A stable range for homology localization
E. Dror, W. G. Dwyer · Illinois Journal of Mathematics · 1977
A. K. Bousfield has recently shown how to construct a canonical integral homology localization Xz for any space X.The aim of this paper is to show that in a natural range of dimensions the homotopy groups of Xz are related in a stable way to the homotopy groups of X itself.In one case this relationship is direct enough to give a novel form of the Whitehead theorem.Our technique is to construct a first quadrant spectral sequence which con- verges to n.(Xz).We assume that the homotopy groups rcX are nilpotent rclX-modules [1, 4.2] for 2 1), and then show that in dimensions less than 2n the E2-term of this spectral sequence depends only on nix and on the action of nX upon the individual higher homotopy groups of X. More- over, the influence of a given nX-module on the tractable part of E 2 is both additive in nature and independent of the particular dimension in which the module appears as a higher homotopy group.This is what stability means.Sometimes the spectral sequence allows some homotopy groups of Xz to be computed explicitly.For instance"1.1 THEOREM.Let X be a connected space with finite skeleta.Suppose that rcX is a nilpotent group and that rcX acts nilpotently on nX for 2 1).Then there are natural isomorphisms r(Xz) ztX,i < n and zt(Xz) (rcX) ^,n < <_ 2n 1.Here (ztiX) denotes the lower central series completion (4.3) of rcX with respect to the action of ztlX.The space X has finite skeleta if it has a finite number of simplices or cells in each dimension.Since the integral homology localization functor converts homology equiv- alences into homotopy equivalences, we immediately obtain" 1.2 COROLLARY.Suppose that X and Y are spaces as in 1.1, and that f: X --, Y is a map which induces an isomorphism on integral homology.Then f induces isomorphisms niX' ziY, <_ n and (rciX) (ztiY) ^,n < < 2n 1.Organization of the paper.In Section 2 we prove a technical lemma which is at the foundation of everything that follows, in Sections 3 and 4 we define and