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*
* Ktheory.
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 ! Pn1X (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
APn1X _____//_Pn1X_____//_(Pn1X)+
The fibers F and are (n1)connected, and their lowest nonvanishing 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 2acyclic group (that is, H1(G; Z) = 0 = H2(G; Z)), th*
*en
every Gmodule 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 CWspace 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 CWspace 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 2s*
*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 Gmodules
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 Gmodule 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 nacyclic if Hk(G; Z) = 0 fo*
*r 1
k n.
Thus 1acyclic groups are known as perfect groups. 2acyclic groups are sometim*
*es called
üs perperfect".
2.2 Definition Let G be a group and n 0. A Gmodule M is called nacycli*
*c if
Hk(G; M) = 0 for 0 k n.
In analogy with the group theoretic terminology, we sometimes refer to a 0acyc*
*lic G
module as a "perfect Gmodule".
2.3 Lemma A group G is nacyclic if and only if its augmentation ideal I*
* is an
(n  1)acyclic Gmodule.
Proof Apply H*(G; ) to I æ Z[G] i Z.
2.4 Corollary A group G is 1acyclic if and only if the multiplication map*
* ~ :
I G I ! I is an epimorphism. G is 2acyclic 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 nacyclic group, and let M be an ar*
*bitrary
Gmodule; then the Gmodule 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 nacyclic group and A is an abelian group with tr*
*ivial
Gaction, 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 2acyclic 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 Gmodules onto the*
* category
A1Z[G]Mod of 1acyclic Gmodules.
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 Gmodule N is the submodule of elements on *
*which
G acts trivially. A central extension of a Gmodule 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 2acyclic group G, a central extension A æ fM i M of
Gmodules is initial amongst all central extensions of M if and only if fM is 1*
*acyclic .
Proof Assume fM is 1acyclic. 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 0acyclic. *
* Part (i)
implies that fM ~=EM which is 1acyclic.
We call a sequence of Gmodules, as in Theorem 2.10, the universal central exte*
*nsion of
M.
2.11 Theorem Given a 2acyclic group G, the following hold:
(i)For every 0acyclic Gmodule M
H1(G; M) æ I G M = EM i M
is a universal central extension of M.
8
(ii)A Gmodule M has a universal central extension if and only if M is 0acycl*
*ic;
compare [8, 5.7].
Proof (i) The given sequence is (MS), taking into account that M is 0acycli*
*c. EM
is 1acyclic by Corollary 2.5. So the claim follows from Theorem 2.10.
(ii) Suppose M is not 0acyclic, 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 Gtrivial 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 1acyclic by Corollary 2.5. So
P M = im(~) is 0acyclic by 2.12. It is a maximal 0acyclic submodule of M bec*
*ause
any module N with P M < N < M yields a quotient N=P M < H0(G; M) with trivial
Gaction. 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 nacyclic
modules:
2.12 Proposition For any group G, and n 0, the class of perfect Gmodul*
*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 Gmodules is closed under direct sums. Further, any quotient *
*M of a
perfect Gmodule N is again perfect because 0 = H0(G; N) i H0(G; M).
2.13 Proposition Given a 2acyclic group G, the class of 1acyclic Gmodu*
*les is
closed under extensions and arbitrary colimits.
Proof If M0 æ M i M00is an extension of Gmodules with M0 and M001acyclic,
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 1acyclic*
* Gmodules.
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(Pn1X, PnX; Z) ! Hk((Pn1X) , (PnX) ; Z).
3.2 Lemma For n 2, F is (n  1)connected.
Proof F is at least (n2)connected because K(ßnX, n) and are (n1)connec*
*ted.
We must show that ßn1F = 0 as well. First of all, we have an epimorphism ßn *
* i
~=
ßn1F . So ßn1F is abelian, and the Hurewicz map ßn1F ! Hn1F is an isomor*
*phism
even for n = 2. Next, by applying the Serre spectral sequence to the fibration *
*APnX !
APn1X, we see that Hn1F is a 1acyclic Gemodule. Furthermore, the commutati*
*ve
diagram
ßn1 _________////ßn1F
 
