The plus construction, Postnikov towers and universal central module extensions David Blanc and George Peschke* June 18, 2002 Abstract Given a connected space X, we consider the effect of Quillen's plus constr* *uction on the homotopy groups of X in terms of its Postnikov decomposition. Specifi* *cally, using universal properties of the fibration sequence AX ! X ! X+ , we ex* *plain the contribution of ßnX to ßnX+ , ßn+1X+ and ßnAX, ßn+1AX explicitly in terms of the low dimensional homology of ßnX regarded as a module over ß1X* *. Key ingredients developed here for this purpose are universal -central fibrat* *ions and a theory of universal central extensions of modules, analogous to universa* *l central extensions of perfect groups. 1 Introduction Quillen's plus construction (cf. [10]), applied to a space X, * *yields a universal map j : X ! X+ , which is characterized by the fact that it quotient* *s out the maximal perfect subgroup of ß1X and induces isomorphisms in all homology th* *eories (including homology with twisted coefficients). In general, a map between conn* *ected spaces satisfies this homological condition if and only if its homotopy fiber i* *s acyclic; see [11] and compare [5]. We denote the homotopy fiber of j : X ! X+ by AX. Understanding the map ß*j : ß*X ! ß*X+ is helpful in studying the effect of h* *omolog- ical localization functors on homotopy groups (see 4.5) and in higher algebraic* * K-theory. Such understanding was obtained early on in low dimensions and, except for spec* *ial cases, this has remained the extent of our knowledge. With the following result we cla* *rify com- pletely the contribution of ßnX to ßnX+ and ßn+1X+ , for each n 2. Theorem A Let X be a connected CW complex. Applying the plus construction to the Postnikov section K(ßnX, n) ! PnX ! Pn-1X (n 2), yields the commutative ________________________________ *Research partially supported by NSERC of Canada 1Key words: Quillen's plus construction, localization, colocalization, univ* *ersal central extension, acyclic space; Subject class: 19D06, 20C99, 55S45, 55P60, 55Q99 1 diagram of fibrations whose properties are formulated below: F _______//_K(ßnX, n)______// (FD) | | | | | | | fflffl| fflffl| fflffl| APnX _______//_PnX_______//_(PnX)+ | | | | | | fflffl| fflffl| fflffl| APn-1X _____//_Pn-1X_____//_(Pn-1X)+ The fibers F and are (n-1)-connected, and their lowest non-vanishing homotopy* * groups fit into the natural commutative diagram of exact sequences in which every vert* *ical arrow is an isomorphism. ~ H1(Ge; ßnX) //__//I[Ge] GeßnX___//ßnX___////_H0(Ge; ßnX) (UCE) ~=|| ~=|| |||| ~=|| | | || | fflffl| fflffl| || fflffl| ßn+1 //_________//ßnF________//ßnX_______////_ßn Moreover, there is an epimorphism ~= ~= ßn+2 -! ßn+1F -!! Hn+1F - H2(Ge; I[Ge] GeßnX). Here eGis the universal central extension of the maximal perfect subgroup G of * *ß1X, and I[Ge] is the augmentation ideal of the integral group ring of eG. On the background of Theorem A: Our approach to Theorem A is guided by prop- erties of the homotopy fibration sequence of j: '' + X+ - i!AX -! X -! X As noted in [9, 0.1.iv], the acyclic space AX is just the acyclization of X, as* * defined by Dror in [3] (see also [4]). The following two theorems express the universal pr* *operties of the plus construction and acyclization in a form which lends itself better to an in* *terpretation in terms of homotopy groups. Theorem B [9, 7.7] The fibration X+ !i AX ! X is -central, in the sense that all Whitehead products [i*ff, fi] vanish, where