Cellular approximations using Moore spaces
Jose L. Rodriguez and Jer^ome Scherer *
Abstract
For a two-dimensional Moore space M with fundamental group G, we identify
__
the effect of the cellularization CWM and the fiber P M of the nullificat*
*ion on an
Eilenberg-Mac Lane space K(N; 1), where N is any group: both induce on the
fundamental group a group theoretical analogue, which can also be describe*
*d in
terms of certain universal extensions. We characterize completely M-cellul*
*ar and
M-acyclic spaces, in the case when M = M(Z =pk; 1).
0 Introduction
Let M be a pointed connected CW -complex. The nullification functor PM and the*
* cel-
lularization functor CWM have been carefully studied in the last few years (se*
*e e.g. [8],
[17], [18], [14]). These are generalizations of Postnikov sections and connect*
*ive covers,
where the role of spheres is replaced by a connected CW -complex M and its susp*
*ensions.
This list of functors also includes plus-constructions and acyclic functors ass*
*ociated with a
homology theory, for which M is a universal acyclic space ([2], [13], [22], [24*
*]). Recall that
a connected space X is called M-cellular if CWM X ' X, or, equivalently, if it *
*belongs to
the smallest class C(M) of spaces which contains M and is closed under homotopy*
* equiv-
alences and pointed homotopy colimits. Analogously, X is called M-acyclic if PM*
* X ' *
__
or, equivalently, P M X ' X. It was shown in [14] that the class of M-acyclic *
*spaces
______
is the smallest class C(M) of spaces which contains M and is closed under homo*
*topy
equivalences, pointed homotopy colimits, and extensions by fibrations.
________________________________
*The first author was partially supported by DGICYT grants PB94-0725 and PB9*
*7-0202. The second
by the grant 81LA-51213 of the Swiss National Foundation for Science and the CR*
*M.
1
Very interesting examples are given by the family of Moore spaces M(Z=p; n),*
* the
homotopy cofiber of the degree p self-map of Sn. For n 2, these spaces are the*
* "building
blocks" for simply-connected p-torsion spaces. More precisely, it is shown in *
*[3] (see
also [11]) that the M(Z=p; n)-cellular spaces are exactly the (n - 1)-connected*
* spaces X
such that p . ssnX = 0 and sskX is p-torsion for k > n. However the methods use*
*d in those
papers can not handle the case n = 1. In this paper we introduce the group theo*
*retical
tools that are necessary to deal with this case. They apply to the more general*
* situation
when M is a two-dimensional CW -complex with fundamental group G. As we will se*
*e in
Proposition 2.10 and in the introduction of Section 3, the interesting phenomen*
*a occur
when H2(M; Z) = 0. In that case we say that M is a Moore space of type M(G; 1),*
* and
we shortly write M = M(G; 1).
The G-socle of a group N, which we denote by SG N, is the subgroup of N gene*
*rated
by the images of all homomorphisms from G into N. We introduce the class C(G) f*
*or any
group G. It is the smallest class of groups containing G which is closed under *
*isomorphisms
and colimits. We construct explicitely the right adjoint CG to the inclusion o*
*f C(G) in
the category of groups and show the following (see also Theorem 2.3).
Theorem 2.9 Let M be a two dimensional CW -complex with fundamental group G. Let
X = K(N; 1) where N is any group. Then we have a natural isomorphism
ss1(CWM X) ~=CG N:
Moreover, the action of CG N on the higher homotopy groups of CWM X is trivial.
We further prove the existence of a central extension
0 ! A ! CG N ! SG N ! 1
which is universal in the sense explained in Theorem 2.7.
The proof of such results uses a description of Chacholski, exhibiting CWM X*
* as the
fibre of a map X ! LX, where LX is obtained from X by first killing all maps fr*
*om M,
and then applying M-nullification.
This leads us, in the case when G = Z=p, to the following result. We must no*
*te that
our proof is also valid for M(Z=p; n) with n 2, cases which were previously de*
*alt with
in [3] or [11].
2
Theorem 6.2 Let M = M(Z=p; 1) be the cofiber of the degree p self-map of S1 and*
* X be
a connected space. Then X is M-cellular if and only if ss1X is generated by ele*
*ments of
order p and Hn(X; Z) is p-torsion for n 2.
In particular, a nilpotent space X is M(Z=p; 1)-cellular if and only if ss1X is*
* generated by
elements of order p and ssn(X) is p-torsion for n 2.
Of course, the homotopy groups of non-nilpotent M-cellular spaces need not b*
*e p-
groups. For instance, the universal cover of M(Z=2; 1) is S2. Likewise, a space*
* all whose
homotopy groups are p-groups need not be M-cellular, as shown by Example 6.4, w*
*here
we compute the M(Z=2; 1)-cellularization of K(3; 1).
Nullifications with respect to Moore spaces are better understood. Our aim h*
*ere is to
investigate the homotopy fibre of such nullifications.
Recall that the G-radical of a group N, which we denote by TG N as in [10] o*
*r [8], is the
smallest subgroup of N such that Hom (G; N=TG N) = 0: It is known [10] that whe*
*n M is a
two dimensional CW -complex with fundamental group G, then ss1PM X ~=ss1X=TG (s*
*s1X).
If in addition M is a M(G; 1), the space PM X can be viewed as the fibrewise R-*
*completion,
in the sense of Bousfield and Kan [9], of a covering fibration associated to th*
*e G-radical
subgroup (for a suitable coefficient ring R); see [9], [10], and [12].
_____
We introduce the class C(G) , for any group G. It is the smallest class of *
*groups
containing G which is closed under isomorphisms, colimits, and extensions. We s*
*how in
Proposition 3.11 that the fundamental group of any M-acyclic space belongs to t*
*his class.
