W. G. Dwyer

Pre-nilpotent spaces

Math Reviews: 53:11604

A space X is said to be pre-nilpotent if there is a homology isomorphism from X to a nilpotent space; equivalently, X is pre-nilpotent if the Bousfield-Kan Z-completion of X is nilpotent. Under the blanket assumption that the integral homology groups of X are finitely generated, this paper gives necessary and sufficient conditions for X to be pre-nilpotent:

1. The fundamental group of X must be pre-nilpotent, in the sense that its lower central series stabilizes after a finite number of steps.
2. Let Y be the regular covering space of X corresponding to the intersection of all of the lower central series subgroups of the fundamental group of X. Let G be the quotient of the fundamental group of X by the intersection of its lower central series subgroups, so that G acts on Y. Then the action of G on each integral homology group of Y must be pre-nilpotent, in the sense that the filtration of each homology group by powers of the augmentation ideal in the group ring of G must stabilize after a finite number of steps.

In retrospect, the finiteness condition could have been eliminated by expressing the result a little differently. Say that a map of projective chain complexes over Z[G] is an H-equivalence if it becomes a quasi-isomorphism after the chain complexes are tensored over Z[G] with the trivial module Z. (A quasi-isomorphism is a map of chain complexes which induces isomorphisms on the homology groups.) Then condition (1) is unchanged, but condition (2) can be replaced by the requirement that the chains on Y be H-equivalent over Z[G] to a chain complex with the property that the action of G on each homology group is nilpotent.