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. |__| References [1]A. K. Bousfield, The localization of spaces with respect to homology, Topol* *ogy 14 (1975), 133-150. [2]A. K. Bousfield, On homology equivalences and homological localizations of * *spaces, Amer. J. Math. 104 (1982), 1025-1042. [3]A. K. Bousfield, On the telescopic homotopy theory of spaces, Trans. Amer. * *Math. Soc. (to appear). [4]A. K. Bousfield and D. M. Kan, Homotopy Limits, Completions and Localizatio* *ns, Lecture Notes in Math. vol. 304, Springer-Verlag, Berlin Heidelberg New York, 1972. [5]C. Casacuberta, J. L. Rodriguez, and J.-Y. Tai, Localization of abelian Eil* *enberg-Mac Lane spaces of finite type, preprint, 1998. [6]E. Devinatz, Hopkins' proof that 1 CP1 is Bousfield equivalent to S0, lett* *er, 1999. 6 CARLES CASACUBERTA AND JER^OME SCHERER [7]D. K. Harrison, Infinite abelian groups and homological methods, Ann. of Ma* *th. 69 (1959), 366-391. [8]M. J. Hopkins and J. H. Smith, CP1 is Bousfield equivalent to the sphere, * *preprint, 1999. [9]G. Mislin, Localization with respect to K-theory, J. Pure Appl. Algebra 10 * *(1977), 201-213. [10]J. Neisendorfer, Localization and connected covers of finite complexes, in:* * The Cech Centen- nial; A Conference on Homotopy Theory (Boston, 1993), Contemp. Math. vol. 18* *1, Amer. Math. Soc., Providence, 1995, pp. 385-389. [11]J.-Y. Tai, On f-localization functors and connectivity, in: Stable and Uns* *table Homo- topy (Toronto, 1996), Fields Inst. Commun. vol. 19, Amer. Math. Soc., Provid* *ence, 1998, pp. 285-298. 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