__
We define then DG N as the fundamental group of P MK(N; 1), note that the actio*
*n of
__
DG N on the higher homotopy groups of P MK(N; 1) is trivial, and prove:
Theorem 3.13 Let M = M(G; 1) be a two-dimensional Moore space. Then DG is right
_____
adjoint to the inclusion of C(G) in the category of groups.
As in the case of cellularization, there exists a central extension
0 ! B ! DG N ! TG N ! 1 (0.*
*1)
which is universal in the sense explained in Theorem 3.3.
A very enlightening example is given by the acyclic space described by Berri*
*ck and
Casacuberta in [2, Example 5.3], which turns out to be an M(G; 1) for some acyc*
*lic
group G. In this case M(G; 1)-nullification is equivalent to Quillen's plus-co*
*nstruction
3
and the G-radical of any group N is its largest perfect subgroup. Thus, in this*
* case, the
central extension (0.1) is the usual universal central extension of TG N.
Similar results have been obtained by Mislin and Peschke in [22] in the case*
* when PM
is the plus construction associated to a generalized homology theory. In all t*
*hese cases
__
CWM and P M coincide.
Finally, we obtain the following result for G = Z=p (compare with [3] or [11*
*]).
Theorem 6.1 Let M = M(Z=p; 1). Then X is M-acyclic if and only if ss1X coincides
with its Z=p-radical and Hn(X; Z) is p-torsion for n 2.
In particular, a nilpotent space X is M(Z=p; 1)-acyclic if and only if ss1X coi*
*ncides with
its Z=p-radical and ssn(X) is p-torsion for n 2.
Acknowledgements: This paper was mainly elaborated during the 1998 Topology Sem*
*ester
at the CRM. We are especially indebted to Jon Berrick and Wojciech Chacholski f*
*or
several enlightening discussions. We warmly appreciate Carles Casacuberta's adv*
*ice and
encouragement. The second author would like to thank the CRM for making the lif*
*e of a
post-doctoral fellow so pleasant.
1 Preliminary results
We give here a short review of the terminology involved in the theory of homoto*
*pical
localization. We also remind the reader some of the results needed in this pape*
*r. More
details can be found in [17], [18], [14], [7].
By an idempotent augmented functor in the category of spaces, we mean a func*
*tor E
from the category of pointed spaces to itself. It preserves weak homotopy equi*
*valences
and is equipped with a natural transformation c : E ! Idfrom E to the identity *
*functor,
such that cEX : EEX ' EX for all spaces X.
__
The most important examples for us are CWM and P M. Let M be a connected CW*
* -
complex. A map Y ! X is an M-cellular equivalence if it induces a weak equivale*
*nce on
pointed mapping spaces
map *(M; Y ) -~! map *(M; X) :
There exists then, for each connected space X, a map CWM X ! X, called M-cellul*
*ar
approximation, which is universal (initial) among all M-cellular equivalences t*
*o X. The
4
spaces for which CWM X ' X are called M-cellular. The class of M-cellular spac*
*es
has been identified as the smallest class C(M) of spaces containing M and close*
*d under
weak equivalences and pointed homotopy colimits; see [14, Theorem 8.2] and [18,*
* 2.D].
A connected space Z is said to be M-null if map *(M; Z) is weakly contractible,*
* that is,
[kM; X] = * for all k 0. There exists a map X ! PM X, called M-nullification,
which is universal (terminal) among all maps from X to an M-null space. Finally*
* denote
__
by P MX ! X the homotopy fibre of X ! PM X. A connected space X is called M-
__
acyclic if PM X ' *, i.e., P MX ' X. The class of M-acyclic spaces has been ide*
*ntified in
______
[14, Theorem 17.3] as the class C(M) . In addition to being closed under weak e*
*quivalences
and pointed homotopy colimits, it is also closed under extensions by fibrations.
So, every M-cellular space is M-acyclic and furthermore, it is known that each *
*M-acyclic
space is M-cellular ([18, 3.B.3]). Hence by universality we have natural maps
__ ff fi __
P M X -! CWM X -! P MX: (1.*
*2)
Thus CWM X can be thought of as the fiber of a mixing process between M- and M-
nullification. More precisely:
Theorem 1.1 ([14, Theorem 20.5]) Let M be any connected CW -complex. There i*
*s a
fibration
j
CWM X -! X -! LX
where j is the composition of the inclusion X ! X0 of X into the homotopy cofib*
*re of the
evaluation map _[M;X]M ! X, followed by X0! PM (X0).
Note that the inclusion X ! X0is in fact functorial in the homotopy category*
*, and it is
universal (initial) among all maps X ! Z such that M ! X ! Z is homotopically t*
*rivial
(compare with [2, Corollary 2.2]). Hence, X ! LX is also functorial in the hom*
*otopy
category.
The fundamental groups of PM X and LX have a group theoretical meaning in the
case when M is a two-dimensional CW -complex, as we next explain. The G-socle *
*of a
group N, which we denote by SG N, is the subgroup of N generated by the images *
*of all
homomorphisms from G into N. The G-radical of N, which we denote by TG N as in
[10] or [8], is the smallest subgroup of N such that Hom (G; N=TG N) = 0: The G*
*-radical
5
of N can be constructed as a (possibly transfinite) direct limit of subgroups T*
*i where T1
is the G-socle of N and Ti=Ti-1= SG (N=Ti-1). In other words, the groups N such*
* that
TG N = N are precisely the groups which have a normal series whose factors coin*
*cide
with their G-socles. The first link between the topological nullification and t*
*heir discrete
analogues is given by the following result. Its first part is [10, Theorem 3.5*
*] (see also
[6, Theorem 5.2]).