ff 2 ßp X+ and fi 2 ßqAX,* * p, q 1. Theorem C The fibration X+ ! AX ! X is initial amongst -central fibrations in the following sense: given a solid diagram of -central fibrations X+______//_AX___//___X ____ ______ || ____ ___ || fflffl____fflffl___|| F _______//E__q_//_X 2 in which G := im(q) is the maximal perfect subgroup of ß1X, dotted maps exist m* *aking the diagram commute. Moreover, the dotted maps are unique up to vertical homoto* *py. To get a feel for the implications of Theorems B and C, consider first the foll* *owing well known exact sequence ß2X+ ________//ß1AX________//ß1X________//_ß1X+ ?? ""????? "??" ??? """ ??? """ ØØØØ????"" ØØØØ???"" H2G G in which ß1AX is the universal central extension of G. This sequence can be nicely explained as a consequence of Theorems B and C, usi* *ng results on the universal central extension of a perfect group, due to Milnor [8* *, Sect. 5] and Kervaire [6]. As another consequence of Theorems B and C, we obtain Theorem A. It depends upo* *n a new concept from algebra, namely the universal central extension of a perfect m* *odule: Theorem D If G is a 2-acyclic group (that is, H1(G; Z) = 0 = H2(G; Z)), th* *en every G-module M fits into an exact sequence ~ H1(G; M) //__//I[G] G M____//M____////_H0(G; M) whose terms have the following properties: (i)im (~) = I[G].M is the unique maximal perfect submodule of M; i.e. I[G].im* * (~) = im(~). (ii)H1(G; M) æ I[G] G M i im(~) is a central extension of im(~) (i.e. G acts t* *rivially on H1(G; M)), and it is initial amongst all such central extensions. Organization of the paper Section 1 supplies some facts on -central fibration* *s, leading up to Theorem C. Section 2 develops material on universal central exten* *sions of a module over a group ring, leading up to Theorem D. In Section 3 we prove Theo* *rem A, and in Section 4 we compute the acyclic Postnikov invariants of AX (cf. [3]) in* * terms of the ordinary Postnikov invariants of X. We thank the referees of this paper for their constructive comments. 3 1 -central fibrations with perfect target Here we develop properties of -central fibrations, leading up to Theorem C whi* *ch, in turn, guides our approach towards analyzing the effect of the plus construction* * on ho- motopy groups. We assume throughout that spaces, maps and homotopies are pointe* *d. Spaces are assumed to be path connected, except possibly those arising as homot* *opy fibers. 1.1 Definition [9, Sect. 7] A fibration sequence F -!i E ! B is called -* *central if all Whitehead products [i*ff, fi] vanish for any ff 2 ßpF and fi 2 ßqE with* * p, q 1. Given a map q : W ! Y , we refer to G := im(q*) < ß1Y as its target in homotopy dimension 1. We say that q has perfect target (in homotopy dimension 1) if G is* * a perfect group. q 1.2 Lemma Let F ! W -! Y be a -central fibration, such that q has pe* *rfect target G < ß1Y , and suppose f : X ! Y is a map for which G0:= (ß1f)-1G i* *s a perfect subgroup of ß1X. Then the pullback fibration __i0//_0__q0//_ F W X || | | || f0| pull f| || | back | || fflffl| fflffl| F __i__//W__q_//_Y is -central, and q0has perfect target G0. Proof The pullback of a -central fibration is again -central by [9, 7.6]. * *An elemen- tary argument shows that im(ß1q0) = G0, which is perfect by assumption. 