the method of computing [X, Y]B used in [ô], [7] as the Adams spectral sequence has to the killing-homotopy method of computing ordinary homotopy groups. This note should be read as a sequel to [7]. A different spectral sequence for [X, Y]B is given by Becker in [3]. A sequence apparently similar to the one to be described here is mentioned in [4] and credited to Becker and Milgram.