Homology equivalences inducing an epimorphism on
the fundamental group
Jose L. Rodriguez and Dirk Scevenels
Abstract
Quillen's plus construction is a topological construction that kills the m*
*aximal per-
fect subgroup of the fundamental group of a space without changing the int*
*egral
homology of the space. In this paper we show that there is a topological c*
*onstruc-
tion that, while leaving the integral homology of a space unaltered, kills*
* even the
intersection of the transfinite lower central series of its fundamental gr*
*oup. More-
over, we show that this is the maximal subgroup that can be factored out o*
*f the
fundamental group without changing the integral homology of a space.
0 Introduction
As explained in [8], [9], Bousfield's HZ-localization XHZ of a space X ([2]) is*
* homotopy
equivalent to its localization with respect to a map Bf : BF1 ! BF2 induced by *
*a certain
homomorphism f : F1 ! F2 between free groups. This means that a space X is HZ-l*
*ocal
if and only if the induced map Bf*: map (BF2; X) ! map (BF1; X) is a weak homot*
*opy
equivalence. Moreover, the effect of Bf-localization on the fundamental group p*
*roduces
precisely the group-theoretical HZ-localization (i.e. f-localization) of the f*
*undamental
group, i.e. ss1LBfX ~=Lf(ss1X) ~=(ss1X)HZ for all spaces X.
A universal acyclic space for HZ-localization (i.e. Bf-localization), in th*
*e sense of
Bousfield ([4]), was studied by Berrick and Casacuberta in [1]. They show that*
* nullifi-
cation with respect to such a universal acyclic space coincides with Quillen's *
*plus con-
struction X+ for a space X. Moreover, this universal acyclic space can be take*
*n to be
a two-dimensional Eilenberg-Mac Lane space K(A(f); 1), where A(f) is a locally *
*free,
universal f-acyclic group, in the sense of [12]. The effect of K(A(f); 1)-null*
*ification on
1
the fundamental group is precisely given by its A(f)-nullification. This A(f)-n*
*ullification
factors out the perfect radical of a group, i.e. the maximal perfect subgroup, *
*which can
be obtained as the intersection of the (transfinite) derived series of the grou*
*p ([7]).
The intersection G of the transfinite lower central series of a group G is a*
*lso a
radical, and, as observed by Bousfield ([3]), it is in fact the maximal G-perfe*
*ct normal
subgroup of G (where a normal subgroup H of G is called G-perfect if H ~=[H; G]*
*). As
explained in [7], there is an epireflection that corresponds to this radical G.*
* By adapting
the methods used by Berrick and Casacuberta in [1], we describe in Theorem 2.6 *
*an
epimorphism g such that LgG ~=G=G for all groups G.
We further construct a localization functor of topological spaces that is in*
*termediate
between Quillen's plus construction and Bousfield's homological localization, a*
*nd induces
localization with respect to g on the fundamental group. More precisely, we des*
*cribe in
Theorem 2.10 a map ' such that '-localization of spaces factors out the interse*
*ction of
the transfinite lower central series of the fundamental group, while preserving*
* the integral
homology of the space.
We furthermore show that this is in some sense the best possible result. In*
*deed,
we show in Proposition 3.2 that the maximal subgroup that can be factored out o*
*f the
fundamental group without altering the integral homology of a space, is precise*
*ly the
intersection of the transfinite lower central series of the fundamental group.
Finally, we turn our attention to the question whether there exists a locali*
*zation
functor of spaces which induces localization with respect to a universal epimor*
*phic HZ-
equivalence of groups (in the sense of [12]) on the fundamental group, without *
*changing
the integral homology of a space. In fact, we show in Proposition 3.3 that the *
*answer to
this question is affirmative if and only if, for all groups G, the kernel of th*
*e HZ-localization
homomorphism G ! GHZ coincides with G.
1 Preliminaries
For the convenience of the reader, we recall some terminology and bacis facts a*
*bout
localization with respect to a given continuous map (see e.g. [4], [8]). Give*
*n a map
f : A ! B, a space X is called f-local if the induced map
f*: map (B; X) ! map (A; X)
2
is a weak homotopy equivalence. For every space X there is a map lX : X ! LfX, *
*which
is initial among all maps from X into f-local spaces. Lf is called the localiza*
*tion functor
with respect to f. A map OE is called an f-equivalence if LfOE is a homotopy eq*
*uivalence.
