On the comparison of stable and unstable p -completion
aa r X i v : . [ m a t h . A T ] D ec ON THE COMPARISON OF STABLE AND UNSTABLE p -COMPLETION TOBIAS BARTHEL AND A. K. BOUSFIELD
Abstract.
In this note we show that a p -complete nilpotent space X has a p -complete sus-pension spectrum if and only if its homotopy groups π ∗ X are bounded p -torsion. In contrast, if π ∗ X is not all bounded p -torsion, we locate uncountable rational vector spaces in the integralhomology and in the stable homotopy groups of X . To prove this, we establish a homologi-cal criterion for p -completeness of connective spectra. Moreover, we illustrate our results bystudying the stable homotopy groups of K ( Z p , n ) via Goodwillie calculus. Introduction
The notion of p -completion plays a fundamental role in algebra and topology, for it provideseffective means to isolate and study p -primary properties. Applied to homotopy theory byBousfield and Kan [BK72] as well as Sullivan [Sul74] and developed further in [Bou75, Bou79], ithas since become one of the standard tools in the hands of algebraic topologists. However, thereappears to be no general account of the comparison between unstable and stable p -completionin the literature, which is the question we address in the present note.Our main goal is to characterize p -complete spaces which have p -complete suspension spectra: Theorem 4.7. If X is a p -complete nilpotent space, then Σ ∞ X is p -complete if and only if π n X is bounded p -torsion for each n . In fact, we exhibit a sharp dichotomy of p -complete nilpotent spaces: if X is a p -completenilpotent space whose homotopy groups are not all bounded p -torsion, then the integral homologygroups and stable homotopy groups of X both contain an uncountable rational vector space. Asa consequence, we deduce that a nilpotent space X with derived p -complete integral homologyand unstable homotopy must have both H n ( X ; Z ) and π n X of bounded p -torsion for all n .In a first step towards the proof of the theorem, we complement the second author’s char-acterization of p -complete spectra in terms of homotopy groups with an integral homologicalcriterion, using a mild generalization of Serre classes appropriate for stable homotopy theory.This is in sharp contrast to the aforementioned fact that the integral homology of p -completespaces is not well-behaved, and thus cannot be used to characterize p -completeness of spaces. Corollary 3.3.
A bounded below spectrum X is p -complete if and only if H ∗ ( X ; Z ) is derived p -complete in each degree. In order to use this result to prove the theorem, we need to detect rational classes in thehomology of p -complete spaces whose homotopy is not bounded p -torsion. This rests on the studyof the integral homology of p -complete spheres. We end this note with a sample computation,illustrating how Goodwillie calculus allows us to detect rational classes in the stable homotopygroups of the Eilenberg–MacLane space K ( Z p , n ). Proposition 5.3.
For n ≥ and k > , the stable homotopy group π nk Σ ∞ K ( Z p , n ) containsan uncountable rational vector space. In particular, Σ ∞ K ( Z p , n ) is not p -complete. In fact, we also give a short alternative argument based on the integral homology of K ( Z p , n ). Date : December 21, 2017.
1N THE COMPARISON OF STABLE AND UNSTABLE p -COMPLETION 2 Conventions.
Throughout this paper, p will be a fixed prime number and Z p denotes the p -adicintegers. We say that a nilpotent group N is bounded p -torsion if there exists an m such thatfor all x ∈ N , we have x p m = 1. A graded nilpotent group N ∗ is said to be of bounded p -torsionif N k is bounded p -torsion for each k ; however, we do not require a uniform bound. Wheneverwe are in a graded context, we indicate the degree of an abelian group A by square brackets, i.e., A [ n ] refers to A placed in degree n . If X is a topological space, then H ∗ ( X ; A ) is the reducedhomology of X with coefficients in A . For a space or spectrum X , we write τ ≤ n X = τ
We are grateful to Peter May for suggesting the authors get in touch overthis problem. Furthermore, the first author would like to thank Bjørn Dundas, Frank Gounelas,Jesper Grodal, and Thomas Nikolaus for helpful conversations about p -completion, and has beenpartially supported by the DNRF92.2. Preliminaries on p -completion We briefly recall the basic properties of p -completion for nilpotent groups, topological spaces,and spectra. With the exceptions of Lemma 2.2 and Proposition 2.4, this material is mostlytaken from [BK72, Bou75, Bou79], and we refer to these sources as well as [HS99, MP12] forfurther references.2.1. Algebraic p -completion for abelian groups. In general, the p -completion functor M lim i M/p i M on the category of abelian groups is neither left nor right exact, so one studies itszeroth and first left derived functors L and L , respectively. An abelian group M is calledderived p -complete (or Ext- p -complete or L -complete) if the natural completion map M → L M is an isomorphism. For each abelian group M , the map M → L M will then be the universalhomomorphism from M to a derived p -complete abelian group by [BK72, Ch. VI, 3.2]. We willdenote the full subcategory of derived p -complete abelian groups by C p . Proposition 2.1.