1.3 Example For every connected CW-space X, the sequence X+ ! AX -c! X is a -central fibration, and c has perfect target equal to the maximal perfect* * subgroup of ß1X. q 1.4 Definition A -central fibration F ! W -! X such that q has perfect * *target G is universal if, under conditions (i) and (ii) below, for every solid diagram q F______//_W____//__X ___ ____ | ____ ef____ |f ____ ____ | fflffl____fflffl___fflffl| F1 ____//_W1q1_//Y 4 there exists a morphism efof fibrations, unique up to homotopy, which makes the* * diagram commute. Diagram conditions: (i)The bottom row is a -central fibration such that q1 has perfect target G1; (ii)f*(G), the image of G under ß1f, is contained in G1. 1.5 Theorem For every connected CW-space X, the -central fibration X+* * ! AX -c! X is universal, and c has target G (=the maximal perfect subgroup of ß1X* *). Proof According to Definition 1.4, suppose we are given a solid diagram __c_//_ X+_______//AX__ X ___ _____ | ____ fe____ f| ____ ___ | fflffl____fflffl___fflffl| F1______//W1_q1_//_Y We use obstruction theory to obtain the required morphism of fibrations. At the* * level of fundamental groups we have the diagram of central extensions i1c H2G_ //___//_ß1AX____////__G ____ _____ | ____ ____ i1f| ____ ____ | fflffl____ fflffl___fflffl| ker(ß1q1)//__//_ß1W1___////G1 with universal top row. Thus there exists a lift ef2: (AX)2 ! W1 from the 2-s* *keleton of AX to W1, and its restriction to (AX)1 is homotopically unique. The existenc* *e and homotopical uniqueness of effollow because AX is acyclic and the action of ß1Y * *on ß*F1 is trivial. Theorem C follows as a special case of Theorem 1.5. 2 Universal central extensions of perfect G-modules In this section we develop the concept of perfect modules and their central ext* *ensions, and prove Theorem D. We assume some background material on perfect groups and t* *heir universal central extensions from [8, Sect. 5]. Given a group G and a left G-module M, we often use the exact üm ltiplication s* *equence" ~ H1(G; M) //__//I G M____//_M___////_H0(G; M) (MS) 5 which comes from applying TorZ[G]-(-, M) to I æ Z[G] i Z. Here Z[G] is the i* *ntegral group ring of G and I is its augmentation ideal. All tensor products are over * *Z[G], and ~ is the multiplication map. 2.1 Definition For n 1, a group G is called n-acyclic if Hk(G; Z) = 0 fo* *r 1 k n. Thus 1-acyclic groups are known as perfect groups. 2-acyclic groups are sometim* *es called üs perperfect". 2.2 Definition Let G be a group and n 0. A G-module M is called n-acycli* *c if Hk(G; M) = 0 for 0 k n. In analogy with the group theoretic terminology, we sometimes refer to a 0-acyc* *lic G- module as a "perfect G-module". 2.3 Lemma A group G is n-acyclic if and only if its augmentation ideal I* * is an (n - 1)-acyclic G-module. Proof Apply H*(G; -) to I æ Z[G] i Z. 2.4 Corollary A group G is 1-acyclic if and only if the multiplication map* * ~ : I G I ! I is an epimorphism. G is 2-acyclic if and only if ~ is an isomorphis* *m. Proof Apply Lemma 2.3 to (MS), using M = I. 2.5 Corollary For n = 1, 2 let G be an n-acyclic group, and let M be an ar* *bitrary G-module; then the G-module I G M is (n - 1)-acyclic. Proof The multiplication map ~0for M0 = I G M is given by the composite ~= ~I M I G (I G M) ____//_(I G I) G M___//_I G M. Thus the claim follows from Corollary 2.4. 