The Localization of Spaces to Homology

The plausibility and desirability of such a functor E were shown by Adams [2]. To obtain an existence proof (§3), I will construct an appropriate localization functor on the category of simplicial sets and will show that it induces the desired h,-localization functor on Ho. The backbone of this proof is in an appendix ($10~$12), where I introduce a version of simplicial homotopy theory in which the h,-equivalences play the role of weak homotopy equivalences. I show that this theory fits into Quillen’s framework of homotopical algebra rm [I 11. Special cases of the h,-localization, X + EX, are familiar. If X is simply connected (or nilpotent) and h, = H*( ; Z[J-‘1) where Z[J-‘1 denotes a subring of the rationals,then Xi EX is the usual Z[J-‘]-localization with II, EX %Z[J-l] 0 II, X. This case was discovered by Barratt-Moore (co 1957, unpublished) and has subsequently been discovered and/or studied by various others, e.g. [4], [5], [7], [9], [I 11, [14], [15]. If Xis simply connected (or nilpotent) and h, = H*( ; Z,) with p prime, then X + EX is the p-completion [5, p. 1861 with IT, EX = Ext(ZpOo II, X) @ Hom(Z,,, II,_, X). If in addition X is of finite type, then X + EX is the pprofinite completion [12], [14] with II, EX given by the p-profinite completion of II, X. In [5] we gave various other examples of H,( ; R)-localizations where R = Z, or R = Z[J-‘1, and we constructed an ” R-completion ” X -+ R, X which coincides with the H,( ; R)-localization provided X is “ R-good “. A major part of this paper is devoted to the study of H,( ; R)-local spaces, i.e. spaces X E Ho satisfying the equivalent conditions :