 
 
fflffl fflffl
ßn1 (PnX)+ ____//_ßn1APnX
tells us that eGacts trivially on the image of ßn1F ! ßn1APnX. 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 @ : ßnAPn1X i ßn1F . On the other hand, eGacts trivia*
*lly
~=
on ßkAPn1X for k n, because we have isomorphisms ßk (Pn1X)+ ! ßkAPn1X
in the central fibration (Pn1X)+ ! APn1X ! Pn1X. Now @ is a morphism of
eGmodules, implying that eGacts trivially on the 1acyclic eGmodule ßn1F . T*
*herefore
ßn1F = 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 eGmodules and ßn+1 is contained in the center*
* of ßnF .
Further, ßnF ~=HnF is seen to be a 1acyclic Gemodule, by using the Serre spe*
*ctral
sequence of the fibration F ! APnX ! APn1X. 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 eGmodule because it fits into the exact sequence*
* ßn+2APn1X !
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 eGmodule 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(Pn1X)+ 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 nstage of Z is simply the ac*
*yclization
APnZ of the usual Postnikov section. The acyclic Postnikov nstage 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 Zn1 to an nstage, one splices into ß*Zn1 *
*a 1
acyclic ß1Zmodule ffn, and there is a corresponding ä cyclic Postnikov invaria*
*nt" ~n 2
Hn+1(Zn1; M). In addition, in dimensions greater than n, one splices into ß*Zn*
*1 certain
ß1Zmodules with trivial action.
In general, starting with an arbitrary space X, Dror's acyclic Postnikov tower *
*of AX has
APnAX as its nth acyclic Postnikov stage. In Theorem A, we were working with a*
* tower
whose nth 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 CWspace. 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 
APn1AXP ________ _______//Pn1AXN 
PPPP  NNN 
PPPP  NNNN 
un1 P'' fflffl N&& fflffl
APn1X _________________//Pn1X
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 ßkisomorphisms 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 CWspace X is acyclic if and only if its fundament*
*al
group G is 2acyclic and, for every Gmodule 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 2acyclic; 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))


Hr1(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 2acyclic 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 CWspace with nth kinvariant kn in
Hn+1(Pn1X; ßnX). Then the nth acyclic kinvariant of AX (see [3]) is ~1 O cn*
*1(kn):
cn1 n+1 ~1 n+1
Hn+1(Pn1X; ßnX) ! H (APn1X; ßnX) ! H (APn1X; I G ßnX).
Here cn1 : APn1X ! Pn1X 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 ! APn1X obtained from the proposed
acyclic kinvariant. 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 hhom*
*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 . The map ß*v is completely under*
*stood
by [2], and here we provide new information on ß*u.
13
References
[1]A.K. Bousfield, "Types of acyclicity", J. Pure Appl. Algebra 4 (1974) 293*
*298.
[2]A.K. Bousfield, "The localization of spaces with respect to homology", Top*
*ology 14
(1975) 135150.
[3]E. Dror, Ä cyclic spaces", Topology 11 (1972) 339348.
[4]E. Dror Farjoun. Cellular Spaces, Null Spaces, and Homotopy Localization. *
*Springer
Verlag Lect. Notes in Math. 1622, BerlinNew York 1995.
[5]J.C. Hausmann, D. Husemoller, Ä cyclic Maps", L'Enseignement Math'ematique*
* 25
(1979) 5375.
[6]M.A. Kervaire, üM ltiplicateurs de Schur et Kth'eorie", in A. Haefliger*
* and
R. Narasimhan, eds., Essays on Topology and related topics: Memoires d'ed*
*i'es `a
Georges de Rham (1903 ), SpringerVerlag, BerlinNew York, 1970, 212225.
[7]W. Meier, R. Strebel, öH motopy Groups of Acyclic Spaces", Quart. J. Math.
Oxford 32 (1981) 8195.
[8]J.W. Milnor, Introduction to Algebraic KTheory. Ann. 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) 585608.
[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, 4751.
[11]D.G. Quillen, "Higher Ktheory 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, 95103.
[12]J.Y. Tai, "Generalized plusconstructions and fundamental groups". J. Pur*
*e Appl.
Alg. 132 (1998) 207220.
David Blanc George Peschke
Department of Mathematics Department of Mathematical Sciences
University of Haifa University of Alberta
31905 Haifa Edmonton
Israel Canada T6G 2G1
email: blanc@math.haifa.ac.il email: George.Peschke@ualberta.ca
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