aa r X i v : . [ m a t h . A T ] J a n p -hyperbolicity of homotopy groups via K -theory Guy BoydeJanuary 13, 2021
Abstract
We show that S n ∨ S m is Z /p r -hyperbolic for all primes p and all r ∈ N , provided n, m ≥ S n ∨ S m as a p -local retract are Z /p r -hyperbolic.We then give a K -theory criterion for a suspension Σ X to be p -hyperbolic, and use it to deducethat the suspension of a complex Grassmannian Σ Gr k,n is p -hyperbolic for all odd primes p when n ≥ < k < n . We obtain similar results for some related spaces. A space X is called rationally elliptic if π ∗ ( X ) ⊗ Q is finite dimensional, and rationally hyperbolic if thedimension of L i ≤ m π i ( X ) ⊗ Q grows exponentially in m . It was proved in [FHT15, Chapter 33] thatsimply connected CW -complexes with rational homology of finite type and finite rational categoryare either rationally elliptic or rationally hyperbolic. In order to study the p -torsion analogue of thisdichotomy, Huang and Wu [HW] introduced the definitions of Z /p r - and p -hyperbolicity.For p prime, by a p -torsion summand in an abelian group A , we mean a direct summand isomorphicto Z /p r for some r ≥ Definition 1.1.
Let X be a space, and let p be a prime. We say that X is p - hyperbolic if the numberof p -torsion summands in π ∗ ( X ) grows exponentially, in the sense thatlim inf m ln( T m ) m > , where T m is the number of p -torsion summands in L i ≤ m π i ( X ).The above definition counts Z /p r -summands for all values of r . It is also possible to consider onlya single r , and by doing so we obtain the definition of Z /p r -hyperbolicity. Definition 1.2.
Let X be a space, let p be a prime, and fix r ∈ N . We say that X is Z /p r - hyperbolic if the number of Z /p r -summands in π ∗ ( X ) grows exponentially, in the sense thatlim inf m ln( t m ) m > , where t m is the number of Z /p r -summands in L i ≤ m π i ( X ).Note that Z /p r -hyperbolicity for any r implies p -hyperbolicity. It follows immediately from a resultof Henn [Hen86, Corollary of Theorem 1] that the lim infs appearing in the above definitions must befinite if X is a simply connected finite CW -complex.Huang and Wu show that for n ≥ r ≥ p any prime, the Moore space P n ( p r ) is Z /p r -hyperbolic and Z /p r +1 -hyperbolic, and that P n (2) is also Z / X to be Z /p r -hyperbolic.1ore recently, Zhu and Pan [ZP21] use a classification of ( n − CW -complexes ofdimension at most n + 2, due to Chang [Cha50], to show that, for n ≥
4, such a complex is Z /p -hyperbolic, provided that it is not contractible or a sphere after p -localization. They also provehyperbolicity results for several so-called elementary Chang complexes.This paper studies p - and Z /p r -hyperbolicity of certain suspensions. Our first result is as follows. Theorem 1.3.
Let q , q ≥
1. Then S q +1 ∨ S q +1 is Z /p r -hyperbolic for all primes p and all r ∈ N .Let p be a prime. If a space X contains a wedge of two spheres as a p -local retract, then Theorem1.3 implies that X is Z /p r -hyperbolic for all r . Various spaces have been shown to contain such awedge - examples of this sort are given in Section 2.1. A summary is as follows: • for n, k ≥
3, the configuration space Conf k ( R n ) is Z /p r -hyperbolic for all p and r (Example 2.1); • an ( n − n -dimensional manifold M , where H n ( M ) is of rank at least 3, is Z /p r -hyperbolic for all p and r (Example 2.2); • a generalized moment-angle complex on a simplicial complex having two minimal missing faceswhich are not disjoint is Z /p r -hyperbolic for all p and r (Example 2.3); • Σ C P is Z /p r -hyperbolic for all p = 2 and all r , and Σ H P is is Z /p r -hyperbolic for all p = 2 , r (Example 2.4).Our other result is as follows. Theorem 1.4.