2.6 Corollary If G is an n-acyclic group and A is an abelian group with tr* *ivial G-action, then TorZ[G]k(I, A) = 0, for 0 k n - 1. Proof Use the long exact sequence obtained by applying TorZ[G]*(-, A) to I * * æ Z[G] i Z. 6 2.7 Definition A colocalizing functor on a category C is a functor C : C * *! C, together with a natural transformation " : C ! IdC making the diagram below com* *mu- tative. __C"_// C O C ~= C Ö C|~=| |"| fflffl| fflffl| C __"___//Id 2.8 Theorem For a 2-acyclic group G, the functor E := I G -, together w* *ith the natural transformation ~ : E ! Id defined by multiply ~M : I G M --! M , is a colocalizing functor from the category Z[G]-Mod of left G-modules onto the* * category A1Z[G]-Mod of 1-acyclic G-modules. Proof E takes values in A1Z[G]-Mod by Corollary 2.5. The colocalizing proper* *ties of E require that (1) the diagram E~M EEM __~=//_EM | | ~EM |~=| ~M|| fflffl| fflffl| EM __~M__//M be commutative and natural in M; and (2) the designated arrows in this diagram be isomorphisms. (1) follows from basic properties of the tensor product. For (2), use Corollar* *y 2.4 to deduce that ~EM is an isomorphism. To see that E~M = I G ~M is an isomorph* *ism, too, we break the sequence (MS) up into short exact sequences: ~M H1(G; M) æ I G M -!! P M and P M æ M i H0(G; M). Apply TorZ[G]-(I, -) to these sequences, and use Corollary 2.6 to see that I G* * ~M is the composite of the two isomorphisms ~= ~= I G (I G M) -! I G P M and I G P M -! I G M. The claim follows. 7 2.9 Definition The center of a G-module N is the submodule of elements on * *which G acts trivially. A central extension of a G-module M is a short exact sequenc* *e of G- modules A æ N i M so that A maps into the center of N. In analogy with universal central extensions of perfect groups we prove 2.10 Theorem Given a 2-acyclic group G, a central extension A æ fM i M of G-modules is initial amongst all central extensions of M if and only if fM is 1* *-acyclic . Proof Assume fM is 1-acyclic. In the diagram below, we assume the solid part* * of the front face is given. ~= 0Ø___________//fM_________//EM "" """"Ø """|| """ Ø """""Ø """ |||| """"" Ø """ Ø """" || A_ //___Ø____//_fM____Ø___////_M || ______ Ø ______ Ø |||| |||| ____ ____ || || ____ fflfflØ____ fflfflØ||~ || __ 0 _____ ____//_EN____||=__//EM ____ " ____ " || " ____""" ____""" || """ ____"" ___"" || "" fflffl____""""fflffl___""""||"""" B //________//_N____q____////_M The solid part of the back face results from applying the colocalizing functor * *E. We find EA = 0 = EB by Corollary 2.6. Thus the back rows are exact, being the ends* * of TorZ[G]*(I, -)-long exact sequences. So there is a map fM! EN which makes the* * right hand back square commute. This yields a map f : fM ! N making the vertical sq* *uare in the center, as well as the right front face, commute. To see that it is uniq* *ue, assume g : fM ! N is another such map. Then q O (f - g) : fM ! M is the zero map, so* * (f - g) lifts to B. This implies that (f - g) = 0, because H0(G; M) = 0 and G acts * *trivially on B. Thus f = g, implying that the sequence is initial amongst all central ext* *ensions of M. To see the converse, we invoke Theorem (2.11) which, of course, does not depend* * on the part of (2.10) we are going to prove now: Part (ii) shows that M is 0-acyclic. * * Part (i) implies that fM ~=EM which is 1-acyclic. We call a sequence of G-modules, as in Theorem 2.10, the universal central exte* *nsion of M. 2.11 Theorem Given a 2-acyclic group G, the following hold: (i)For every 0-acyclic G-module M H1(G; M) æ I G M = EM i M is a universal central extension of M. 8 (ii)A G-module M has a universal central extension if and only if M is 0-acycl* *ic; compare [8, 5.7]. Proof (i) The given sequence is (MS), taking into account that M is 0-acycli* *c. EM is 1-acyclic by Corollary 2.5. So the claim follows from Theorem 2.10. (ii) Suppose M is not 0-acyclic, and A æ N i M is