Further a space X is called f-acyclic if LfX is contractible. In the special ca*
*se where f
is of the form f : A ! *, the f-localization of a space X is also denoted by PA*
*X and it
is called the A-nullification of X. The localization class of f, denoted by *
*, is defined
as the collection of all maps g such that Lg is naturally equivalent to Lf. Whe*
*n f is of
the form f : A ! *, the class of f is simply denoted by and is called the n*
*ullification
class of A. One says that if and only if there is a natural transform*
*ation of
localization functors Lf ! Lg. Recall that this is equivalent to every g-local *
*space being
f-local, to every f-equivalence being a g-equivalence, or to f being a g-equiva*
*lence. As
shown by Bousfield in [4], the collection of localization functors with respect*
* to maps is a
small-complete lattice for this partial order relation. Note that a space X is *
*f-acyclic if
and only if . Furthermore, in [4] Bousfield proved that every localizat*
*ion class
has a best possible approximation by a nullification class. More precisely, for*
* any map f,
there is a maximal nullification class which is smaller than in the*
* lattice of
localization classes. The space A(f) is called a universal f-acyclic space, sin*
*ce a space X
is f-acyclic if and only if X is A(f)-acyclic.
An important algebraic tool in studying localization with respect to a given*
* map is
given by its discrete analogue in the category of groups (see e.g. [4], [5]). *
* For a given
group homomorphism f : A ! B, a group G is called f-local if the induced map of*
* sets
f*: Hom (B; G) ! Hom (A; G)
is a bijection. For every group G there is a homomorphism lG : G ! LfG, which i*
*s initial
among all homomorphisms from G into f-local groups. This allows to introduce t*
*he
localization functor Lf with respect to f. In an obvious way, one speaks of f-e*
*quivalences
and f-acyclic groups, and, if f is of the form f : A ! 1, of A-nullification. *
*It makes
also sense to speak of the localization class of a homomorphism f, denoted by <*
*f>, which
in the special case when f is of the form f : A ! 1, is simply denoted by a*
*nd is
called the nullification class of A. Furthermore, the collection of all locali*
*zation classes
of homomorphisms is again a small-complete lattice for an obvious partial order*
* relation.
In [12] it was proved that every localization class of a homomorphism has a bes*
*t possible
approximation by a nullification class and by the localization class of an epim*
*orphism.
3
More precisely, for any homomorphism f, there is a maximal nullification class *
*
and a maximal class where E(f) is an epimorphism such that
:
The group A(f) is called a universal f-acyclic group, since a group G is f-acyc*
*lic if and
only if G is A(f)-acyclic. Accordingly, E(f) is called a universal epimorphic f*
*-equivalence,
since an epimorphism g is an f-equivalence if and only if g is an E(f)-equivale*
*nce.
It was shown in [12] (see also [7]) that to any localization class of an*
* epimorphism f,
there is associated a radical Rf on the category of groups such that LfG ~= G=R*
*fG
for all groups G. (Recall that a radical R is a functor assignating to every g*
*roup G
a normal subgroup RG in such a way that every homomorphism G ! K restricts to
RG ! RK and such that R(G=RG) = 1.) In fact, there is a bijective corresponden*
*ce
between epireflections (i.e. idempotent functors L on the category of groups f*
*or which
G ! LG is an epimorphism for any group G) and radicals. Furthermore, if the cla*
*ss
is actually a nullification class, then the associated radical Rf is idempotent*
*, meaning
that RfRfG = RfG for all groups G.
2 A universal epimorphic HZ-map
It is well known that Bousfield's HZ-localization XHZ of a space X ([2]) is ho*
*motopy
equivalent to its localization with respect to a map Bf : BF1 ! BF2 induced by *
*a certain
homomorphism f : F1 ! F2 between free groups ([8], [9]). In fact, the homomorph*
*ism f
can be taken to be the free product of a set of representatives of isomorphism *
*classes
of homomorphisms between countable, free groups inducing an isomorphism on the *
*first
integral homology group. Furthermore, the effect of Bf-localization on the fund*
*amental
group is to produce its f-localization, which coincides with the group-theoreti*
*cal HZ-
localization of the fundamental group. In [1, Proposition 4.2] the authors sho*
*w that a
universal f-acyclic group A(f) (in the sense of [12]) can be taken to be the fr*
*ee product
of a set of representatives of all isomorphism classes of countable, locally fr*
*ee, perfect
groups. The key lemma here is a result due to Heller ([10]), which states that *
*for every
element x in any perfect group P , there exists a countable, locally free, perf*
*ect group D
and a homomorphism D ! P containing x in its image. We first prove a "relative"*
* version
of [10, Lemma 5.7]. Recall that a normal subgroup H of a group G is called G-pe*
*rfect if
4
H ~=[H; G].