Let X be a path connected space having the homotopy type of a finite CW -complex,and let p be an odd prime. Suppose that there exists a map µ ∨ µ : S q +1 ∨ S q +1 → Σ X with q i ≥
1, such that the map e K ∗ (Σ X ) ⊗ Z /p ( µ ∨ µ ) ∗ −−−−−−→ e K ∗ ( S q +1 ∨ S q +1 ) ⊗ Z /p ∼ = Z /p ⊕ Z /p is a surjection. Then Σ X is p -hyperbolic.This criterion is quite different to that given by Huang and Wu [HW, Theorem 1.5]. Their criterionis homotopical, using hypotheses on X to produce retracts of ΩΣ X , wheras ours is cohomological,which makes it easier to check. On the other hand, their criterion is stronger, since it gives Z /p r -hyperbolicity, rather than just p -hyperbolicity. The examples they give, primarily various Moorespaces, differ from those we obtain, which are the suspensions of spaces related to complex projectivespace. More precisely, in Section 2.2, we show that the following spaces are p -hyperbolic for all p = 2: • suspended complex projective space Σ CP n for n ≥ • the suspended complex Grassmannian ΣGr k,n for n ≥ < k < n (Example 2.6); • the suspended Milnor Hypersurface Σ H m,n for m ≥ n ≥
3, (Example 2.7); • the suspended unitary group Σ U ( n ) for n ≥ S n ∨ S m [Hil55]. For Theorem 1.4, we employ K -theoretic methods originally used bySelick [Sel83] to prove one direction of Moore’s conjecture for suspensions having torsion-free homology.If the map µ ∨ µ of Theorem 1.4 induces a surjection on e K ∗ ( ) ⊗ Z /p , then so does its suspensionΣ µ ∨ Σ µ . The conclusion of Theorem 1.4 may therefore be strengthened in the following way.2 orollary 1.5. With the hypothesis of Theorem 1.4, Σ n X is p -hyperbolic for all n ≥ X satisfy the growth conditions of Definition 1.1 or 1.2. In theproofs of both Theorem 1.3 and 1.4, the classes that witness the hyperbolicity are composites involvingWhitehead products. The suspension of a Whitehead product is always trivial [Whi46, Theorem 3.11],so the classes we detect cannot be stable. Therefore, Corollary 1.5 does not suggest that the stablehomotopy of Σ X should be p - or Z /p r -hyperbolic. On the other hand, it follows from our methodsthat, under the hypotheses of Theorem 1.4, ΩΣ X is stably p -hyperbolic.By a result of Henn [Hen83], any co- H space, and in particular any suspension, decomposes ratio-nally as a wedge of spheres. It then follows from the Hilton-Milnor theorem [Hil55] and the computa-tion of the rational homotopy groups of spheres [Ser51] that such a suspension is rationally hyperbolicprecisely when there are at least two spheres (of dimension ≥
2) in this decomposition.We will see in Corollary 7.12 that if Σ X satisfies the hypotheses of Theorem 1.4 for any prime(including 2), then Σ X is rationally hyperbolic, hence is rationally a wedge of at least two spheres bythe preceding discussion. This rational equivalence is a local equivalence at all but perhaps finitelymany primes, so by Theorem 1.3, Σ X is Z /p r hyperbolic for all r at all but finitely many primes p . One might therefore conjecture that the conclusion of Theorem 1.4 can be strengthened to give Z /p r -hyperbolicity for all r rather than p -hyperbolicity, but we do not know whether this is possible.We now discuss situations in which it is adequate to consider ordinary cohomology, rather than K -theory. If Σ X has torsion-free integral (co)homology, or if its cohomology is concentrated in evendegrees, then the Atiyah-Hirzebruch spectral sequence for K ∗ (Σ X ) collapses on the E page [Hus+08].It follows by naturality that the image of the map induced by µ ∨ µ : S q +1 ∨ S q +1 → Σ X on K -theory is identified with the image of the induced map on cohomology. We may therefore replace K -theory with cohomology in Theorem 1.4, as follows. Corollary 1.6.