a universal central extension* * of M. Then M, and hence N, have H0(G; M) 6= 0 as a G-trivial quotient. Therefore * *there are at least two distinct morphisms from the assumed universal central extensio* *n to the central extension H0(G; M) //_//_H0(G; M) M____////_M, a contradiction. We remark that [7, Thm. 1] can be regarded as a precursor of Theorem 2.11. Proof of Theorem D (i) The module I G M is 1-acyclic by Corollary 2.5. So P M = im(~) is 0-acyclic by 2.12. It is a maximal 0-acyclic submodule of M bec* *ause any module N with P M < N < M yields a quotient N=P M < H0(G; M) with trivial G-action. However, N=P M is again perfect by Proposition 2.12 below. So N = P* * M. That P M is the unique maximal perfect submodule of M also follows from Proposi* *tion 2.12. (ii) follows from Theorem 2.10. We conclude this section by formulating some closure properties of the classes * *of n-acyclic modules: 2.12 Proposition For any group G, and n 0, the class of perfect G-modul* *es is closed under quotients and arbitrary colimits. Proof The natural isomorphism H*(G; ~2 M~) ~= ~2 H*(G; M~) shows that the class of perfect G-modules is closed under direct sums. Further, any quotient * *M of a perfect G-module N is again perfect because 0 = H0(G; N) i H0(G; M). 2.13 Proposition Given a 2-acyclic group G, the class of 1-acyclic G-modu* *les is closed under extensions and arbitrary colimits. Proof If M0 æ M i M00is an extension of G-modules with M0 and M001-acyclic, then inspection of the associated long exact sequence in homology shows that M * *is 1- acyclic as well. By Corollary 2.5 I G - takes values in the class of 1-acyclic* * G-modules. Moreover, I G - commutes with arbitrary colimits. 9 3 Proof of Theorem A By passing to the appropriate covering space of X, if necessary, we can assume * *that ß1X is perfect. So X+ and each Postnikov section (PnX)+ (n 1) are simply connec* *ted. 3.1 Lemma For n 2, is (n - 1)-connected. Proof This follows from the fact that, for k n, ~= + + 0 = Hk(Pn-1X, PnX; Z) -! Hk((Pn-1X) , (PnX) ; Z). 3.2 Lemma For n 2, F is (n - 1)-connected. Proof F is at least (n-2)-connected because K(ßnX, n) and are (n-1)-connec* *ted. We must show that ßn-1F = 0 as well. First of all, we have an epimorphism ßn * * i ~= ßn-1F . So ßn-1F is abelian, and the Hurewicz map ßn-1F -! Hn-1F is an isomor* *phism even for n = 2. Next, by applying the Serre spectral sequence to the fibration * *APnX ! APn-1X, we see that Hn-1F is a 1-acyclic Ge-module. Furthermore, the commutati* *ve diagram ßn-1 _________////ßn-1F | | | | | | fflffl| |fflffl ßn-1 (PnX)+ ____//_ßn-1APnX tells us that eGacts trivially on the image of ßn-1F ! ßn-1APnX. But the class* * of 0- acyclic modules is closed under quotients by Proposition 2.12. So this image is* * trivial, and we have an epimorphism @ : ßnAPn-1X i ßn-1F . On the other hand, eGacts trivia* *lly ~= on ßkAPn-1X for k n, because we have isomorphisms ßk (Pn-1X)+ -! ßkAPn-1X in the -central fibration (Pn-1X)+ ! APn-1X ! Pn-1X. Now @ is a morphism of eG-modules, implying that eGacts trivially on the 1-acyclic eG-module ßn-1F . T* *herefore ßn-1F = 0, as claimed. Thus we have established the first part of Theorem A. We now turn to diagram (U* *CE) of the Theorem and its properties: The bottom row comes from the fibration F ! K(ßnX, n) ! , using Lemma 3.2. The terms ßn+1 and ßn are trivial eG-modules and ßn+1 is contained in the center* * of ßnF . Further, ßnF ~=HnF is seen to be a 1-acyclic Ge-module, by using the Serre spe* *ctral sequence of the fibration F ! APnX ! APn-1X. Thus N := im(ßnF ! ßnX) = I[Ge].