Lemma 1.2 If M is a two-dimensional CW -complex with fundamental group G and*
* X
is any space, we have isomorphisms
ss1(PM X) ~=ss1X=TG (ss1X) and ss1X0~= ss1LX ~=ss1X=SG (ss1X);
where X0 and LX are as in Theorem 1.1. 2
Such a two-dimensional CW -complex M is called a Moore space if H2(M; Z) = 0*
*. It
has type M(G; 1) if ss1M ~=G. Using Theorem 1.1 it is easy to see that two Moor*
*e spaces
of type M(G; 1) determine the same cellularization and nullification functors.
Bousfield computed in [8, Section 7] the effect of the nullification functor*
* with respect
to a two-dimensional Moore space M(G; 1) on nilpotent spaces; see also [10, The*
*orem 4.4]
for G = Z=p, and [12, Theorem 2.4] when G = Z[1=p]. Let J be the set of primes *
*p for
which Gab is uniquely p-divisible. Define R = Z(J), the integers localized at *
*J, if Gab
is torsion, and R = p2JZ=p otherwise. Let R1 X be the Bousfield-Kan R-completi*
*on
of X; see [9]. Then PM X can be obtained as the fibrewise R-completion of the c*
*overing
fibration associated to the G-radical of ss1X. That is, we have a diagram of fi*
*brations
"X -! X -! K(ss1X=TG (ss1X); 1)
# # #id (1.*
*3)
R1 "X -! PM X -! K(ss1X=TG (ss1X); 1);
where R1 "X is simply connected, as the fundamental group of X" is R-perfect, i*
*.e.,
H1(X"; R) = H1(TG (ss1X); R) = 0. Hence R1 "Xcoincides with "X+HR, the plus-con*
*struction
with respect to ordinary homology with coefficients in R; see [9, VII.6] and [1*
*2]. The uni-
versal cover of PM X is thus equivalent to the three following spaces:
PM X"' R1 "X' "X+HR:
Let ARX denote the R-acyclic functor, that is, the homotopy fiber of X ! X+H*
*R (cf.
[9, VII, 6.7]). Then, a connected space X is HR-acyclic, i.e., H"*(X; R) = 0, i*
*f and only
if ARX ' X. The above remark immediately implies the following.
6
Proposition 1.3 Let M = M(G; 1) be a two-dimensional Moore space, and X be any
connected space. If X" denotes the covering of X corresponding to the subgroup *
*TG (ss1X),
then
__ __
PM X ' ARX" ' PM X";
where R is the ring associated to G as above. *
* 2
Corollary 1.4 Let M = M(G; 1) be a two-dimensional Moore space, and X be any
connected space. Then, X is M-acyclic if and only if
TG (ss1X) = ss1X and Hk(X; R) = 0 for k 2 : *
* 2
2 The fundamental group of M (G; 1)-cellular spaces
In this section we define algebraically a G-cellularization functor CG in the *
*category
of groups. We show that CG N coincides with the fundamental group of the M(G; *
*1)-
cellularization of K(N; 1). This yields a characterization of CG N as a certain*
* universal
central extension of the G-socle of N. We also prove that the action of CG N is*
* trivial on
the higher homotopy groups of CWM(G;1)K(N; 1).
As suggested by Dror-Farjoun, we introduce the closed class of groups C(G). *
*It is the
smallest class of groups containing G, and closed under isomorphisms and taking*
* colimits.
In other words, if F : I ! Groups is a diagram with F (i) 2 C(G) for any i 2 I,*
* then
colimIF should again belong to C(G).
The following proposition gives the explicit construction of a G-cellulariza*
*tion functor.
The existence of such a functor is also ensured by [5, Corollary 7.5].
Proposition 2.1 Let G be a group. The inclusion C(G) Groups has a right adjo*
*int
CG : Groups ! C(G).
Proof. For any group N, the map CG N ! N is constructed by induction as follow*
*s.
First define C0 = *h:G!N G, the free product of as many copies of G as there ar*
*e morphisms
from G to N, and let h0 : C0 ! N be the evaluation (so that h0(C0) = SG N). Now*
* take
the free product *(h0;h00)G, where h0; h00: G ! C0 is any pair of morphisms coe*
*qualized
by h0. Define C1 as the coequalizer of *(h0;h00)G C0, and repeat this process*
* (maybe
transfinitely). Notice that this inductive construction of CG N shows that we *
*have a
7
natural epimorphism CG N!! SG N for any group N. The group CG N is in C(G) and
the morphism c : CG N ! N is universal (terminal), i.e., c induces a bijection *
*of sets
Hom (G; c) : Hom (G; CG N) ~=Hom (G; N). *
* 2
By analogy to the case of spaces, a group N in C(G) is called G-cellular.
Lemma 2.2 Let M be a two-dimensional CW -complex with fundamental group G. A
group homomorphism N ! N0 induces an isomorphism CG N ~= CG N0 if and only if
CWM K(N; 1) ' CWM K(N0; 1).