Let X be a path connected space having the homotopy type of a finite CW -complex,such that the Atiyah-Hirzebruch spectral sequence for K ∗ (Σ X ) collapses on the E page. Let p be anodd prime. Suppose that there exists a map µ ∨ µ : S q +1 ∨ S q +1 → Σ X with q i ≥
1, such that themap induced by µ ∨ µ on e H ∗ ( ) ⊗ Z /p is a surjection. Then Σ X is p -hyperbolic.One advantage of ordinary cohomology is that it is connected to the homotopy groups integrally,via the universal coefficient theorem and Hurewicz map. We can exploit this as follows. Example . Suppose that the Atiyah-Hirzebruch spectral sequence for K ∗ (Σ X ) collapses (for exam-ple, if Σ X has torsion-free homology) and that there exists q ∈ N so that e H i (Σ X ) = 0 for i ≤ q , anddim Q ( e H q +1 (Σ X ) ⊗ Q ) ≥
2. The Hurewicz map π q +1 (Σ X ) → e H q +1 (Σ X ) is an isomorphism, so thereexists a map µ ∨ µ : S q +1 ∨ S q +1 → Σ X inducing the inclusion of a Z -summand in e H q +1 (Σ X ).By the universal coefficient theorem relating ordinary homology and cohomology, µ ∨ µ induces asurjection on integral cohomology, so by Corollary 1.6, Σ X is p -hyperbolic for all odd primes p .I would like to thank my PhD supervisor, Stephen Theriault, for suggesting the problems that thispaper tries to address, and for many helpful conversations along the way. From a technical point ofview, much is owed to papers of Huang and Wu [HW], and of Selick [Sel83]. Neil Strickland’s ‘Bestiary’was extremely helpful in providing examples of Theorem 1.4. Conversations with Sam Hughes werevery useful in formulating Corollary 1.6. Theorem 1.3 immediately implies that any space X which has S q +1 ∨ S q +1 as a retract after p -localization is Z /p r -hyperbolic for that p and all r . This implies that for all n ≥
1, Σ n X contains3 q + n +1 ∨ S q + n +1 as a p -local retract, and so is Z /p r -hyperbolic for all r . We first consider examplesof this form. Example . It is known [Knu18, Section 3.1] that Conf k ( R n ), the ordered configuration space of k points in R n , contains W k − S n − as a retract. It follows that, when n, k ≥
3, Conf k ( R n ) is Z /p r -hyperbolic for all p and r . Example . Let M be an ( n − n -dimensional manifold. By the universal coefficienttheorem, there can be no torsion in H n ( M ). Suppose that the rank of H n ( M ) is at least 3. By workof Beben and Theriault [BT14, Theorem 1.4], Ω M contains a wedge of two spheres as a retract afterlooping. Thus, M is again Z /p r -hyperbolic for all p and r . Example . Let K be a simplicial complex on the vertex set [ m ] = { , . . . , m } , and let ( X, A ) be anysequence of pairs ( D n i , S n i − ) with n i ≥ ≤ i ≤ m . If there exist two distinct minimal missingfaces of K which are not disjoint, then by work of Hao, Sun and Theriault [HST19, Theorem 4.2] thepolyhedral product ( X, A ) K contains a wedge of two spheres as a retract after looping, and hence is Z /p r -hyperbolic for all p and all r . Example . Localized away from 2, Σ C P ≃ S ∨ S . To see this, note that Σ C P has a CW -structure consisting of one 3-cell and one 5-cell, and that π ( S ) ∼ = Z / C P is Z /p r -hyperbolic for all r when p = 2.Similarly, Σ H P admits a cell structure with one 5-cell and one 9-cell, and π ( S ) ∼ = Z /
24. Thus,Σ H P is Z /p r -hyperbolic for all r when p = 2 , C P n Suppose that one has verified the hypotheses of Theorem 1.4 for a given space X and odd prime p ,using a map µ ∨ µ : S q +1 ∨ S q +1 → Σ X . If another space Y admits a map σ : Σ X → Σ Y whichinduces a surjection on e K ∗ ( ) ⊗ Z /p , then it is immediate that σ ◦ ( µ ∨ µ ) satisfies the hypothesesof Theorem 1.4, and hence that Σ Y is p -hyperbolic. The slogan is that K -theory surjections allow usto generate new examples from old ones.In this section, we will apply this idea. We will first show that Σ C P satisfies the hypotheses ofTheorem 1.4 at all odd primes p . We will then consider spaces X which are known to admit maps C P → X which induce surjections on integral K -theory, and hence on e K ∗ ( ) ⊗ Z /p for all odd p . Itfollows in each case that Σ X is p -hyperbolic, and further by Corollary 1.5, that Σ n X is p -hyperbolicfor all n ≥ η be the Hopf invariant one class which is the attaching map for the top cell of C P .Since Σ η lies in π ( S ) ∼ = Z /
