W. G. Dwyer

Exotic convergence of the Eilenberg-Moore spectral sequence

Illinois J. Math. (19), 1975, 607-617.

Math Reviews: 52:4290

This paper uses machinery from Strong convergence of the Eilenberg-Moore spectral sequence to study the ordinary homology Eilenberg-Moore spectral sequence (EMSS) of a fibration in some cases in which the spectral sequence does not converge strongly.

Let B be the base of the fibration and F the fibre. The fundamental group of B acts by monodromy on the homology of F. Let I denotes the augmentation ideal in the group ring of the fundamental group of B. The main theorem says that if

  1. B is a nilpotent space (i.e, its fundamental group is a nilpotent group and acts nilpotently on the higher homotopy groups) and
  2. certain finiteness conditions are satisfied
then the EMSS of the fibration converges to homology of F with the I-adic filtration. This does not mean that at E-infinity the spectral sequence displays exactly the graded object associated to the filtration of the homology of F by powers of I, but it does mean that the topology on each homology group of F induced by the abutment filtration is equivalent to the I-adic topology. The exact statement is easiest to express in terms of towers of abelian groups: the natural map from the I-adic completion tower of each homology group of F to the corresponding abutment tower of the spectral sequence is a pro-isomorphism.

One application is the computation in certain cases of the homology of the universal cover of the Bousfield-Kan completion of a space.