Contemporary Mathematics
Homological localizations preserve 1-connectivity
Carles Casacuberta and Jer^ome Scherer
Abstract.Every generalized homology theory E yields a localization func-
tor LE that sends the E-equivalences to homotopy equivalences. We prove
that if X is any 1-connected space, then LEX is also 1-connected, for ev*
*ery
generalized homology theory E. This is deduced from a result by Hopkins *
*and
Smith stating that if K(Z; 2) is E-acyclic then E is trivial.
Introduction
A number of results in the literature suggest that idempotent functors in the
homotopy category of spaces preserve 1-connectivity, although no proof of this *
*fact
has so far been given. One of the earliest examples is localization with respec*
*t to
ordinary homology, which in fact preserves n-connectivity for all n; see [1].
In the same article [1], Bousfield proved the existence of localization with*
* re-
spect to any generalized homology theory E; that is, a functor LE which assigns
to every space X a space LE X together with a natural map X ! LE X which
is terminal in the homotopy category among E-equivalences with source X. (An
E-equivalence is a map X ! Y inducing isomorphisms En(X) ~=En(Y ) for all n.)
In [9], Mislin showed that K-theory localization does not preserve n-connec-
tivity in general, since for example ss3(LK S2p+2; Z=p) 6= 0 for every odd prim*
*e p.
However, Mislin also proved in [9] that the K-localization of every 1-connected
space is 1-connected. Further evidence of the fact that 1-connectivity could be
preserved by arbitrary idempotent functors in the homotopy category was given by
Neisendorfer in [10] and by Tai in his detailed study of the problem in [11].
It is therefore natural to address the question of whether or not localizati*
*ons
with respect to generalized homology theories preserve 1-connectivity. Such lo-
calizations were thoroughly discussed by Bousfield in [2], where a description *
*was
given of their effect on abelian Eilenberg-Mac Lane spaces. The main tool was an
arithmetic square, already exploited by Mislin in [9], allowing one to determin*
*e the
E-localization of a space (with some restrictions on the fundamental group) from
its EZ=p-localizations and rational coherence data.
____________
2000 Mathematics Subject Classification. Primary 55P60, 55N20; Secondary 20*
*K21.
The first-named author was supported by DGES grant PB97-0202.
The second-named author was supported by the Centre de Recerca Matematica, *
*Barcelona,
and Swiss NSF grant 81LA-51213.
Oc0000 (copyright holder)
1
2 CARLES CASACUBERTA AND JER^OME SCHERER
Our main result is that LE X is 1-connected if X is 1-connected, for any gen-
eralized homology theory E. This follows by combining the methods of Bousfield
in [2] with a result proved by Hopkins and Smith in [8], according to which a
K(Z; 2) is never E-acyclic if E is nontrivial. We note, however, that K(Z; 3) *
*is
KZ=p-acyclic for all p, by [9, Corollary 2.3]. It is known that, if L is any ho*
*motopy
idempotent functor, then LK(Z; n) is necessarily a K(A; n) where A is either ze*
*ro
or a commutative ring with 1, for all n; see [5]. If L = LE for some nontrivial
homology theory E, then the possibility that A = 0 has been discarded for n = 2
in [8], and this opens the way to substantial improvements of earlier results o*
*r to
new results as in this article.
Acknowledgements The plausibility of the main result in this article was commu-
nicated to us by A. K. Bousfield, to whom we are indebted. We first learned a
proof of the non-acyclicity of K(Z; 2), due to M. J. Hopkins and J. H. Smith, f*
*rom
a very helpful letter written by E. Devinatz [6]. We also thank W. Chacholski f*
*or
informing us and for several conversations on this subject.