Lemma 2.1 Let A be any group with an A-perfect normal subgroup K. Let F be*
* a
countable free group and let F 0be a subgroup. Let : F ! A be a homomorphism
such that the restriction |F 0is a homomorphism into K. Then there exists a co*
*untable,
locally free group D with a D-perfect normal subgroup D0 such that can be fac*
*torized
as F ! D ! A and moreover this factorization restricts to a factorization of |*
*F 0as
F 0! D0! K.
Proof. Choose free generators xi; yj for F such that the generators xi freely *
*gener-
ate F 0. For each generator xi of F 0, we can find finitely many elements ai;`*
*of A and
Q
ki;`of K, such that (xi) = `[ai;`; ki;`]. Let F1 be the free group generate*
*d by the
set {x(i; `; 1); x(i; `; 2); y(j)}, and let F10be the free subgroup generated b*
*y the subset
{x(i; `; 1); x(i; `; 2)}. Define a homomorphism 1: F1 ! A, by setting 1(x(i; *
*`; 1)) = ai;`,
1(x(i; `; 2)) = ki;`and 1(y(j)) = (yj). Define a homomorphism ' : F ! F1 by
Q
'(xi) = `[x(i; `; 1); x(i; `; 2)] and '(yj) = y(j). Then 1O ' = and the im*
*age of 1 re-
stricted to F10is contained in K. Iterating this construction by similarly fact*
*orizing 1, we
arrive at a sequence F ! F1 ! F2 ! ::: and a sequence of subgroups F 0! F10! F2*
*0! :::,
whose colimits are the required D, resp. D0. 2
This immediately implies the following result.
Lemma 2.2 Let A be any group with an A-perfect normal subgroup K. Let x be *
*any
element in K. Then there exist a countable, locally free group D with a D-perfe*
*ct normal
subgroup D0 and a homomorphism : D ! A such that x belongs to the image of the
restriction |D0. 2
Recall ([3]) that any group G has a maximal G-perfect subgroup, which we den*
*ote
by G, and that can be obtained as the intersection of the transfinite lower cen*
*tral series
of G. Lemma 2.2 enables us to prove the following result.
Proposition 2.3 Let G be any group. Then the following assertions are equivale*
*nt:
(i)For every group A and every A-perfect normal subgroup K of A, the restrict*
*ion of
any homomorphism A ! G to K is trivial;
(ii)For every countable, locally free group A and every A-perfect normal subg*
*roup K
of A, the restriction of any homomorphism A ! G to K is trivial;
5
(iii)For every countable, locally free group A, the restriction of any homomor*
*phism
A ! G to A is trivial. 2
The subgroup G actually defines a radical on the category of groups, and, he*
*nce,
by [7], the assignation G ! LG = G=G is an epireflection. Our aim is to show th*
*at this
epireflection is singly generated. More precisely, we want to exhibit a homomor*
*phism g
such that LgG ~= LG = G=G for any group G. Moreover, by [12] we know that it is
possible to choose g to be an epimorphism.
Let be a set of representatives of isomorphism classes of countable, locall*
*y free
S
groups. If D0denotes the normal closure of D in the free product D of all gr*
*oups D
in , which we denote by Fr D, then we have a short exact sequence (cf. [11, Exe*
*r-
cise 6.2.5])
D0ae D = Fr D i D=D0~= Fr (D=D); (2.*
*1)
where Fr (D=D) denotes the free product of all groups D=D for which (the isomor-
phism class of) D belongs to .
Proposition 2.4 Localization with respect to the epimorphism
h: D i D=D0
given in (2.1) satisfies LhG ~=G=G for all groups G.
Proof. Observe that a group G is f-local for any given epimorphism f : A i B if*
* and
only if the restriction of any homomorphism A ! G to the kernel of f is trivial*
*. The
proof is now completed by using Proposition 2.3. 2
Note that we can partition into 1, containing all the representatives of th*
*e isomor-
phism classes of countable, locally free, perfect groups, and its complement c1*
*. We then
can write
= *
= *
= * ;
where F is the universal HZ-acyclic group defined in [1]. (Here we denote by *
the least upper bound of the classes and in the lattice of localizati*
*on classes, and
6
we have used the fact that the free product f1 * f2 is a representative of this*
* least upper
bound.)