The category C p is a full abelian subcategory of Mod Z closed under extensionsand limits. Furthermore, for any M ∈ Mod Z there is a short exact sequence / / lim i Hom Z ( Z /p i , M ) / / L M / / lim i M/p i M / / relating derived p -completion to ordinary p -completion.Proof. This is essentially proven in [BK72, Ch. VI, 2.1], but can also be deduced as a specialcase of [HS99, Thms. A.2 and A.6]. (cid:3)
We will later make use of the following observation.
Lemma 2.2. If A ∈ C p is torsion, then A is bounded p -torsion.Proof. We give two proofs, a conceptual one and an elementary argument. First, any derived p -complete group A is cotorsion, so the Baer–Fomin theorem [Bae36] implies that A is the directsum of a divisible group and a bounded torsion group. Since A is derived p -complete, it must bereduced, hence the divisible summand is trivial.Second, suppose that the conclusion of the lemma is false, i.e., that there exists a sequence( a i ) i ∈ N of elements of A such that the order of a i is p i . Set x j = P j − i =0 a i +1 p i , then the element x = ( x , x , x , . . . ) ∈ Q j ∈ N A lies in lim j A/p j . By construction, x is not p -torsion, whichcontradicts the fact that A → lim j A/p j is surjective, forcing lim j A/p j to be p -torsion. (cid:3) Remark . By a theorem of Pr¨ufer, the conclusion of the lemma implies that A must in factbe a direct sum of cyclic p -groups. N THE COMPARISON OF STABLE AND UNSTABLE p -COMPLETION 3 Algebraic p -completion for nilpotent groups. Recall from [BK72, Ch. VI, §
2] that thenotion of derived p -completion can be extended to nilpotent groups, as follows: If X ∧ p denotesthe Bousfield–Kan p -completion of a nilpotent space X as recalled in the next subsection, thenwe define the derived p -completion of the nilpotent group N as L N = π ( K ( N, ∧ p ) and L N = π ( K ( N, ∧ p ). A nilpotent group N is called derived p -complete if the completion map N → L N is an isomorphism; for each nilpotent group N , the map N → L N will then be the universalhomomorphism from N to a derived p -complete nilpotent group by [BK72, Ch. VI, 3.2]. Wedenote the category of derived p -complete nilpotent groups by N p .The inclusion functor C p → N p has a left adjoint given by taking a derived p -completenilpotent group N to the derived p -completion of its abelianization L ( N/ [ N, N ]). Note thatthe unit of this adjunction is surjective, i.e., for any derived p -complete nilpotent group N , thecanonical map N → L ( N/ [ N, N ]) is surjective. Indeed, since L preserves epimorphisms ofnilpotent groups, all maps in the following commutative diagram are surjective: N / / ∼ = (cid:15) (cid:15) N/ [ N, N ] (cid:15) (cid:15) L N / / L ( N/ [ N, N ]) . We obtain the following generalization of Lemma 2.2:
Proposition 2.4.