Proof. The pointed mapping space map *(M; K(N; 1)) is weakly equivalent to the *
*dis-
crete set Hom (G; N). *
* 2
Since ss1 commutes with homotopy colimits, the fundamental group of any M-ce*
*llular
space is ss1M-cellular, for any M. Furthermore, the following holds (this could*
* also have
been taken as definition of CG ):
Theorem 2.3 Let M be a two-dimensional CW -complex with fundamental group G. *
*Let
X = K(N; 1) where N is any group. Then we have a natural isomorphism
ss1(CWM X) ~=CG N:
Proof. By the previous observation, ss1(CWM X) is G-cellular. It thus only rema*
*ins to
prove that c : CWM X ! X induces a bijection of sets ss1(c)* = Hom (G; ss1(c)).*
* Consider
the following commutative diagram (of sets)
[M; CWM X] -e! Hom (G; ss1(CWM X))
#c* #ss1(c)*
0
[M; X] -e! Hom (G; N):
Since M is two-dimensional, e is surjective and X being a K(N; 1), e0 is biject*
*ive. On
the other hand, c* is also bijective by the universal property of CWM . Thus, *
*ss1(c)* is
bijective, as desired. *
* 2
Lemma 2.4 Let "Xdenote the covering of X corresponding to the subgroup SG (s*
*s1X) and
let M be a two-dimensional CW -complex with fundamental group G. Then
(i)(X")0is the universal cover of X0,
(ii)LX" is the universal cover of LX,
where X0 and LX are defined in Theorem 1.1.
8
p
Proof. The covering fibration X" ! X ! K(ss1X=SG (ss1X); 1) induces a biject*
*ion
[M; "X] ~= [M; X] because the map [M; p] is trivial. Apply now Mather-Puppe th*
*eorem
(see [16, Proposition 6.1]) saying that "the fiber of the push-out is the push-*
*out of the
fibers" when the base space is fixed, to get a fibration X"0! X0 ! K(ss1X=SG (s*
*s1X); 1).
This is the universal cover fibration by Lemma 1.2. So part (i) holds. For pa*
*rt (ii) we
note that the previous fibration has a M-null base, and is therefore preserved *
*under
M-nullification. 2
The following corollary could also have been proved directly by checking tha*
*t the
covering X"! X is indeed an M-equivalence.
Corollary 2.5 Let X" denote the covering of X corresponding to the subgroup SG*
* (ss1X)
and let M be a two-dimensional CW -complex with fundamental group G. We have a
homotopy equivalence CWM X ' CWM X". 2
Lemma 2.6 Let G be any group, and 0 ! A ! E ! N ! 1 be a central extension
of groups. Then CG E ~= CG N if and only if Hom (Gab; A) = 0 and the natural *
*map
Hom (G; N) -! H2(G; A) is trivial.
Proof. Apply map *(K(G; 1); -) to the fibration
K(E; 1) ! K(N; 1) ! K(A; 2) :
This gives a new fibration, whose homotopy sequence
0 ! Hom (G; A) ! Hom (G; E) ! Hom (G; N) ! H2(G; A)
is exact as in [9, IX, 4.1]). The lemma is proved. *
* 2
We now know enough to describe our first universal central extension. It is *
*nothing
but a universal central G-cellular equivalence.
Theorem 2.7 Let G be any group. Then, for each group N, there is a central ex*
*tension
0 ! A ! CG N ! SG N ! 1
such that Hom (Gab; A) = 0 and the natural map Hom (G; N) - ! H2(G; A) is tri*
*vial.
Moreover, this extension is universal with respect to these two properties.
9
Proof. Let M be a two-dimensional CW -complex with fundamental group G and let
X = K(SG N; 1). The space LX is 1-connected by Lemma 1.2 and define then
A = ss2LX ~=ss2X0=TGab(ss2X0)
(this follows from [8, Theorem 7.5]). The long exact sequence in homotopy of t*
*he fibration
CWM X ! X ! LX produces now the desired central extension, where we identify
CG (SG N) with CG N. This can be deduced from Corollary 2.5. The G-cellulari*
*zation
of CG N ! SG N is also an isomorphism since Hom (G; CG N) ~= Hom (C; SG N). H*
*ence
Hom (Gab; A) = 0 and the natural map Hom (G; N) -! H2(G; A) is trivial by Lem*
*ma 2.6.
The universal property is a direct consequence of the same lemma. *
* 2
Corollary 2.8 Let M be a two-dimensional CW -complex with fundamental group G.
Then SG N = N if and only if LK(N; 1) is 1-connected, and CG N ~= N if and onl*
*y if
LK(N; 1) is 2-connected. In particular
(1) ss2LK(N; 1) ~=H2LK(SG N; 1);
(2) ss3LK(N; 1) ~=H3LK(CG N; 1). *
* 2
In the next theorem we identify the universal cover of CWM K(N; 1) and rema*
*rk that
the action of the fundamental group is trivial. This could also be seen as a p*
*articular case
of [20, Proposition A.1] or the even more general [22, Corollary 7.7].
Theorem 2.9 Let M be a two-dimensional CW -complex with fundamental group G.
Then CWM K(N; 1) ' CWM K(CG N; 1) and the universal cover fibration is given by
LK(CG N; 1) -! CWM K(N; 1) -! K(CG N; 1):
Moreover the action of CG N on ssnCWM K(N; 1) for n 2 is trivial.
Proof. The first part follows from Lemma 2.2. The proof of the second part has*
* been
suggested by Carles Casacuberta. The fibration
CWM K(N; 1) ! K(CG N; 1) ! LK(CG N; 1)
induces a long exact sequence of CG N-modules in homotopy. But the space LK(CG*
* N; 1)
is 2-connected by Corollary 2.8, so that the action of CG N on the higher homo*
*topy groups
of CWM K(N; 1) is trivial. *
* 2
10
Proposition 2.10 Let M be a two-dimensional CW -complex with fundamental group*
* G.
