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Decompositions of Certain Loop Spaces View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Pure and Applied Algebra 185 (2003) 289–304 www.elsevier.com/locate/jpaa Decompositions of certain loop spaces Manfred Stelzer Sesenheimer Stra e 20, 10627 Berlin, Germany Received 14 May 2001; received in revised form 7 March 2003 Communicated by J. Huebschmann Abstract In this note we will study, for a space taken from two di3erent classes of spaces, the decom- position of the loop space on such a space into atomic factors. The ÿrst class consists of certain mapping cones of maps between wedge products of Moore spaces. The second one is a p-local version of the class of coformal spaces from rational homotopy theory. As an application we verify Moore’s conjecture about homotopy exponents for these spaces. c 2003 Elsevier B.V. All rights reserved. MSC: Primary: 55P35; secondary: 55Q52 1. Introduction and statement of results In homotopy theory decompositions of loop spaces into atomic factors proved to be important. The theorem of Hilton [21] is a famous example of such a result. It gives a decomposition of the loop space of a ÿnite wedge product of spheres. In [6] Anick proved a secondary p-local version. Theorem 1 (Anick). Let W0 and W1 be ÿnite wedge products of simply connected 1 p-local spheres and f : W1 → W0 a map with mapping cone Cf. Let p¿max (3; 2 (dimCf − 1)) and suppose that H∗(Cf; Z(p)) is torsion free. Then Cf is of the homotopy type of a weak product having as factors p-local spheres of odd dimension and loop spaces of such spheres. E-mail address: [email protected] (M. Stelzer). 0022-4049/03/$ - see front matter c 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0022-4049(03)00106-3 290 M. Stelzer / Journal of Pure and Applied Algebra 185 (2003) 289–304 A p-local space is called atomic if the ÿrst non-vanishing homotopy group of X is cyclic and if any self-map of X which induces an isomorphism on this homotopy group is a homotopy equivalence. Our ÿrst result is a torsion version of Theorem 1.To state it we recall the following facts about the atomic factors of the loops on a Moore space Pm(pn), of type pn with top cell in dimension m ¿ 3 where p ¿ 3, see [14]. 2‘+1 n n 2‘+1 For ‘; n ∈ N+ let S {p } be the homotopy ÿber of the degree p map on S . Moreover, let T 2‘+1{pn} denote the Cohen–Moore–Neisendorfer factor of P2‘+1(pn). These spaces are atomic except for T 3{p}T 2p+1{p}×U with U atomic. The loop space of Pm(pn) decomposes into a weak product with factors of these types. Let 2‘+1 n 2‘+1 n Tn = {S {p };T {p }|‘ ¿ 1} if n¿1 and add U to this set if n = 1. We write T for the union of all the Tn. 2‘+1 2‘+1 2‘+1 Furthermore, let S = {S(p) ;S(p) | ‘ ¿ 1} where S(p) is the p-local sphere. Theorem 2. Let r ¿ 1 and p ¿ 5. Let W0 and W1 be ÿnite wedge products of r- n connected Moore spaces of type p and let f : W1 → W0 be a map with mapping ˜ cone Cf. Let dimCf 6 N =min(r +2p−3;rp−1) and suppose that H ∗(Cf; Z(p)) is n a free Z=p Z-module. Then Cf is of the homotopy type of a weak product having as factors spaces in Tn. Remark 1. The loop space of Cf where f is the Whitehead product of id and the Bockstein of id on P4n(p) has a decomposition in which a more complicated factor shows up [9]. So a condition on H∗(Cf; Z(p)) is necessary in order for the conclusion of the theorem to hold. The next class of spaces whose loop space decompositions will be considered is quadraticin another way. This classis a p-local version of the class of coformal spaces in rational homotopy theory [26]. To deÿne it we recall another result of Anick. Let m R ⊆ Q be a subring. Denote by CWr (R) the category of R-local CW -complexes of dimension 6 m which are (r +1)-reduced, i.e. which have trivial r-skeleton. Moreover, m−1 let Lr−1 (R) denote the category of di3erential graded Lie algebras over R which are free as graded Lie algebras on a free graded module V . This module is concentrated in dimensions s for r 6 s 6 m − 1. By the main result of [5], if the smallest prime p not invertible