Exotic convergence of the Eilenberg-Moore spectral sequence

Let p:E B be a fibration over a connected space B with fiber F. The Eilenberg-Moore spectral sequence of p is a second quadrant spectral sequence which tries and sometimes fails to converge strongly to the homology of F (see 1-53). The purpose of this paper is to determine what the spectral sequence does converge to. An abstract answer (Theorem 1.1) is that the spectral sequence almost always converges to the homology of the fiber of the nilpotent completion of the map p. A concrete answer (Theorem 2.1) is that under certain natural conditions on B and certain finiteness hypotheses the spectral sequence converges weakly to the homology of F with the nl(B) filtration. As an aid to understanding these theorems, recall the similar behavior of the mod q Adams spectral sequence of a spectrum X. In absolute generality the spectral sequence converges only to the homotopy groups of some completion of X [1]. However, if X is connected and suitable finiteness conditions are satisfied, the spectral sequence converges to the actual homotopy groups of X with the "power of q" filtration. Note also that the spectral sequence converges strongly to the homotopy of X only in the rare case that each rX is a q-group of finite exponent. Sections 3 and 4 are devoted to applications of the convergence theorems in Section and Section 2. In Section 3 we compute, in a certain sense, the homology of the universal cover of the nilpotent completion of a space X. In Section 4 we show that the cohomology of certain nilpotent groups is generated , in the sense of matric Massey products, by classes of degree one. Throughout Section and Section 2 we will work with the fixed fibration p described above. R will be a ring of the form Z/qZ (q prime) or a subring of the rationals, and A will be a fixed R-module. We will freely use the ideas and conventions of [5]. In particular, we will associate to the fibration p a certain augmented cosimplicial space F F, called the Eilenberg-Moore object of p. The mod A Eilenberg-Moore spectral sequence of p is understood to be the homo-topyspectral sequence ofthe augmented tower offibrations A (R) F { TotsA (R) F}s.