However, it is possible to give another description of , which will be mo*
*re useful
later on. Indeed, note that, if f1: A i B is an epimorphism, and f2: B ! C is *
*any
homomorphism, then = = * . This enables us to pro*
*ve the
following preliminary result.
Lemma 2.5 Let f : A i B be any epimorphism. Then = , where lA : A ! *
*LfA
denotes the localization homomorphism.
Proof. Since f is an epimorphism, we infer from [12, Theorem 2.1] that lA is a*
*n epi-
morphism. Hence, = * = = = * = . 2
The above lemma now proves the following alternative description of .
Theorem 2.6 Consider the natural homomorphism g: D ! D=D, where D is as defin*
*ed
in (2.1). Then = , where h is defined in (2.1). In other words, LgG ~=G=*
*G for
all groups G. 2
In fact, the homomorphism g is a "universal epimorphic HZ-map", as we next s*
*how.
Recall from [3] that a group homomorphism g is called an HZ-map if H1g (i.e. t*
*he
homomorphism induced by g on the first integral homology group) is an isomorphi*
*sm and
H2g is an epimorphism. We first need a characterization of the epimorphisms th*
*at are
HZ-maps (cf. [3]).
Lemma 2.7 Let h be an epimorphism h: A i B with kernel K. Then h is an HZ-map
if and only if K is A-perfect.
Proof. This is an obvious consequence of the 5-term exact sequence
H2(A) ! H2(B) ! K=[K; A] ! H1(A) ! H1(B) ! 0: 2
Proposition 2.8 Let G be any group and let g : D ! D=D, where D is as defined
in (2.1). The homomorphism G ! LgG ~= G=G is terminal among all epimorphic
HZ-maps going out of G. 2
7
Observe that the epimorphism g that we have constructed is not a universal e*
*pi-
morphic HZ-equivalence (in the sense of [12]). Indeed, there are "more" epimor*
*phic
HZ-equivalences than epimorphic HZ-maps. However, if we denote by A(f), resp. E*
*(f)
a universal HZ-acyclic group, resp. a universal epimorphic HZ-equivalence, the*
*n there
are natural homomorphisms
G ! PA(f)G ! LgG ~=G=G ! LE(f)G ! GHZ ;
for any group G, where G ! LE(f)G ! GHZ is an epi-mono factorization. Moreover,*
* for
many groups G (e.g. finite groups, nilpotent groups, or more generally, groups *
*for which
the lower central series stabilizes), we have isomorphisms LgG ~=G=G ~=LE(f)G ~*
*=GHZ
(cf. [3]).
We now want to realize localization with respect to g : D i D=D topologicall*
*y, by
exhibiting a localization functor of spaces that induces g-localization on the *
*fundamental
group and which is intermediate between Quillen's plus construction and Bousfie*
*ld's HZ-
localization (and, hence, does not change the integral homology of a space).
We will need the following proposition, which is similar to results obtained*
* in [5]
and [6].
Proposition 2.9 Let : A ! B be any map which induces an epimorphism * = ss1(*
* ):
ss1(A) i ss1(B) and suppose that A is a CW-complex of dimension at most two. T*
*hen
-localization of spaces is ss1-compatible, i.e.
ss1(L X) ~=L *(ss1X);
for all spaces X.
Proof. For any space X, the map X ! L X is a -equivalence. Hence, by [5, Propo*
*si-
tion 3.3], the induced homomorphism ss1(X) ! ss1(L X) is a *-equivalence. More*
*over,
we claim that ss1(L X) is *-local. To see this, it suffices to prove that the *
*restriction of
any homomorphism `: ss1A ! ss1(L X) to ker * is trivial. However, since the dim*
*ension
of A is at most two, there exists a map : A ! L X inducing ` on the fundamental*
* group.