1. Torsion homology theories
Throughout the paper we denote by E a spectrum or the associated homology
theory. For an abelian group R, the corresponding spectrum with coefficients in*
* R is
defined as ER = E ^ SR where SR is the Moore spectrum of type (R; 0). The only
cases of interest in this article are R = Z=p and R a subring of Q. A spectrum E
is called torsion if EQ is contractible. The ordinary Eilenberg-Mac Lane spectr*
*um
with coefficientsSin R is denoted by HR. We denote by Z^pthe p-adics, by Z(p1 )*
* the
Pr"ufer group 1n=1Z=pn and, for a set of primes P , we denote by ZP the integ*
*ers
localized at P .
In this first section we concentrate on mod p homology theories, where p is *
*any
prime. Using the Atiyah-Hirzebruch spectral sequence, one sees that if E is any
homology theory, then every HZ=p-equivalence is an EZ=p-equivalence; details are
given in [9, x 1]. Hence, all EZ=p-local spaces are HZ=p-local and there is a n*
*atural
transformation of functors : LHZ=p ! LEZ=p.
We next prove that, if X is connected, then the induced homomorphism
*: ss1(LHZ=pX) ! ss1(LEZ=pX)
is surjective. This result is essentially contained in the proof of Propositio*
*n 7.1
in [2], as we next recall for the sake of completeness. The argument is based on
Bousfield's version of the Whitehead theorem (cf. [2, Theorem 5.2]), stating th*
*at
if R is Z=p or a subring of Q, and f :X ! Y is a map inducing isomorphisms
Hi(X; R) ~=Hi(Y ; R) for i < n and an epimorphism Hn(X; R) i Hn(Y ; R), where
n 1, then f also induces isomorphisms ssi(LHR X) ~=ssi(LHR Y ) for i < n and an
epimorphism ssn(LHR X) i ssn(LHR Y ).
Theorem 1.1. Let E be any homology theory and p any prime. Then, for every
connected space X, the natural homomorphism *: ss1(LHZ=pX) ! ss1(LEZ=pX) is
surjective.
Proof. The claim is obvious if EZ=p is trivial. If EZ=p is not trivial, th*
*en
K(Z=p; 1) is not EZ=p-acyclic, as shown in [2, Proposition 2.2]. Since the natu*
*ral
map : LHZ=pX ! LEZ=pX is an EZ=p-equivalence, we obtain an isomorphism
(1.1) *: H1(LHZ=pX; Z=p) ~=H1(LEZ=pX; Z=p)
HOMOLOGICAL LOCALIZATIONS PRESERVE 1-CONNECTIVITY 3
using [2, Proposition 2.1] or [5, Theorem 1.3], according to which K(Z=p; 1) is
EZ=p-local. By the generalized Whitehead theorem stated above, induces then_
an epimorphism ss1(LHZ=pX) i ss1(LEZ=pX), since LEZ=pX is HZ=p-local. |__|
Corollary 1.2. If E is any torsion homology theory and X is 1-connected,
then LE X is also 1-connected.
Proof. As in [2], we denote by PE the set of primes p such that ss*(E) is n*
*ot
uniquely p-divisible. By [2, Proposition 7.1], for each torsion homology theory*
* E
and every 1-connected space X, we have a homotopy equivalence
Y
LE X ' LEZ=pX:
p2PE
Now recall from [1] that LHZ=pX is 1-connected if X is 1-connected. Therefore,_
Theorem 1.1 tells us that LE X is 1-connected. |__|
Before discussing non-torsion homology theories, we need to study the second
homotopy group ss2(LEZ=pX) when X is 1-connected. The following result is the
main input in our discussion.
Theorem 1.3. Let E be a homology theory and p any prime. Suppose that
EZ=p is nontrivial. Then either K(Z=p; 2) or K(Z^p; 2) is EZ=p-local.
Proof. The classification of acyclicity patterns for Eilenberg-Mac Lane spa*
*ces
given by Bousfield in [2, x 4] implies that LEZ=pK(Z; n) = K(A; n) for each n *
*1,
where the group A can be Z^p, or Z=pi for some i 1, or zero. In [8], it is sho*
*wn
that if a reduced homology theory vanishes on K(Z; 2), then it is trivial. (Thu*
*s,
nontrivial mod p homology theories of type IV-1 as defined in [2, x 4] do not e*
*xist.)