Assume that H2(M; Z) 6= 0. Then
CWM K(N; 1) ' K(CG N; 1) :
Proof. Choose a presentation OE : *Z ! *Z of G, and realize it as a map f betw*
*een
wedges of circles having M as its homotopy cofiber. Note that a simply connecte*
*d space
Y is M-null if and only if the G-radical of ss2Y is trivial, and sskY is OEab-l*
*ocal for any
k 3, i.e., Hom (OEab; sskY ) is bijective (see [23, Theorem 4.3.6]). When H2(*
*M; Z) 6= 0,
the homomorphism OEab is not injective and any OEab-local group is trivial. So *
*LK(CG N; 1)
is the trivial space, as it is already 2-connected (see [23, Corollary 4.3.9]).*
* 2
Example 2.11 If G = C2 and N is nilpotent then CG N = SG N (by Corollary 6.3
below). However, CG N 6~=SG N in general, as shown by the following example, wh*
*ich was
suggested by Alejandro Adem:
N = :
In other words N is the push-out of the diagram C2 * C2 Z ! C2 * C2 where both
arrows send the generator of Z to the commutator. The Mayer-Vietoris sequence s*
*hows
then that H2N ~= Z ~=A. Thus CG N is an extension of N by Z. This also provides*
* an
example of a quotient of a free product of copies of G which is not cellular.
3 The fundamental group of M (G; 1)-acyclic spaces
__
We imitate now the preceding section, replacing CWM by P M . First we change *
*our
*
* _____
closed class. In addition to being closed under isomorphisms and colimits, the *
*class C(G)
is assumed to be closed under taking arbitrary extensions. That is, if N ,! E!*
*! Q is
_____ _____
an extension with N; Q 2 C(G) , then E belongs to C(G) as well. The right adj*
*oint of
_____
the inclusion of C(G) in the category of groups is denoted by DG and we constr*
*uct it
__
by topological means, namely as the fundamental group of P M(G;1)K(N; 1). It co*
*uld be
interesting to have an algebraic description of DG N, similar to that of CG N, *
*in terms of
colimits and extension by short exact sequences.
A well known topological proof of the existence of the universal central ext*
*ension over
a perfect group N uses Quillen's plus-construction. We will follow exactly the *
*same line
11
of proof here, the plus-construction being replaced by a nullification functor *
*with respect
to a Moore space. This is a true generalization of this old result in light of *
*[2], where it
is proven that the plus-construction is indeed the nullification with respect t*
*o a Moore
space. Another approach is taken in [22], where the plus-construction associate*
*d to any
homology theory determines a universal central extension.
When M is a two-dimensional CW -complex with fundamental group G which is not
a Moore space, i.e., H2(M; Z) 6= 0, the effect of PM is drastic. As in the pr*
*oof of
Proposition 2.10, one shows that
PM X ' K(ss1X=TG (ss1X); 1):
__
Hence PM X is the covering of X corresponding to the subgroup TG (ss1X). From n*
*ow on,
we will therefore only consider Moore spaces.
Define, for a two-dimensional Moore space M = M(G; 1),
__
DG N := ss1(P M K(N; 1)):
This does not depend on the choice of M by the observation made after Lemma 1.2.
Lemma 3.1 Let M be a two-dimensional Moore space with fundamental group G an*
*d let
B denote the group ss2PM K(TG N; 1). The space K(B; 2) is then M-null, or equiv*
*alently,
Hom (Gab; B) = 0 = Ext(Gab; B).
Proof. The space PM K(TG N; 1) is 1-connected by Lemma 1.2. The second Postnikov
section K(ss2X; 2) of a simply connected M-null space X is M-null as well, sinc*
*e ss2PM X
only depends on ss2X by [8, Theorem 7.5]. *
* 2
The groups B satisfying Hom (Gab; B) = 0 = Ext (Gab; B) can only be of the t*
*wo
following forms ([8, 7.1]):
Fact 3.2 Let J denote the set of primes p such that Gab is uniquely p-divisibl*
*e, and
J0 the complementary set of primes. Then, either Gab is J0-torsion and B is J-*
*local,
or Gab is uniquely J-divisible and B is Ext-J-complete (in the sense of [9]). *
*In other
words, Hom (Gab; B) = 0 = Ext(Gab; B) if and only if Hom (H; B) = 0 = Ext(H; B)*
* where
H = p2J0Z=p if Gab is torsion, or H = Z[J-1] otherwise.
We are now ready to prove the existence of our second universal central exte*
*nsion.
12
Theorem 3.3 Let G be the fundamental group of a two-dimensional Moore space, *
*which
we denote by M. Then, for each group N, there is a central extension
0 ! B ! DG N ! TG N ! 1
such that Hom (Gab; B) = 0 = Ext(Gab; B). Moreover, this extension is universal*
* (initial)
with respect to this property.
Proof. The idea of the proof is analogous to that of Theorem 2.7. Let X = K(TG *
*N; 1).
__
The fibration P MX ! X ! PM X produces the desired extension using Proposition *
*1.3,
where B = ss2PM K(TG N; 1) satisfies the property by Lemma 3.1.
We check now the universal property. Let B0 be an abelian group having the a*
*bove
property, and 0 ! B0 ! E ! TG N ! 1 a central extension. Realize it as a fibra*
*tion
K(E; 1) -! X -! K(B0; 2), where the base space is M-null. There exists therefor*
*e a
map PM X ! K(B0; 2), unique up to homotopy, inducing a map of fibrations. *
* 2
Example 3.4 Let G = C2 and N be the group described in Example 2.11. Then DG N
is an extension of N by Z[1=2].
Remark 3.5 The group TG N is R-perfect and the central extension of Theorem 3*
*.3 is
the universal central extension of TG N with coefficients in R. By this we mean*
* the cen-
tral extension induced from the fibration ARK(TG N; 1) ! K(TG N; 1) ! K(TG N; 1*
*)+HR.