The following conditions are equivalent for N ∈ N p :(1) N is torsion.(2) L ( N/ [ N, N ]) is torsion.(3) N is bounded p -torsion.Proof. The surjectivity of the map N → L ( N/ [ N, N ]) observed above immediately gives theimplication (1) ⇒ (2), while (3) ⇒ (1) is trivial.Assume that L ( N/ [ N, N ]) is torsion and thus bounded p -torsion by Lemma 2.2. Considerthe lower central series of N , N = γ N ⊇ γ N ⊇ . . . ⊇ γ m N = 1 , with successive abelian quotients Q i ( N ) = γ i N/γ i +1 N . We claim that, for each i ≥ Q i ( N )is a direct sum of a p -divisible group and a bounded p -torsion group. Indeed, we start with theabelianization Q ( N ) = N/ [ N, N ] of N . Lemma 3.7 in [BK72, Ch. VI] implies that the kernelof the completion map Q ( N ) → L Q ( N ) is p -divisible, so the claim holds for Q ( N ). Thegeneral case follows from this, because L i ≥ Q i ( N ) is generated as a Lie algebra by Q ( N ). By[BK72, Ch. VI, 2.5], there is an exact sequence L Q i ( N ) / / L ( N/γ i +1 N ) / / L ( N/γ i N ) / / i ≥
1. Using the previous claim, L Q i ( N ) is bounded p -torsion, so we see inductivelythat L ( N/γ i N ) is bounded p -torsion for all i ≥
1, hence (3) holds. (cid:3)
Remark . The implication (1) ⇒ (3) in the previous proposition could also be proven moredirectly via the upper central series of N , whose quotients are known to be derived p -completeby [BK72, VI. 3.4(ii)], but this result would be insufficient for our later use.2.3. Topological p -completion. In [BK72], Bousfield and Kan introduced the notion of p -completion for topological spaces, lifting the algebraic notion defined above to topology. Ingeneral, the p -completion of a space is difficult to describe, but the theory simplifies significantlyfor nilpotent spaces; in particular, in this case p -completion coincides with H F p -localization[Bou75]. Furthermore, for nilpotent spaces with F p -homology of finite type, p -completion can be N THE COMPARISON OF STABLE AND UNSTABLE p -COMPLETION 4 identified with p -profinite completion due to Sullivan [Sul74]. Similarly, the category of spectraadmits (at least) two notions of p -completion, given either by H F p -localization or, the one wewill use here, localization at the mod p Moore spectrum S /p , see [Bou79]. The next resultsummarizes the relation between these constructions and lists their basic properties. Theorem 2.6 (Bousfield, Kan) . (1) A nilpotent space X is p -complete if and only if π n X is derived p -complete for all n ∈ N .Moreover, the notions of p -completion and H F p -localization coincide for nilpotent spaces.(2) A spectrum X is p -complete if and only if π n X is derived p -complete for all n ∈ Z . If X is bounded below, then X is p -complete if and only if X is H F p -local.Moreover, if X is a nilpotent space or spectrum, then there exists a short exact sequence computingthe unstable or stable homotopy groups of its p -completion, respectively: / / L π n X / / π n ( X ∧ p ) / / L π n − X / / for any n , where L i ( − ) ∼ = Ext − i Z ( Z /p ∞ , − ) are the derived functors of p -completion. Generalized Serre theory
The full subcategory C p of Mod Z is not closed under subobjects or quotients, and thus doesnot form a Serre class in the usual sense. This necessitates a mild generalization of Serre’s mod C theory which we develop in this section. Definition 3.1.
A weak Serre class is a full subcategory
C ⊆
Mod Z such that if A / / A / / A / / A / / A is an exact sequence in Mod Z with A , A , A , A ∈ C , then also A ∈ C .More explicitly, this means that C ⊆
Mod Z is a full additive subcategory closed under kernels,cokernels, and extensions. It follows that C is also closed under tensoring and Tor Z with respectto finitely generated abelian groups. For instance, any Serre subcategory of Mod Z is a weak Serreclass, but the converse does not hold. The main example of interest to us here is the category C p of derived p -complete abelian groups, see Proposition 2.1. Proposition 3.2.