ßnX = im(~) is the maximal perfect submodule of ßnX; see Theorem D(i). F* *rom Theorem 2.10 we see that ßn+1 æ ßnF i N is the universal central extension of * *N. So 10 the vertical arrows on the left are isomorphisms by Theorem D(ii). The vertical* * arrow on the right is an isomorphism by the Five Lemma. As to ßn+1F , it is a trivial eG-module because it fits into the exact sequence* * ßn+2APn-1X ! ßn+1F ! ßn+1APnX, where eG acts trivially on the outside terms. The Hurewicz m* *ap ßn+1F i Hn+1F is onto and is a eG-module map. Thus eGacts trivially on Hn+1F as* * well. Now the Serre spectral sequence yields an isomorphism ~= H2(Ge; I GeßnX) -! H0(Ge; Hn+1F ) ~=Hn+1F, which proves the claim, and completes the proof of Theorem A. 3.3 Remark By chasing the diagram of homotopy groups coming from the fib* *ration diagram (FD) one can deduce further that the maximal perfect submodule of ßnX is always contained in ker(ßnPnX ! ßn(PnX)+). Moreover, the two modules are equal exactly when ßn+1(PnX)+ ! ßn+1(Pn-1X)+ is onto. 4 The acyclic Postnikov tower of AX Already in the early 1970's Dror showed how to use the acyclic Postnikov tower * *[3] to analyze an acyclic space Z. The acyclic Postnikov n-stage of Z is simply the ac* *yclization APnZ of the usual Postnikov section. The acyclic Postnikov n-stage need not h* *ave trivial homotopy groups above dimension n. Instead, the only requirement is th* *at the fundamental group must act trivially on these higher homotopy groups. When passing from an (n - 1)-stage Zn-1 to an n-stage, one splices into ß*Zn-1 * *a 1- acyclic ß1Z-module ffn, and there is a corresponding ä cyclic Postnikov invaria* *nt" ~n 2 Hn+1(Zn-1; M). In addition, in dimensions greater than n, one splices into ß*Zn* *-1 certain ß1Z-modules with trivial action. In general, starting with an arbitrary space X, Dror's acyclic Postnikov tower * *of AX has APnAX as its n-th acyclic Postnikov stage. In Theorem A, we were working with a* * tower whose n-th stage is APnX. Below, we establish explicitly a natural equivalence* * between these towers. With the aid of Theorem A, we express the acyclic Postnikov invar* *iants of AX in terms of the ordinary Postnikov invariants of X. 4.1 Lemma Let X be a connected CW-space. Applying successively the appro* *pri- 11 ate functors to the map AX ! X yields the commutative cube APnAX ___________________//PnAX | PPPPu NNNN | PPnPP | NNN | PP(( | N''N | __________|________// | APnX PnX | | | | | | | | fflffl| || fflffl| || APn-1AXP ________ |_______//Pn-1AXN | PPPP | NNN | PPPP | NNNN | un-1 P'' fflffl| N&& fflffl| APn-1X _________________//Pn-1X whose left hand face is a homotopy equivalence of acyclic Postnikov towers. Proof To see that each un is a homotopy equivalence, we argue as follows. Ap* *plying A to the commutative diagram ~= AX ______//_X AAX ______//_AX | | | | | | yields | | | | | | fflffl| fflffl| fflffl| fflffl| PnAX ____//_PnX APnAX _un_//_APnX For k n, the right hand square induces ßk-isomorphisms because the maps on th* *e top and the sides do. This follows from Lemma 3.2. By [3, 3.4], un is a homotopy eq* *uivalence. 