Therefore, if LEZ=p is nontrivial, then the localization LEZ=pK(Z; 2) is necess*
*arily
K(Z^p; 2) or K(Z=pi; 2) for some i 1. In the latter case, K(Z=p; 2) cannot be
EZ=p-acyclic, as one sees by induction using the fibre sequences
K(Z=p; 2) ! K(Z=pi; 2) ! K(Z=pi-1; 2):
Hence, K(Z=p; 2) is EZ=p-local, by [2, Proposition 2.1] or [5, Lemma 1.4]. *
*|___|
If K(Z=p; 2) is EZ=p-local and X is 1-connected, then, using the fact that
: LHZ=pX ! LEZ=pX is an EZ=p-equivalence, we obtain as in (1.1) an isomor-
phism
(1.2) *: H2(LHZ=pX; Z=p) ~=H2(LEZ=pX; Z=p):
Thus, the homomorphism ss2(LHZ=pX) ! ss2(LEZ=pX) induced by is surjective,
by the generalized Whitehead theorem.
Now suppose that K(Z^p; 2) is EZ=p-local and X is 1-connected. Similarly as
in the previous case, since is an EZ=p-equivalence, we have an isomorphism
(1.3) *: Hom (ss2(LEZ=pX); Z^p) ~=Hom (ss2(LHZ=pX); Z^p):
In order to use this information, we recall the following concept from [4, V*
*I.3]
and [7]. An abelian group A is called Ext-p-complete if the natural homomorphism
A ! Ext(Z(p1 ); A) derived from the short exact sequence
0 ! Z ! Z[1=p] ! Z(p1 ) ! 0
is an isomorphism. Equivalently, an abelian group A is Ext-p-complete if and on*
*ly
if both Hom (Z[1=p]; A) = 0 and Ext(Z[1=p]; A) = 0. As explained in [4, VI.4],
4 CARLES CASACUBERTA AND JER^OME SCHERER
Ext-p-complete abelian groups are uniquely q-divisible for primes q 6= p, and t*
*hey
admit a canonical Z^p-module structure.
An Ext-p-complete abelian group A is called adjusted if the quotient A=T A
of A by its torsion subgroup T A is p-divisible (hence divisible). Thus, A is a*
*djusted
if and only if A does not admit any torsion-free Ext-p-complete quotients other
than zero. Since T A Z(p1 ) = 0, it also follows that an Ext-p-complete abelian
group A is adjusted if and only if A Z(p1 ) = 0.
Theorem 1.4. Let E be a homology theory and p a prime. Suppose that EZ=p
is nontrivial. Then, for every 1-connected space X, the cokernel of the natural
homomorphism *: ss2(LHZ=pX) ! ss2(LEZ=pX) is an adjusted Ext-p-complete
abelian group, which is zero if K(Z=p; 2) is EZ=p-local.
Proof. The spaces LHZ=pX and LEZ=pX are HZ=p-local. The abelian groups
ss2(LHZ=pX) and ss2(LEZ=pX) are thus Ext-p-complete, by [1, Theorem 5.5]. Hence,
Coker* is Ext-p-complete, since the cokernel of any homomorphism between
Ext-p-completeabelian groups is Ext-p-complete. If K(Z=p; 2) is EZ=p-local, then
we already proved, by means of (1.2), that Coker* is zero. Thus, we assume that
K(Z^p; 2) is EZ=p-local. In this case, the isomorphism displayed in (1.3) shows*
* that
Hom (Coker *; Z^p) = 0. For an abelian group A, if Hom (A; Z^p) = 0 then we have
Hom (AZ(p1 ); Z(p1 )) = 0 by adjunction. Since AZ(p1 ) is a p-torsion divisible
abelian group, we may infer that A Z(p1 ) = 0 and this implies that A=T A is
p-divisible, as we needed. (In fact, an Ext-p-complete abelian group A is adjus*
*ted
if and only if the condition Hom (A; Z^p) = 0 holds. This has also been pointed*
*_out
in [3, Lemma 7.7].) |__|
2. Non-torsion homology theories
In this section we deal with non-torsion homology theories. In this case, th*
*ere is
an arithmetic square allowing one to compute E-localizations of 1-connected spa*
*ces
by combining mod p data and rational data. Specifically, the following diagram *
*is
a homotopy pull-back square if X is 1-connected (and also under less restrictive
conditions; see [2, Proposition 7.2]). Recall that PE denotes the set of primes*
* p
such that ss*(E) is not uniquely p-divisible.