2, we have that 2Σ η = 0. This gives the following map of cofibre sequences. S η / / S / / Σ C P / / S / / S S O O ∗ / / S / / S ∨ S µ O O / / S O O ∗ / / S . Let the restrictions of µ to S and S be µ and µ respectively, so that µ = µ ∨ µ . The map Σ η induces zero on reduced integral K -theory for degree reasons, so we obtain a diagram of short exactsequences: 0 e K ∗ ( S ) o o e K ∗ (Σ C P ) o o ( µ ∨ µ ) ∗ (cid:15) (cid:15) e K ∗ ( S ) o o (cid:15) (cid:15) o o e K ∗ ( S ) o o e K ∗ ( S ) ⊕ e K ∗ ( S ) o o e K ∗ ( S ) o o . o o
4e have obtained a map µ ∨ µ : S ∨ S → Σ C P which induces a surjection on e K ∗ ( ) ⊗ Z /p for all odd primes p . We now seek maps of spaces which allow us to extend to Σ C P n .The inclusion of C P n into C P n +1 induces a surjection on K -theory, so it must still induce asurjection after suspending. Composing these inclusions with µ ∨ µ gives, for each n ≥
2, a map S ∨ S → Σ C P n which still induces a surjection on e K ∗ ( ) ⊗ Z /p for all odd primes p . ApplyingTheorem 1.4 to this map gives the following. Example . For n ≥
2, Σ C P n is p -hyperbolic for all p = 2.Now let Gr k,n be the Grassmannian of k -dimensional complex subspaces of C n . First note thatorthogonal complement gives a homeomorphism Gr k,n ∼ = Gr n − k,n . In particular Gr n − ,n ∼ = Gr ,n ∼ = C P n − , so ΣGr n − ,n is p -hyperbolic. Other Grassmannians can be treated more uniformly, as follows.Let γ k,n denote the tautological bundle over Gr k,n . Consider the inclusion ι n : C n → C n +1 ( x , x , . . . , x n ) ( x , x , . . . , x n , . This inclusion induces a map i k,n : Gr k,n → Gr k,n +1 , defined on V ∈ Gr k,n by V ι n ( V ). Itfollows from this definition that i ∗ k,n ( γ k,n +1 ) = γ k,n . Letting e i denote the i -th standard basis vectorin C n , we also have a map j k,n : Gr k,n → Gr k +1 ,n +1 , defined on V = Span( v , v , . . . , v k ) ∈ Gr k,n by V Span( ι ( v ) , ι ( v ) , . . . , ι ( v k ) , e n +1 ). It follows from this definition that j ∗ k,n ( γ k +1 ,n +1 ) = γ k,n ⊕ C ,where C is the 1-dimensional trivial bundle.Since K ∗ ( C P n ) is generated by the class of the tautological bundle, composing the maps i k,n and j k,n for different values of k and n will give maps C P = Gr , → Gr k,n for all 1 ≤ k ≤ n − n ≥ K -theory. As in Example 2.5, this implies the following (the case k = n − n − ,n , which was treated first). Example . For n ≥ < k < n , the suspended complex Grassmannian ΣGr k,n is p -hyperbolicfor all p = 2.For m ≤ n , the Milnor Hypersurface H m,n is defined by H m,n = { ([ z ] , [ w ]) ∈ C P m × C P n | m X i =0 z i w i = 0 } . Suppose that m ≥ n ≥
3. Then there is an inclusion ι : C P → H m,n , defined by ι ([ z : z : z ]) = ([ z : z : z : 0 : · · · : 0] , [0 : · · · : 0 : 1]) . Write π for the projection H m,n → C P m . Then the inclusion C P → C P m factors as C P ●●●●●●●●● ι / / H m,nπ (cid:15) (cid:15) C P m . This implies that ι induces a surjection on integral K -theory, so we obtain the following. Example . For m ≥ n ≥
3, the suspended Milnor Hypersurface Σ H m,n is p -hyperbolic for all p = 2.Let U ( n ) denote the unitary group. There is a well-known map r : Σ C P n − → U ( n ) (see forexample [Whi78]) which induces a surjection on K -theory. From this we obtain Example . For n ≥
3, the suspended unitary group Σ U ( n ) is p -hyperbolic for all p = 2.5 .3 Quantitative lower bounds on growth In Section 4, we will derive the following simple lower bound for the lim inf in the definition of Z /p r -hyperbolicity, for the space S q +1 ∨ S q +1 . Corollary 2.9.
