Math Reviews: 52:15464
This paper proves that the ordinary homology Eilenberg-Moore spectral sequence (EMSS) of a fibration converges strongly if and only if the monodromy action of the fundamental group of the base on the homology of the fibre is nilpotent. Strong convergence means that the spectral sequence gives each homology group of the fibre up to a finite filtration. The monodromy action is nilpotent if it is trivial up to a finite filtration.
One direction is easy. The EMSS operates by resolving the original fibration by product fibrations. This implies that the fundamental group of the base acts trivially on the EMSS. It follows that if the EMSS gives each homology group of the fibre up to a finite filtration, then the fundamental group of the base acts trivially on each homology group up to a finite filtration.
The proof of the other implication proceeds by showing that the abutment of the EMSS fits into a spectral sequence equation: there is a Serre spectral sequence converging from the homology of the base with coefficients in the EMSS abutment to the homology of the total space. This can be compared with the usual Serre spectral sequence of the fibration. The Zeeman comparison theorem shows that under the nilpotency hypothesis the EMSS abutment is isomorphic to the homology of the fibre.
A technical point which comes up in the paper is that the EMSS abutment is most conveniently treated as a collection of towers of abelian groups, not as the collection of their inverse limits.
The techniques used here are applied again in Exotic convergence of the Eilenberg-Moore spectral sequence to obtain some information about the abutment of the EMSS in cases in which it does not converge strongly. B. Shipley has gone much further along these lines. She and T. Goodwillie have also shown that if coefficients are taken mod p then the abutment of the EMSS always vanishes in negative total degree.