4.2 Corollary The functors APnA and APn are naturally equivalent. In order to determine the acyclic Postnikov invariants of AX, we require the fo* *llowing cohomological recognition tool for acyclic spaces: 4.3 Lemma A connected CW-space X is acyclic if and only if its fundament* *al group G is 2-acyclic and, for every G-module M, the morphism ~ : I G M ! M ind* *uces isomorphisms ~= r ~* : Hr(X; I G M) -! H (X; M) for r 2. Proof If X is acyclic, then G is 2-acyclic; see [3, 4.1]. To see that ~* is * *an isomorphism, we split the sequence (MS) up into short exact sequences H1(G; M) æ I G M i P M and P M æ M i H0(G; M), 12 where P M denotes the maximal perfect submodule of M; see Theorem D. We then get coefficient sequences of the form Hr(X; H1(G; M)) ! Hr(X; I G M) ____//_Hr(X; P M)_________//Hr+1(X; H1(G; M)) || || Hr-1(X; H0(G; M)) ___________//_Hr(X; P M)___//_Hr(X; M) ! Hr(X; H0(G;* * M)) The coefficient map ~* appears as a composite in the middle of the diagram. If* * X is acyclic, then the end terms of both rows are 0. So ~* is an isomorphism. Now suppose G is 2-acyclic and ~* is an isomorphism for all M and r 2. With M* * = Z[G] ~= we have H1(G; M) = 0 and, consequently, isomorphisms Hr(X; I) -! Hr(X; P M) for ~= all r 2. So Hr(X; P M) -! Hr(X; Z[G]) are isomorphisms for r 2 as well. We* * have H1(X; Z) = H1(G; Z) = 0. But then Hr(X; Z) = 0 for r 1. So X is acyclic. 4.4 Proposition Let X be a connected CW-space with n-th k-invariant kn in Hn+1(Pn-1X; ßnX). Then the n-th acyclic k-invariant of AX (see [3]) is ~-1 O cn* *-1(kn): cn-1 n+1 ~-1 n+1 Hn+1(Pn-1X; ßnX) -! H (APn-1X; ßnX) -! H (APn-1X; I G ßnX). Here cn-1 : APn-1X ! Pn-1X is the colocalizing map, G is ß1AX, I is the augment* *ation ideal of Z[G], and ~-1 is the coefficient isomorphism of Lemma 4.3. Sketch of Proof Consider the fibration Y ! APn-1X obtained from the proposed acyclic k-invariant. There is a morphism of fibrations ' : APnX ! Y over APn-* *1X. With the methods supplied in the previous discussion it is possible to show that (1) ßr' is an isomorphism for 1 r n; (2) ßr(AY ! Y ) is an isomorphism for 1 r n; (3) the unique lift f : APnX ! AY of ' is a weak homotopy equivalence. This implies the claim. 4.5 Remark In many situations our work can be used to clarify the effect* * on homo- topy groups of plus constructions and localizations with respect to more genera* *l homology theories h. For example, let h be connective. Note first that X ! Xh (the h-hom* *ology localization of X) factors through X ! X+ . If X+ is simply connected, then the* * canon- ical map X+h ! Xh is a homotopy equivalence; see [9, 1.7]. Now X ! X+h agrees w* *ith X ! X+HR for a suitable ring R of the form ZP or p2PZ=p, where P is a set of * *primes; see [1, 1.1] and compare [12, Sect. 4]. Consequently, the four localization maps X ! Xh, X ! X+h , X ! XHR , X ! X+HR all agree and factor as X -u! X+ - v!(X+ )HR . 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Math. Studies 72, Pr* *inceton Univ. Press, Princeton, NJ 1971. [9]G. Mislin, G. Peschke. "Central extensions and generalized plus construct* *ions". Trans. AMS 353 No. 2 (2001) 585-608. [10]D.G. Quillen, öC homology of groups" In Actes du Congr`es International d* *es Math'ematiciens (Nice, 1970), volume II, Gauthier Villars 1971, 47-51. [11]D.G. Quillen, "Higher K-theory for categories with exact sequences", in Ne* *w de- velopments in topology (Proc. Sympos. Algebraic Topology, Oxford 1972, Lon* *don Math. Soc. Lect. Notes 11, Cambridge Univ. Press 1972, 95-103. [12]J.-Y. Tai, "Generalized plus-constructions and fundamental groups". J. Pur* *e Appl. Alg. 132 (1998) 207-220. David Blanc George Peschke Department of Mathematics Department of Mathematical Sciences University of Haifa University of Alberta 31905 Haifa Edmonton Israel Canada T6G 2G1 e-mail: blanc@math.haifa.ac.il e-mail: George.Peschke@ualberta.ca 14