Let p be a prime and r ∈ N . Let t m be the constants of Definition 1.2 for X = S q +1 ∨ S q +1 . Then lim inf m ln( t m ) m ≥ ln(2)max( q , q ) . This implies that the t m eventually exceed ((1 − ε )2) m max( q ,q for any ε >
0. The constant 2 reflectsthe number of wedge summands. Note that this bound is independent of p and r . Example . Taking ε = , we find that for all r ∈ N and all primes p the number of Z /p r -summandsin L i ≤ m π i ( S ∨ S ) eventually exceeds ( ) m .One can produce an analogous quantitative bound on the lim inf in the case of Theorem 1.4, butthis bound is very weak. In particular, it depends on knowledge of the Adams operations on K ∗ ( X ),and is at best ln(2)2( p − . Both Theorem 1.3 and Theorem 1.4 will be proven by means of Lemma 3.3. Our first goal is toestablish this lemma.Let L be the free Lie algebra over Q on basis elements x , . . . , x n . Write L k for the subset of L consisting of the basic products of the x i of weight k , in the sense of [Hil55], where the basic productsof weight 1 are taken to be the x i , ordered by x < x < · · · < x n . The union L = S ∞ k =1 L k is a vectorspace basis for L (see for example [Ser06, Theorem 5.3], but note that what we call basic products,Serre calls a Hall basis ).Let µ : N −→ {− , , } be the M¨obius inversion function, defined by µ ( s ) = s = 10 s > − ℓ s > ℓ distinct primes.The Witt Formula W n ( k ) is then defined by W n ( k ) = 1 k X d | k µ ( d ) n kd . Theorem 3.1. [Hil55, Theorem 3.3] Let L be the free Lie algebra over Q on basis elements x , . . . , x n .Then | L k | = W n ( k ). Lemma 3.2. [BO15, Introduction] The ratio W n ( k ) k n k tends to 1 as k tends to ∞ . (cid:3) In particular, this implies that for n ≥
2, the Witt function W n ( k ) grows exponentially in k . Itshould follow that if the number of p -torsion summands in L i ≤ k π i ( Y ) exceeds W ( k ), then Y is p -hyperbolic. The following lemma makes a slightly generalised form of this idea precise. Lemma 3.3.
Let Y be a space. Suppose that there exist a, b ∈ N such that the number of p -torsionsummands (respectively, Z /p r -summands) in L i ≤ ak + b π i ( Y ) exceeds W ( k ), for all k large enough.Then Y is p -hyperbolic (respectively, Z /p r -hyperbolic).6 roof. The proofs for p - and Z /p r -hyperbolicity are identical, so we give only the former. Reframing thehypothesis in terms of the sequence T m of Definition 1.1, we are assuming precisely that T ak + b > W ( k )for sufficiently large k . We then have thatlim inf m ln( T m ) m = lim inf k ln( T ak + b ) ak + b ≥ lim inf k ln( W ( k )) ak + b . It then follows from Lemma 3.2 that if 1 > ε >
0, once k is large enough, we have W ( k ) > (1 − ε ) 1 k k . This implies that lim inf k ln( W ( k )) ak + b ≥ lim inf k ln((1 − ε ) k k ) ak + b , and since this holds for all ε > m ln( T m ) m ≥ lim inf k ln( W ( k )) ak + b ≥ lim inf k ln( k k ) ak + b = lim inf k ln( k ) + k ln(2) ak + b = ln(2) a , which is greater than zero, as required. We write π Sj for the j -th stable stem in the homotopy groups of spheres, that is π Sj := lim n →∞ π n + j ( S n ) . The proof of Theorem 1.3, depends on having, for each p and r , some j such that π Sj contains a Z /p r -summand. The purpose of this subsection is to show that the existence of such a j follows fromexisting work of Adams and others. Lemma 3.4.
For any prime p and any r ∈ N , there exists j such that Z /p r is a direct summand in π Sj . That is, for a fixed choice of such a j , Z /p r is a direct summand in π n + j ( S n ) for all n ≥ j + 2. Proof.