These two extensions coincide by Proposition 1.3. Even though this "R universal*
* central
extension" seems to be classical, we do not know any other reference than [22].
Example 3.6 Let G = Z[1=p], so that R = Z=p. In [22, Proposition 5.4] Mislin*
* and
Peschke computed that
B ~=Ext (Z(p1 ); H2(TG N; Z)) Hom (Z(p1 ); H1(TG N; Z)) ;
where Z(p1 ) is the p-torsion subgroup of Q=Z. Let G = Z=p, so that R = Z[1=p].*
* Then
B = H2(TG N; R).
An interesting consequence of the previous result is that the functor DG is *
*idempo-
tent. It is worth noting that it seems rather difficult to prove this fact dire*
*ctly from the
definition.
13
__ __
Theorem 3.7 Let M be as above. Then P MK(DG N; 1) ' PM K(N; 1) and in partic*
*ular
the functor DG is idempotent. The universal cover fibration is given by
__
PM K(DG N; 1) -! PM K(N; 1) -! K(DG N; 1):
__
Moreover the action of DG N on ssnP M K(N; 1) for n 2 is trivial.
Proof. The functor PM preserves the fibration
K(DG N; 1) ! K(TG N; 1) ! K(B; 2)
*
* __
of Theorem 3.3 since K(B; 2) is M-null by Lemma 3.1. Thus so does the functor*
* P M.
__ __
That is, we have P M K(DG N; 1) ' P M K(TG N; 1). The later space is equivale*
*nt to
__
P M K(N; 1) by Proposition 1.3. The statements about the universal cover fibra*
*tion follow
as in Theorem 2.9. *
* 2
Remember that the ring R is determined by the group G as follows: R = Z(J)i*
*f Gab
is torsion, and R = p2JZ=p otherwise. We say that a group N is super R-perfec*
*t if
H1(N; R) = 0 = H2(N; R).
Proposition 3.8 Let G be the fundamental group of a two-dimensional Moore spa*
*ce M.
The following statements are equivalent:
(1) DG N ~=N.
(2) The space PM K(N; 1) is 2-connected.
(3) H2(N; B) = 0 for any B such that Hom (Gab; B) = 0 = Ext(Gab; B).
(4) TG N = N and N is super R-perfect.
Proof. Theorem 3.3 implies that (1), (2), and (3) are equivalent. We only prov*
*e that
(4) implies (2). Since N coincides with its G-radical, PM K(N; 1) ' K(N; 1)+H*
*R (see
diagram (1.3)), and it is 1-connected. Thus K(N; 1)+HR' K(N; 1)HR the HR-homol*
*ogical
localization by [22, Proposition 1.6]. Moreover, by [4, Theorem 5.5], ss2K(N; *
*1)HR is an
HR-local group. But H1(ss2K(N; 1)HR ; R) = 0, so it has to be trivial. *
* 2
As a consequence, we obtain the following formulae for the low-dimensional *
*homotopy
groups of PM K(N; 1); cf. [1, Corollary 8.6].
14
Corollary 3.9 Let M be a two-dimensional Moore space with fundamental group *
*G.
Then
(1) ss2PM K(N; 1) ~=H2PM K(TG N; 1);
(2) ss3PM K(N; 1) ~=H3PM K(DG N; 1). *
* 2
We end this section by proving that this topological construction gives not*
*hing else
_____
but the right adjoint of the inclusion of C(G) into the category of groups. We*
* denote the
class {N | DG N ~=N} by D(G).
Proposition 3.10 Let M be a two-dimensional Moore space with fundamental grou*
*p G.
The class D(G) is closed under arbitrary extensions and colimits.
Proof. The class of G-radical groups is closed under colimits and extensions, *
*and so is
the class of super R-perfect groups: An easy Hochschild-Serre spectral sequenc*
*e argument
shows that an extension of super R-perfect groups is again super R-perfect, an*
*d a Mayer-
Vietoris argument shows it for a push-out. Since homology commutes with teles*
*copes,
the proposition is proved. *
* 2
Proposition 3.11 Let M be a two-dimensional Moore space with fundamental grou*
*p G.
_____
Then C(G) = D(G).
_____
Proof. By Proposition 3.10, C(G) D(G). To show the converse we prove that t*
*he
______ _____ ______
fundamental group of any space in C(M) is in C(G). But C(M) is the smallest *
*class con-
taining M which is closed under homotopy colimits and extensions by fibrations*
*. Clearly
the fundamental group of the homotopy colimit of a diagram all whose values ha*
*ve ss1 in
_____ _____
C(G) is again in C(G). So assume we have a fibration F ! E ! B of connected sp*
*aces,
_____ _*
*____
where the fundamental groups of F and B are in C(G). We have to prove ss1E 2 C*
*(G).
The cokernel of the boundary morphism ss2B ! ss1F is isomorphic to the coinvar*
*iants
_____
(ss1F )ss2B= colimss2B(ss1F ) and thus belongs to C(G). Therefore ss1E is an e*
*xtension of
_____
two groups of C(G). *
* 2
Corollary 3.12 Let G be the fundamental group of a two-dimensional Moore spac*
*e M.
A group N belongs then to D(G) if and only if there exists an M-acyclic space *
*X with
ss1X ~=N. 2
15
Theorem 3.13 Let M be a two-dimensional Moore space with fundamental group G.
_____
The augmented functor DG is then right adjoint to the inclusion of C(G) in the *
*category
of groups, i.e., we have an isomorphism Hom (L; DG N) ~= Hom (L; N) for any g*
*roup
_____
L 2 C(G).