Suppose C is a weak Serre class. If X is a bounded below spectrum, then thefollowing two conditions are equivalent:(1) π n X ∈ C for all n ∈ Z .(2) H n ( X ; Z ) ∈ C for all n ∈ Z .Proof. Assume the first condition holds; we will argue via the Postnikov tower ( τ ≤ n X ) of X . Forsimplicity, we will write H ∗ ( Y ) for the integral homology of a spectrum Y throughout this proof.To start with, we need to show that H ∗ ( HA ) ∈ C for A ∈ C . Using the isomorphisms H ∗ ( HA ) ∼ = H ∗ ( H Z ; A ), the universal coefficient theorem gives a short exact sequence0 / / H ∗ ( H Z ) ⊗ Z A / / H ∗ ( HA ) / / Tor Z ( H ∗− ( H Z ) , A ) / / . In each degree, the integral Steenrod algebra H ∗ ( H Z ) is finitely generated over Z , as followsfrom Serre theory for the class of finitely generated abelian groups. Therefore, the outer termsof this sequence are in C . This shows H ∗ ( HA ) ∈ C as well.Given n ∈ Z , we will now prove that H n ( X ) ∈ C . Since H n ( τ >n X ) = 0 = H n − ( τ >n X )by connectivity, we see that H n ( X ) ∼ = H n ( τ ≤ n X ). This reduces the claim to proving that H ∗ ( τ ≤ n X ) ∈ C . This follows inductively, using the exact sequence H ∗ +1 ( τ ≤ n − X ) / / H ∗ (Σ n Hπ n X ) / / H ∗ ( τ ≤ n X ) / / H ∗ ( τ ≤ n − X ) / / H ∗− (Σ n Hπ n X ) N THE COMPARISON OF STABLE AND UNSTABLE p -COMPLETION 5 associated to the fiber sequence Σ n H ( π n X ) → τ ≤ n X → τ ≤ n − X . Since H k ( Hπ n X ) ∈ C for all k ∈ Z , this gives the implication (1) ⇒ (2).For the converse, consider the convergent Atiyah–Hirzebruch spectral sequence E s,t ∼ = H s ( X ; π t S ) = ⇒ π s + t X. Since π t S is finitely generated over Z for each t ∈ Z , H s ( X ; π t S ) ∈ C for each bidegree ( s, t ),hence π n X is also in C for all n ∈ Z . (cid:3) When applied to the weak Serre class C p , we obtain a homological characterization of p -completeness for bounded below spectra. Corollary 3.3.
For a bounded below spectrum X , the following conditions are equivalent:(1) X is p -complete.(2) π n X is derived p -complete for all n .(3) H n ( X ; Z ) is derived p -complete for all n .Proof. The equivalence of (1) and (2) is the content of Theorem 2.6(2), while (2) is equivalentto (3) by Proposition 3.2. (cid:3)
We deduce that the integral homology of p -complete spaces is well-behaved in the stable range. Corollary 3.4.
Suppose X is p -complete space. If X is n -connected, then H k ( X ; Z ) is derived p -complete for all k ≤ n .Proof. Since π k Σ ∞ X ∼ = π k X for k ≤ n by the Freudenthal suspension theorem, Theorem 2.6implies that π ∗ τ ≤ n Σ ∞ X is derived p -complete in each degree, hence so is H ∗ ( τ ≤ n Σ ∞ X ; Z )by Corollary 3.3. We thus get that H k ( X ; Z ) ∼ = H k (Σ ∞ X ; Z ) ∼ = H k ( τ ≤ n Σ ∞ X ; Z ) is derived p -complete for k ≤ n . (cid:3) The comparison
In this section, we first study the relation between p -completion for spectra and spaces underthe infinite loop space functor Ω ∞ , and then prove our main theorem.4.1. Infinite loop spaces.
It is easy to deduce from Theorem 2.6 the following relation betweenunstable and stable p -completion under Ω ∞ . Proposition 4.1.
