Strong Convergence of the Eilenberg-moore Spectral Sequence

LET p : E-t B be a fibration of pointed spaces with fiber F. Let A be any abelian group, and suppose that the base B is connected. Our main result is: THEOREM. The mod A Eilenberg-Moore spectral sequence of p cotverges strongly to H,(F, A) ifand oni,v if z,(B) acts nilpotently on H,(F, A) for each i 2 0. This statement has to be explained. First of all, the theorem refers to the general " Eilenberg-Moore " spectral sequence of p, with arbitrary coefficients, constructed in $1. Secondly, strong concergence of this second-quadrant spectral sequence means that (I) for each pair (i, j) such thatj + i 2 0, i < 0, there is an R with the property that (2) for all n 2 0, {E,I; : i +j = n} is the set of filtration quotients from ajnite filtration of H,(F, A). Lastly, the action of a group rt on an abeiian group M is said to be nilpotent (i.e. M is a nilpotent rc-module) if there is afinite n-filtration of 1M with the property that n acts trivially on the filtration quotients. In other words, a nilpotent n-module is one which can be constructed from trivial n-modules by a finite number of extensions. The motivation for the proof below comes from an old idea, due apparently to Adams, for proving the convergence of the rational cobar spectral sequence. The idea consisted in filtering an auxiliary cobar construction to get the Serre spectral sequence, and applying the Zeeman comparison theorem. Here a geometric varient of the cobar construction is used (see $1) and the spectral sequence comparison techniques of Bousfield and Quillen replace the classical Zeeman result. In addition, " pro " arguments are used to avoid the extraneous lim problems that can arise when B is not simply connected. e-Previous work on the convergence of the Eilenberg-Moore spectral sequence has been doneheim 161, among others. Some of these authors have obtained results in the case in which z,(B) acts trivially on H,(F, A).