Proof. The map K(DG N; 1) ! K(N; 1) induces a weak equivalence
__ __
PM K(DG N; 1) ' PM K(N; 1)
_____
by Theorem 3.7. Let L 2 C(G). Then
__ __
map *(P M K(L; 1); K(DG N; 1)) ' map *(P M K(L; 1); K(N; 1)) ;
i.e., Hom (L; DG N) ~=Hom (L; N). *
* 2
4 Acyclic spaces
We illustrate the preceding sections by the case when the Moore space M is acyc*
*lic. We
__
identify the functors P M and CWM . The motivating example is the universal ac*
*yclic
group F of Berrick and Casacuberta [2, Example 5.3]. It satisfies SF N = TF N =*
* PN,
the maximal perfect subgroup of N. In this case CF N = DF N = gPN the univers*
*al
central extension of PN and the two central extensions coincide. If M = M(F; 1)*
*, the
__
functors P M and CWM coincide with Dror's acyclic functor A, the fiber of Qui*
*llen's
plus-construction.
We want to consider now an arbitrary acyclic group G, and an acyclic complex*
* M
with fundamental group G. This space M is of course not determined by the group*
*. Since
M ' *, the fibration of Theorem 1.1 has the form
CWM X ! X ! X0
where X0 is the homotopy cofibre of the map _[M;X]M ! X. Let X+Ndenote the plus-
construction of X with respect to a perfect, normal subgroup N of ss1X, and let*
* AN X
be the homotopy fibre of the natural map X ! X+N. The universal property of the*
* plus-
construction ensures that X0 ' X+S, where S = S(M; X) is the topological socle,*
* i.e.,
the subgroup generated by the images of all homomorphisms ss1(M) ! ss1(X) which*
* are
induced by maps M ! X (see [2, Section 2]). This subgroup of ss1X is sometimes *
*also
16
called the subgroup swept by M. Arguing similarly with PM X we deduce the follo*
*wing;
cf. [2, Corollary 2.2].
__
Theorem 4.1 Let M be an acyclic CW -complex. Then the map fi : CWM X ! P MX
is equivalent to ASX ! ATX where S is the subgroup swept by M and T is such that
ss1PM X ~=ss1X=T . *
* 2
When M is two-dimensional, S = SG N and T = TG N for all X, and N denotes
the fundamental group of X. We also have S = SG N if M is any CW -complex and
X = K(N; 1). Therefore, we deduce the following.
Corollary 4.2 Let X be a space with fundamental group N and M an acyclic CW -
complex with fundamental group G. Suppose that M is of dimension two or X = K(N*
*; 1).
Then ss1(CWM X) ~=CG N is the universal central extension of SG N. *
* 2
5 Nilpotent spaces
When X is a nilpotent space, the homotopy long exact sequence associated to the*
* fibration
__
P M X ! X ! PM X yields the homotopy groups of the M-acyclic part of X, as foll*
*ows:
Proposition 5.1 Let n 1, M be a Moore space M(G; n), and let X be any connect*
*ed
space. Suppose that X is nilpotent if n = 1. Let J be the set of primes p such *
*that G is
__
uniquely p-divisible and J0 be the complementary set of primes. Then P MX is (n*
* - 1)-
connected and for k n
(I)if Gab is torsion, then
8
>< 0(Z(p1 ) ss X Tor(Z(p1 ); ss X)) if k n + *
*1;
__ p2J k+1 k
ssk(P M X) ~=
>: 1
p2J0(Z(p ) ssn+1X TG (ssnX)) if k = n;
(II)if Gab is not torsion, then
8
>< (Ext (Z[1=p]; ss X) Hom (Z[1=p]; ss X)) if k n *
*+ 1;
__ p2J k+1 k
ssk(P M X) ~=
>:
p2J(Ext (Z[1=p]; ssn+1X) DG (ssnX) if k = n.
17
Proof. We use from [8, Theorem 7.5] that in the first case we have
8
>< ssk(X) Z(J0) if k n + 1;
ssk(PM X) ~=
>:
ssnX=TG (ssnX) if k = n;
In the second case we have:
8
>< p2J(Ext (Z(p1 ); sskX) Hom (Z(p1 ); ssk-1X)) if k n + 1;
ssk(PM X) ~=
>:
ssnX=TG (ssnX) if k = n.
2
Example 5.2 Let G be a rational group of rank 1 of type (r2; r3; r5; : :):. T*
*hat is, G is the
additive subgroup of Q generated by the fractions 1=ps, for s rp (we write rp *
*= 1 if G
is uniquely p-divisible). Note that if rp < 1 then the G-radical contains the Z*
*=p-radical.
Moreover, in the category of abelian groups, the G-socle coincides with the G-r*
*adical if
and only if G = Z[J-1] (see [19]).
This allows us to construct two subgroups of Q having the same set of primes*
* for
which they are uniquely p-divisible, but with distinct radical, and thus distin*
*ct acyclic
approximation. Fix a prime p and define G by rp = 1 and rq = 1 when q 6= p, so *
*that
__
H = Z[1=p]. Let M = M(G; 1) and M0 = M(H; 1). Then we have P MK(Z[1=p]; 1) ' *,
__
while P M0K(Z[1=p]; 1) ' K(Z[1=p]; 1).
We next give a description of the class of nilpotent M-acyclic spaces; compa*
*re with
Corollary 7.9 in [8], see also [21]. The case when n = 1 gives a less general r*
*esult than
Corollary1.4, but gives a characterization of M(G; 1)-acyclic spaces in terms o*
*f their
homotopy groups rather than their homology groups.