For -connected spectra X and Y , we have:(1) X is p -complete if and only if Ω ∞ X is p -complete.(2) A map f : X → Y is an H F p -equivalence if and only if Ω ∞ f is an H F p -equivalence.(3) The canonical comparison map (Ω ∞ X ) ∧ p → Ω ∞ ( X ∧ p ) is an equivalence.Proof. Since π ∗ Ω ∞ X ∼ = π ∗ X and Ω ∞ X is nilpotent, the first claim is a direct consequence ofTheorem 2.6. In order to prove (2), note that f is an H F p -equivalence if and only if the homotopygroups π ∗ cof( f ) of the cofiber of f are uniquely p -divisible. This is equivalent to the statementthat the F p -homology H ∗ (Ω ∞ cof( f ); F p ) is trivial. The Serre spectral sequence associated to thefiber sequence Ω ∞ X Ω ∞ f / / Ω ∞ Y / / Ω ∞ cof( f )thus shows that this happens if and only if Ω ∞ f is an H F p -equivalence. N THE COMPARISON OF STABLE AND UNSTABLE p -COMPLETION 6 Statement (1) implies that Ω ∞ ( X ∧ p ) is p -complete, so the map Ω ∞ ( X ) → Ω ∞ ( X ∧ p ) factorscanonically through φ : (Ω ∞ X ) ∧ p → Ω ∞ ( X ∧ p ), making the following diagram commute:Ω ∞ X / / $ $ ❏❏❏❏❏❏❏❏❏❏❏ (Ω ∞ X ) ∧ p (cid:15) (cid:15) Ω ∞ ( X ∧ p ) . By Statement (2), both the horizontal and the diagonal map are H F p -equivalences, hence so isthe vertical comparison map. (cid:3) Remark . Let Ω ∞ be the 0-component of Ω ∞ . The last part of the proposition can bestrengthened to an equivalence (Ω ∞ X ) ∧ p → Ω ∞ ( X ∧ p ) for any connective spectrum X such that π X does not contain any copies of Z /p ∞ . To prove this directly, one may use the short exactsequences displayed at the end of Theorem 2.6.4.2. Suspension spectra.
We now turn to the comparison under Σ ∞ . In odd dimensions, thenext result has also been observed in [BK72, Rem. VI.5.7], see also [MP12, Rem. 11.1.5]. Lemma 4.3.
Let n ≥ and write S np for the p -completion of S n . There exists an uncountablerational vector space in H n ( S np ; Z ) which injects into H n ( K ( Z p , n ); Z ) under the map S np → τ ≤ n S np ≃ K ( Z p , n ) .Proof. Consider the following segment of the Serre long exact sequence for the fibration F → S np → K ( Z p , n ): H n ( F ; Z ) / / H n ( S np ; Z ) / / H n ( K ( Z p , n ); Z ) / / H n − ( F ; Z ) / / . . . . Corollary 3.4 implies that H n ( F ; Z ) and H n − ( F ; Z ) are derived p -complete. Recalling thatHom Z ( Q , A ) = 0 = Ext Z ( Q , A ) whenever A is derived p -complete, we see that the natural mapHom Z ( Q , H n ( S np ; Z )) → Hom Z ( Q , H n ( K ( Z p , n ); Z )) is surjective. Thus, it will suffice to showthat H n ( K ( Z p , n ); Z ) contains an uncountable rational vector space, which will be verified inthe homological proof of Proposition 5.3 below. (cid:3) Note that, because H ∗ ( S np ; F p ) ∼ = H ∗ ( S n ; F p ) ∼ = F p [ n ], an application of the universal coeffi-cient theorem shows that H k ( S np ; Z ) is rational for all k > n . Lemma 4.4.