in R satisÿes m 6 min (r +2p − 3;rp− 1), there is an equivalence m m−1 between the homotopy category of CWr (R) and that of Lr−1 (R). We will apply this for R = Z(p), the localization of Z at p. m−1 Deÿnition 1. An object (LV; @)ofLr−1 (Z(p)) is called Z=pZ-coformal if there is a homomorphism of dg Lie algebras ’ :(LV ⊗ Z=pZ;@⊗ 1) → (H∗(LV ; Z=pZ);@=0) which induces an isomorphism on homology. m Deÿnition 2. A space X in CWr (Z(p)) with m 6 N is called Z=p-coformal if a Lie m−1 model from Lr−1 (Z(p)) representing X in homotopy is Z=pZ-coformal. M. Stelzer / Journal of Pure and Applied Algebra 185 (2003) 289–304 291 Remark 2. Using the homotopy theory of simplicial spaces in a form initiated by Dwyer et al. [16,17], one can give a more geometric interpretation of the class of coformal spaces. I hope to come back to this point in a later paper. m Theorem 3. Let X in CWr (Z(p)) be Z=pZ-coformal of ÿnite type. Suppose that ˜ H ∗(X ; Z(p)) is torsion free. Then X is of the homotopy type of a weak prod- uct having as factors spaces in S. m Theorem 4. Let X in CWr (Z(p)) be Z=pZ-coformal of ÿnite type. Suppose that ˜ n H ∗(X ; Z(p)) is a free Z=p Z-module. Then X is of the homotopy type of a weak product having as factors spaces in Tn. A space X is said to have a homotopy exponent at p if, for some n, pn annihilates the p-torsion in ∗(X ). According to a conjecture of Moore, a ÿnite 1-connected complex X admits an exponent at p if and only if dim ∗(X ) ⊗ Q ¡ ∞. Corollary 1. Let X be as in Theorems 1–4. Then the Moore conjecture is true for X . Proof. This follows from the known homotopy exponents of the spaces in S ∪ T [13,14,25]. Remark 3. It has been conjectured by Anick that the loop space of any 1-connected ÿnite complex decomposes into a weak product with factors in S ∪ T at almost all primes. An example in [4] shows that the conjecture is false if one uses only S.In view of a result in [22], the conjecture is true for spaces with dim ∗(X ) ⊗ Q ¡ ∞. The present paper is motivated by Anick’s question. As has been noted above, one has to make simplifying assumptions on the Bockstein spectral sequence to avoid the factors in [9]. We will deal with them in a later paper. The study of two cones is the second step, if one tries to attack Anick’s question by induction on the cone-length. Coformal spaces are related to another inductive scheme. Their Z=pZ-minimal model, as constructed in [19], is purely quadratic. Note that every Z=pZ-minimal model of a m space in CWr (Z(p)) can be seen as a deformation of a quadratic model (see Section 4). This paper is organized as follows: In Section 2 we prove Theorem 2 up to certain technical results on Adams–Hilton models which are given in Section 3. Section 4 is devoted to the proof of Theorems 3 and 4. Examples are studied in Section 5. 2. Proof of Theorem 2 r r Denote by EH and E the homology and homotopy Bockstein spectral sequences, respectively, at p [22]. By Anick [8] (Theorem 9), Theorem 2 is proved if we can r show that, for each r,asaZ=pZ-algebra, EH (Cf) is generated by r r r im(h : E(Cf) → EH (Cf)); where hr is induced by the mod p Hurewicz homomorphism. Next, we recall what is known about the homology of Cf. A model for Cf in m−1 Lr−1 (Z(p)) can be chosen to be of the form (L(V0 ⊕ V1);@)=(LV; @) in the pushout 292 M. Stelzer / Journal of Pure and Applied Algebra 185 (2003) 289–304 diagram + −1 i + −1 s V1 (s V1 V1) Lf (∗) + + V0 (V0 V1) −1 where LV0 and Ls V1 represent W0 and W1, respectively, Lf is induced up to homo- # −1 topy by f , the loops on the adjoint of f, and the complex s V1 ⊕V1 is acyclic. The universal enveloping algebra ULV is a valid Adams–Hilton model [1] for Cf. More- over, the Hopf algebra ULV with coproduct determined by setting V to be primitive is a model for Cf in the sense of [7] where model means model as a Hopf algebra up to homotopy. A theorem proved by Halperin [19] shows that the homology of ULV with coeNcients in Z=pZ is isomorphicas a Hopf algebra to an enveloping algebra UL of a Lie algebra L.
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