Y
LE X _______________//_ LEZ=pX
| p2PE
| |
| |
| fflffl|
| 0 1
fflffl| Y
LHQ X ___________//LHQ @ LEZ=pXA :
p2PE
We also need the following remark.
Lemma 2.1. Suppose given a set of primes P and an adjusted Ext-p-completeij
Q Q
abelian group Ap for all p 2 P . The rationalization p2PAp ! p2PAp Q
is then an epimorphism.
HOMOLOGICAL LOCALIZATIONS PRESERVE 1-CONNECTIVITY 5
Proof. Fixiany prime q 2 P . Then we have Aq Z(q1 ) = 0 since Aq is
Q j Q
adjusted, and p6=qApZ(q1 ) = 0 as well, since p6=qAp is uniquely q-divisib*
*le.
iQ j iQ j
Therefore, p2PAp Z(q1 ) = 0. This shows that p2PAp Q=Z = 0, which
proves our claim. |___|
Our main result is the following.
Theorem 2.2. Let E be any homology theory and let X be 1-connected. Then
LE X is also 1-connected.
Proof. By Corollary 1.2, we may assume that E is not torsion. Our strategy
is to compare the arithmetic squares for E and ordinary homology HZPE . The
natural maps : LHZ=pX ! LEZ=pX yield a commutative diagram
Y ________________//F_0____________________//F
| | |
| | |
| Y fflffl| Y fflffl|
fflffl| L X L X
F 00__________//_ HZ=p ____________//_ EZ=p
| p2PE p2PE
| | |
| | |
| fflffl| fflffl|
| 0 1 0 1
fflffl| Y Y
LHQ F 00___//LHQ @ LHZ=pXA _____//LHQ @ LEZ=pXA
p2PE p2PE
where each row and each column is a fibre sequence. The four spaces in the lower
right square are 1-connected by Corollary 1.2. Therefore, all the fibres except*
* per-
haps Y are connected. The group ss1(F 00) is the product of the cokernels of t*
*he
homomorphisms *: ss2(LHZ=pX) ! ss2(LEZ=pX), so it is a product of adjusted
Ext-p-complete groups, by Theorem 1.4. Hence, Lemma 2.1 tells us that the in-
duced homomorphism ss1(F 00) ! ss1(LHQ F 00) is surjective. This implies that Y*
* is
connected as well, so the homomorphism ss1(F 0) ! ss1(F ) is surjective.
From the arithmetic square for E we see that LE X is 1-connected if and only
if the boundary homomorphism ss2(LHQ X) ! ss1(F ) is surjective. Consider now
the fibre sequence F 0! LHZPE X ! LHQ X appearing in the arithmetic square
for HZPE . Since we know that LHZPE X is 1-connected, the homomorphism
ss2(LHQ X) ! ss1(F 0) is surjective. The composite ss2(LHQ X) ! ss1(F 0) ! ss1(*
*F_)
is thus also surjective, as we needed. |__|
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6 CARLES CASACUBERTA AND JER^OME SCHERER
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Departament de Matematiques, Universitat Autonoma de Barcelona,
E-08193 Bellaterra, Spain
E-mail address: casac@mat.uab.es
Institut de Mathematiques, Universite de Lausanne,
CH-1015 Lausanne, Switzerland
E-mail address: jerome.scherer@ima.unil.ch