Suppose N is a derived p -complete nilpotent (abelian) group and n = 1 ( n ≥ ).If N is not bounded p -torsion, then there exists an element x ∈ N of infinite order inducing amonomorphism H ∗ ( K ( Z p , n ); Q ) → H ∗ ( K ( N, n ); Q ) .Proof. By assumption on N and Proposition 2.4, L ( N/ [ N, N ]) contains elements of infiniteorder. Let x be such an element and let x ∈ N be a lift of x . For the remainder of the proof weassume n = 1; the (easier) case n ≥ N abelian is proven similarly. The element x inducesa map K ( Z p , / / K ( N, / / K ( L ( N/ [ N, N ]) , π . It follows that the rationalization K ( Z p , Q → K ( L ( N/ [ N, N ]) , Q of this map is split, hence the composite H ∗ ( K ( Z p , Q ) / / H ∗ ( K ( N, Q ) / / H ∗ ( K ( L ( N/ [ N, N ]) , Q )is a split monomorphism, which implies the claim. (cid:3) N THE COMPARISON OF STABLE AND UNSTABLE p -COMPLETION 7 Proposition 4.5. If X is a p -complete nilpotent space whose homotopy groups are not allbounded p -torsion, then the integral homology groups H ∗ ( X ; Z ) and the stable homotopy groups π ∗ Σ ∞ X both contain an uncountable rational vector space.Proof. Assume that π ∗ X is not all bounded p -torsion, and let π n X be the lowest such group. Itthen follows from Lemma 4.4 that π n X contains a class x of infinite order inducing a monomor-phism H ∗ ( K ( Z p , n ); Q ) → H ∗ ( K ( π n X, n ); Q ). Since the map τ ≥ n X → X is a rational homologyequivalence, any rational subgroup of H ∗ ( τ ≥ n X ; Z ) must map monomorphically to H ∗ ( X ; Z ), soit suffices to prove the homological claim for τ ≥ n X . The element x yields a map S np → τ ≥ n X such that the composite S np → τ ≥ n X → K ( π n X, n ) factors as τ ≥ n X / / τ ≤ n τ ≥ n X ≃ K ( π n X, n ) S np / / O O τ ≤ n S np ≃ K ( Z p , n ) . O O It follows from Lemma 4.3 and the choice of x that the induced homomorphism in homology H n ( S np ; Z ) / / H n ( τ ≥ n X ; Z ) / / H n ( K ( π n X, n ); Z )maps an uncountable rational vector space monomorphically to H n ( K ( π n X, n ); Z ), hence sodoes the map H n ( S np ; Z ) → H n ( τ ≥ n X ; Z ). This verifies the claim about the integral homologyof X .Recall that, for any connective spectrum Y , the Hurewicz map π ∗ Y → H ∗ ( Y ; Z ) has kernel andcokernel of bounded torsion in each degree. Indeed, the fiber sequence Y ∧ τ > S → Y → Y ∧ H Z reduces this claim to showing that π ∗ ( Y ∧ τ > S ) is bounded torsion in each degree. This followsfrom the convergent Atiyah–Hirzebruch spectral sequence H s ( Y ; π t τ > S ) = ⇒ π s + t ( Y ∧ τ > S ) , because H s ( Y ; π t τ > S ) is bounded torsion for all s and t . Therefore, any rational vector spacein H ∗ ( Y ; Z ) may be lifted back to π ∗ Y . In particular, an uncountable rational vector space in H n ( X ; Z ) may be lifted back to π n (Σ ∞ X ) after suspension. (cid:3) Remark . Suppose X is a p -complete nilpotent space such that π n X is the lowest homotopygroup not of bounded p -torsion. The above argument shows that H n ( X ; Z ) contains an un-countable rational vector space. With more work, we can also show that H k ( X ; Z ) is derived p -complete for k ≤ n − X is( n − H k ( X ; Z ) is in the stablerange.We can now prove our main theorem. Theorem 4.7. If X is a p -complete nilpotent space, then Σ ∞ X is p -complete if and only if π n X is bounded p -torsion for each n . Note that the torsion exponent of π n X may vary with n and does not need to be boundeduniformly for all n . Proof.
First assume that X is a p -complete nilpotent space with π n X of bounded p -torsion foreach n ; we can apply [BK72, Ch. II, 4.7] to see that the Postnikov tower of X can be refined toa tower of principal fibrations whose fibers are Eilenberg–MacLane spaces for bounded p -torsionabelian groups. The category of bounded p -torsion abelian groups forms a Serre class, so Serretheory implies that H ∗ ( X ; Z ) ∼ = H ∗ (Σ ∞ X ; Z ) is degreewise bounded p -torsion. Hence, Σ ∞ X is p -complete as a spectrum by Corollary 3.3. N THE COMPARISON OF STABLE AND UNSTABLE p -COMPLETION 8 The converse is a consequence of Proposition 4.5: if π ∗ X is not all bounded torsion, then H ∗ (Σ ∞ X ; Z ) contains rational classes and thus cannot be derived p -complete, hence Σ ∞ X is not p -complete by Corollary 3.3. (cid:3) Corollary 4.8. If X is a nilpotent space with H n ( X ; Z ) and π n X derived p -complete for all n ,then H n ( X ; Z ) and π n X are bounded p -torsion for all n .Proof. The assumption on π ∗ X implies that X is p -complete by Theorem 2.6, while the as-sumption on H ∗ ( X ; Z ) shows that Σ ∞ X is p -complete, using Corollary 3.3. It thus follows fromTheorem 4.7 that π ∗ X is degreewise bounded p -torsion, hence so is H ∗ ( X ; Z ) by the proof ofTheorem 4.7. (cid:3) The analogue of this corollary does not hold stably, as the following example demonstrates.