Since L X is -local, we infer that there exists a map O: B ! L X such that O O*
* ' ,
which implies that the restriction `| ker * = (ss1(O) O *)| ker * is trivial. *
* 2
Now choose a two-dimensional CW-complex MD such that ss1MD = D. Attach 2-
cells to MD, thereby obtaining a map i: MD ,! C which induces g: D i D=D on the
8
fundamental group. We then obtain a diagram
ss2MD ! ss2C
# #
0 ! H2MD ! H2C ! H2(C; MD) ! 0
# #
H2(ss1MD) ! H2(ss1C) ! 0
# #
0 0
Moreover, since H2(ss1C) = H2(D=D) = 0, we infer that H2C is an epimorphic image
of ss2C. This means that we can kill H2C by attaching 3-cells to C, through a *
*map
j: C ! C0. It is now easily verified that the composition ' of
j 0
MD -i! C -! C (2.*
*2)
is an integral homology equivalence and that ' induces the homomorphism g: D i *
*D=D
on the fundamental group (cf. [2, Lemma 6.1]).
Theorem 2.10 Let ': MD ! C0 be the composition given in (2.2). Then '-locali*
*zation
of spaces induces g-localization on the fundamental group. Moreover, for any s*
*pace X,
there are natural maps
X ! X+ ! L'X ! XHZ :
Proof. Since MD is a two-dimensional CW-complex and since ' induces an epimor-
phism on the fundamental group, we know by Proposition 2.9 that '-localization *
*is ss1-
compatible, so that it induces g-localization on the fundamental group. The sec*
*ond claim
is obvious, since ' is clearly an integral homology equivalence, and the perfec*
*t radical of
ss1(L'X) being trivial (cf. [1]). 2
To see that the natural maps given in Theorem 2.10 are not equivalences in g*
*eneral,
and thus that we have constructed a functor which is really different from Quil*
*len's plus
construction and from HZ-localization, observe the following. On one hand, L'(S*
*1_S1) '
S1 _ S1 ' (S1 _ S1)+. Indeed, the fact that Z * Z is g-local implies that S1 _ *
*S1 is '-local.
On the other hand, ss1(L'B3) ~=Lg(ss1B3) ~=Z=2, while ss1B+3~=PA(f)(ss1B3) ~=3.
9
3 Homology equivalences inducing an epimorphism
on the fundamental group
In this section we want to explore some immediate consequences of our results. *
*In partic-
ular, we want to show that there are some restrictions on (integral) homology e*
*quivalences
that induce an epimorphism on the fundamental group.
Proposition 3.1 Let : X ! Y be an integral homology equivalence of spaces s*
*uch
that the induced homomorphism f = ss1 is an epimorphism. Then RfG G for all
groups G, where Rf denotes the radical associated to the epireflection class .
Proof. By hypothesis we know that f is an epimorphic HZ-map. Hence, ,
where g denotes the universal epimorphic HZ-map of Theorem 2.6. Hence, there a*
*re
natural homomorphisms G i LfG ~=G=RfG i LgG ~=G=G, for any group G. 2
In particular, we have the following result.
Proposition 3.2 Let : X ! Y be an integral homology equivalence of spaces suc*
*h that
the induced homomorphism f = ss1 is an epimorphism. Then kerf ss1X.
Proof. Since f is an epimorphic HZ-map, we know that kerf is ss1X-perfect. 2
In other words, for any space X, the maximal subgroup that can be factored o*
*ut of
ss1X without altering the integral homology of X, is precisely ss1X. In particu*
*lar, this
implies that there is a restriction on the possibility of realizing topological*
*ly a universal
epimorphic HZ-equivalence of groups (i.e. of finding an integral homology equiv*
*alence of
spaces which induces localization with respect to a universal epimorphic HZ-equ*
*ivalence
of groups on the fundamental group).
Proposition 3.3 Let E(f) denote a universal epimorphic HZ-equivalence of group*
*s. Then
the following assertions are equivalent:
(i)There exists an integral homology equivalence : X ! Y of spaces such that*
* ss1 =
E(f) and L is ss1-compatible;
(ii)ker(lHZ : G ! GHZ ) = G for all groups G.
Proof. To see that (i) implies (ii), it suffices to show that kerlHZ G. Howeve*
*r, this
is an immediate consequence of Proposition 3.2, since Lss1G ~=LE(f)G ~=G= kerlH*
*Z for
all groups G. Finally (ii) implies (i), as is shown by our construction of ' in*
* (2.2). 2
10
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izations, preprint (1998).
Departament de Matematiques, Universitat Autonoma de Barcelona,
E-08193 Bellaterra, Spain, e-mail: jlrodri@mat.uab.es
Departement Wiskunde, Katholieke Universiteit Leuven,
Celestijnenlaan 200 B, B-3001 Heverlee, Belgium,
e-mail: dirk.scevenels@wis.kuleuven.ac.be
11