Proposition 5.3 Let M be a Moore space M(G; n) with n 1, and let X be any con-
nected space. Suppose that X is nilpotent if n = 1. Then, X is M-acyclic if a*
*nd only
if X is (n - 1)-connected, ssnX coincides with its G-radical and ssk(X) is J0-t*
*orsion for
k n + 1 in the case when G is torsion, or ssk(X) is uniquely J-divisible for k*
* n + 1
otherwise. *
* 2
18
6 The torsion case
In this section we only deal with the case of the Moore spaces M(Z=pk; 1), for *
*k 1. We
give a characterization of M(Z=pk; 1)-cellular spaces, which holds even for non*
*-nilpotent
spaces. The M(Z=pk; 1)-acyclic spaces have been already identified in Corollary*
* 1.4. The
following reformulation only makes use of the fact that a group A is p-torsion *
*if and only
if A Z[1=p] = 0.
Theorem 6.1 Let M = M(Z=pk; 1), k 1. Then a space X is M-acyclic if and only*
* if
ss1X coincides with its Z=p-radical and Hn(X; Z) is p-torsion for n 2. *
* 2
Theorem 6.2 Let M = M(Z=pk; 1), k 1. Then a space X is M-cellular if and onl*
*y if
ss1X is generated by elements of order pl for l k and Hn(X; Z) is p-torsion fo*
*r n 2.
Proof. We use again the fact that CWM X can be obtained as the fiber of the map
X ! PM(Z=pk;2)X0, where X0 is the cofiber of _M ! X. So we have to find a neces*
*sary
and sufficient condition for PM(Z=p;2)X0 to be trivial. First X0 has to be 1-c*
*onnected,
and this is equivalent to ss1X coinciding with its Z=pk-socle. Knowing that X0 *
*and thus
PM(Z=p;2)X0 are 1-connected, the triviality of the latest is equivalent to its *
*acyclicity. By
Proposition 5.3 the homotopy, or equivalently the reduced integral homology of *
*X0, has
to be p-torsion. The long exact sequence in homology of the cofibration sequenc*
*e defining
X0 shows that this is equivalent to "H*(X; Z) being p-torsion. *
* 2
Corollary 6.3 Let M = M(Z=pk; 1). A nilpotent space X is M-cellular if and on*
*ly if
ss1X is generated by elements of order plfor l k and ssn(X) is p-torsion for n*
* 2. 2
The characterization given in [14, 12.5] or [15] of M(Z=2; n)-cellular space*
*s (ssn is
generated by involutions, and the higher homotopy groups are 2-torsion) is true*
* for n = 1
if we work in the category of nilpotent spaces. An easy counter-example for non*
*-nilpotent
spaces is given by M(Z=2; 1) itself. It is of course an M(Z=2; 1)-cellular spa*
*ce, even
though ss2M(Z=2; 1) ~=Z. We finally consider the following example.
Example 6.4 The symmetric groups n are C2-cellular for n 2. Indeed, using t*
*he
presentation n = ,
one can obtain n as a push-out of a family of homomorphisms between free produc*
*ts of
19
C2. For n - 1 j 1, let Gj be the coproduct of n - 1 - j copies of C2 * C2, an*
*d Hj
the coproduct of as many copies of C2. Define Gj ! Hj to be the coproduct of th*
*e fold
maps C2* C2 ! C2. Let K be the coproduct of n - 1 copies of C2, and call the ge*
*nerators
x1; . .;.xn-1. Define a map G1 ! K by sending the (2i - 1)st generator to xixi+*
*1xi and
the 2ith one to xi+1xixi+1, where n - 1 i 1. For j 2, the map Gj ! K is defi*
*ned
by sending the (2i - 1)st generator to xixi+jxi and the 2ith one to xi+j. The p*
*ush-out of
the diagram
(*n-1j=1Hj) (*n-1j=1Gj) ! K
is then the symmetric group n.
The spaces K(n; 1) have therefore C2-cellular fundamental group, and 2-torsi*
*on
higher homotopy groups. They are however not M(Z=2; 1)-cellular by Theorem 6.2,*
* since
the integral homology of K(n; 1) contains 3-torsion. Actually, we can even com*
*pute
the cellularization of K(3; 1). We know by Theorem 1.1 that it is the fiber of *
*the map
K(3; 1) ! PM(Z=2;2)K(3; 1)0. The later space is simply connected, and is 3-comp*
*lete.
Its mod 3 cohomology is that of K(3; 1), so it is by [9, VII, 4.4] the deloopin*
*g of S3{3},
the fiber of the degree 3 self-map of S3. In other words CWM(Z=2;1)K(3; 1) is *
*a space
whose fundamental group is 3 and whose universal cover is S3{3}, i.e., the univ*
*ersal
cover fibration is
S3{3} ! CWM(Z=2;1)K(3; 1) ! K(3; 1):
The action of 3 on S3{3} is trivial by Theorem 2.9.
In the above example we could identify a certain 3-complete space as the del*
*ooping of
S3{3}. In general it is of course not to expect to find nice and well-known spa*
*ces as the
fiber of the cellularization. However, the same argument as above proves the f*
*ollowing
proposition.
Proposition 6.5 Let ss be a finite Cp-cellular group. The universal cover fib*
*ration of
Theorem 2.9 is then
K(ss; 1)p0^! CWM(Z=p;1)K(ss; 1) ! K(ss; 1)
Y
where Xp0^denotes the completion away from p, i.e., Xp0^= Xq^. *
* 2
q6=p
20
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Departament de Matematiques, Universitat Autonoma de Barcelona,
E-08193 Bellaterra, Spain, e-mail: jlrodri@mat.uab.es
Centre de Recerca Matematica, Institut d'Estudis Catalans,
E-08193 Bellaterra, Spain, e-mail: jerome@crm.es
22