Example 4.9.
Let M ( Z p , n ) be the Moore space for Z p in degree n ≥
2. As H ∗ (Σ ∞ M ( Z p , n ); Z )is isomorphic to Z p [ n ], we see that Σ ∞ M ( Z p , n ) is p -complete and consequently has derived p -complete stable homotopy groups and integral homology groups. However, H n (Σ ∞ M ( Z p , n ); Z ) ∼ = Z p is clearly not bounded p -torsion. In particular, M ( Z p , n ) is not p -complete, so this also showsthat the assumption that X be p -complete cannot be dropped in Theorem 4.7.5. Rational classes in the stable homotopy groups of K ( Z p , n )In this section, we present an example that illustrates how the rational classes in the stablehomotopy groups of p -complete spaces arise. In fact, we present two different approaches: Oneusing the integral homology of K ( Z p , n ), and one using Goodwillie calculus. The latter derivationis entirely stable and might be of independent interest.First, we need a well-known auxiliary result; we outline a proof because we were unable tofind a published reference for it. For an abelian group A and any k ≥
0, let Sym k Z ( A ) and Λ k Z ( A )be the k th symmetric power and the k th exterior power on A , respectively. Lemma 5.1. If k > , then Λ k Z ( Z p ) and the kernel of the multiplication map Sym k Z ( Z p ) → Z p are uncountable rational vector spaces.Proof. Since both symmetric and exterior power commute with base-change along Z → Z /l forany prime l , the indicated maps are isomorphisms mod l . Moreover, Sym k Z ( A ) and Λ k Z ( A ) aretorsion-free whenever A is, so both ker(Sym k Z ( Z p ) → Z p ) and Λ k Z ( Z p ) are rational vector spaces.We may therefore base-change to Q , where it is easy to verify that the Q -dimension of the groupsunder consideration is that of Q p . (cid:3) Remark . A similar argument also shows that Z p / Z ( p ) is a rational vector space with thesame Q -dimension as Q p . Proposition 5.3.
For n ≥ and all k > , the stable homotopy group π nk Σ ∞ K ( Z p , n ) containsan uncountable rational vector space. In particular, Σ ∞ K ( Z p , n ) is not p -complete.First proof. Let A be an abelian group and recall that H ∗ ( K ( A, n ); Z ) equipped with the Pon-tryagin product is a graded commutative algebra such that squares of odd dimensional elementsare zero; in fact, it has the structure of a graded divided power algebra, see [EML54, Car56]or more recently [Ric09]. With notation as in the previous lemma, the canonical isomorphism A → H n ( K ( A, n ); Z ) thus extends to a natural homomorphism φ k ( A, n ) : Λ k Z ( A ) / / H kn ( K ( A, n ); Z ) , if n odd φ k ( A, n ) : Sym k Z ( A ) / / H kn ( K ( A, n ); Z ) , if n even N THE COMPARISON OF STABLE AND UNSTABLE p -COMPLETION 9 for any n, k >
0. Moreover, we know that φ k ( A, n ) ⊗ Z Q is a rational isomorphism. It thenfollows from Lemma 5.1 that, for k >
1, there exists an uncountable rational vector space whichis mapped monomorphically to H kn ( K ( Z p , n ); Z ) via φ k ( Z p , n ). We thus obtain an uncountablerational vector space in H kn ( K ( Z p , n ); Z ) that may be lifted back to give the desired uncountablerational vector space in π nk Σ ∞ K ( Z p , n ) for k >
1, as in the proof of Proposition 4.5. (cid:3)
Second proof.
We will compute the homotopy groups of Σ ∞ K ( Z p , n ) ≃ Σ ∞ Ω ∞ Σ n H Z p usingGoodwillie calculus [Goo03]. To this end, recall that the Goodwillie tower ( P k ) k ≥ associated tothe functor Σ ∞ Ω ∞ : Sp → Sp is assembled from fiber sequences of functors D k / / P k / / P k − (5.4)with layers D k X ≃ X ∧ kh Σ k , where the homotopy orbits are formed with respect to the permutationaction of Σ k (see for example [KM13] and the references given therein). Moreover, the Goodwillietower ( P k ) k ≥ converges for connective spectra, i.e., there is a canonical equivalenceΣ ∞ Ω ∞ X ∼ / / lim k P k X for any connective X ∈ Sp. We will apply this in the case X = Σ n H Z p .In order to understand the layers, we start by analyzing π ∗ (Σ n H Z p ) ∧ k via the universalcoefficient theorem. We claim that, for all k ≥
1, the homotopy groups have the following form π ∗ (Σ n H Z p ) ∧ k ∼ = ∗ < nk Z ⊗ Z kp ∗ = nk finite ∗ > nk. (5.5)By the universal coefficient theorem, we have an isomorphism π ∗ (Σ n H Z p ) ∧ k ∼ = ( π ∗ (Σ n H Z ) ∧ k ) ⊗ Z Z ⊗ kp . In degrees ∗ > nk , the groups π ∗ (Σ n H Z ) ∧ k are torsion, so the only torsion-free summand appearsin degree nk . Since π ∗ (Σ n H Z ) ∧ k is finitely generated over Z in each degree, the claim follows.We now plug the formula (5.5) into the convergent homotopy orbit spectral sequence H s (Σ k , π t (Σ n H Z p ) ∧ k ) = ⇒ π s + t D k (Σ n H Z p ) . There are two cases: If t > nk or t < nk , then the groups H s (Σ k , π t (Σ n H Z p ) ∧ k ) are finite ortrivial for all s , respectively. Let t = nk . By Lemma 5.1 and (5.5), there is an isomorphism H s (Σ k , π nk (Σ n H Z p ) ∧ k ) ∼ = H s (Σ k , Z p ) for s > H (Σ k , π nk (Σ n H Z p ) ∧ k ) contains an un-countable rational vector space V k if k >
1. To see the last statement, it suffices to computethe coinvariants on the rational submodule of Z ⊗ Z kp by choosing a Q -bases, as in the proof ofLemma 5.1. Furthermore, since the integral homology of Σ k is finitely generated over Z in eachdegree and rationally trivial in positive degrees, H s (Σ k , π nk (Σ n H Z p ) ∧ k ) is finite for all s > D Σ n H Z p ≃ Σ n H Z p and for k > π ∗ D k (Σ n H Z p ) ∼ = ∗ < nkV k ⊕ W k ∗ = nk finite ∗ > nk, (5.6)where V k is an uncountable rational vector space and W k is some abelian group.This allows us to derive a structural formula for π ∗ P k Σ n H Z p . Consider the following segmentof the long exact sequence of homotopy groups associated to the fiber sequence (5.4): . . . / / π nk +1 P k − Σ n H Z p / / π nk D k Σ n H Z p / / π nk P k Σ n H Z p / / . . . . N THE COMPARISON OF STABLE AND UNSTABLE p -COMPLETION 10 Because n ≥
1, it follows inductively from (5.6) that the term on the left is finite, hence V k mustbe a summand in π nk P k Σ n H Z p . This yields for all k ≥ π ∗ P k Σ n H Z p ∼ = ∗ < nV l ⊕ W ′ l ∗ = nl with 1 ≤ l ≤ k finite otherwise , (5.7)where V l is as above for l ≥
2, and V and W ′ l are some abelian groups.Finally, since D k Σ n H Z p is nk -connective for all k , the tower ( π ∗ P k Σ n H Z p ) k ≥ stabilizes afterfinally many steps in each degree and hence is Mittag-Leffler. The corresponding Milnor sequencethus degenerates to an isomorphism π ∗ Σ ∞ K ( Z p , n ) ∼ = π ∗ Σ ∞ Ω ∞ Σ n H Z p ∼ = lim k π ∗ P k Σ n H Z p . Therefore, the claim follows from (5.7). (cid:3)
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