We write ν p ( s ) for the largest power of p dividing the integer s .CASE 1 ( p odd): Set t := p r − ( p − p −
1) is even, j := 4 t − π Sj contains a direct summand isomorphic to Z /m (2 t ), for a function m which Adams defines.By decomposing this subgroup into direct summands of prime power order, it suffices to show that ν p ( m (2 t )) = r .The discussion after Theorem 2.5 in [Ada65] gives that since t ≡ p − ν p ( m (2 t )) = 1 + ν p (2 t ) . Now, ν p (2 t ) is equal to ( r − t , so ν p ( m (2 t )) = r , as required.CASE 2 ( p = 2, r ≥ t := 2 r − , and set j := 4 t −
1. From Theorem 1.5, and the discussionfollowing Theorem 1.6 in [Ada66], π Sj has a direct summand isomorphic to Z /m (2 t ), regardless ofwhether j is congruent to 3 or 7 mod 8. Again, referring to the discussion after Theorem 2.5 of[Ada65], we see that ν ( m (2 t )) = 2 + ν (2 t ) = 3 + ν ( t ) = r, as required.CASE 3 ( p r = 2 and p r = 4): It is known from [Fre38] that π S ∼ = Z /
2, and from [BMT70] that π S ∼ = Z / ⊕ ( Z / . 7 Proof of Theorem 1.3
In this section we prove Theorem 1.3, which says that the wedge of two spheres is Z /p r -hyperbolic forall p and r . We also prove Corollary 2.9, which extracts some simple quantitative information fromthe proof of Theorem 1.3. We first record the following simple observation. Remark . Let k , . . . , k n and q , . . . q n be non-negative integers. Suppose that q ≤ q ≤ · · · ≤ q n ,and let k = P ni =1 k i . Then kq ≤ n X i =1 k i q i ≤ kq n . Proof of Theorem 1.3.
Assume without loss of generality that q ≤ q . By Lemma 3.3 It suffices toshow that there exist constants a and b such that the number of Z /p r -summands in M i ≤ ak + b π i ( S q +1 ∨ S q +1 )exceeds W ( k ), for k large enough.We first apply the Hilton-Milnor Theorem. Since we are dealing with spheres, we need only theoriginal form, due to Hilton in [Hil55]:Ω( S q +1 ∨ S q +1 ) ≃ ΩΣ( S q ∨ S q ) ≃ Ω Y B ∈ L S k q + k q +1 , where, as in Section 3, L = S ∞ k =1 L k is Hilton’s ‘basic product’ basis for L , the free Lie Algebraover Q on two generators x and x , and k i is the number of occurrences of the generator x i in thebracket B . Recall also from Section 3 that the weight k of a bracket B is equal to k + k , and thatthe cardinality of L k is given by the Witt formula W ( k ) by Theorem 3.1.For fixed k ∈ N , consider the factor in the above product corresponding to L k ⊂ L : F k := Ω Y B ∈ L k S k q + k q +1 . The associated subgroup of π ∗ ( S q +1 ∨ S q +1 ), M B ∈ L k π ∗ ( S k q + k q +1 ) , is a direct summand.We will first find a Z /p r -summand in the homotopy groups of each of the spheres appearing in F k .Since q ≤ q , Remark 4.1 applies, and we may lower bound the dimensions of spheres appearing in F k by k q + k q + 1 ≥ kq + 1. By Lemma 3.4, there exists j ∈ N such that π j + ℓ ( S ℓ ) has a directsummand Z /p r for ℓ ≥ j +2. Therefore, if k is large enough that kq ≥ j +1, then k q + k q +1 ≥ j +2- that is, the j -th stem is stable on all of the spheres occurring in F k . Thus, for k large enough, thereis a Z /p r summand in π j + k q + k q +1 ( S k q + k q +1 ) whenever k + k = k .We now upper bound the dimension of the homotopy groups in which these summands appear.Since q ≤ q we have by Remark 4.1 that j + k q + k q +1 ≤ kq +1+ j , so each of the Z /p r -summandswe have identified is a distinct direct summand in M i ≤ kq +1+ j M B ∈ L k π i ( S k q + k q +1 ) , hence in M i ≤ kq +1+ j π i ( S q +1 ∨ S q +1 ) .
8e have identified one such summand for each B ∈ L k , so the number of Z /p r -summands in L i ≤ kq +1+ j π i ( S q +1 ∨ S q +1 ) is at least | L k | = W ( k ). Thus, taking a = q and b = 1 + j inLemma 3.3 suffices. Proof of Corollary 2.9.
The last line of the proof of Lemma 3.3 shows that lim inf m ln t m m > ln 2 a . Thelast line of the proof of Theorem 1.3 implies that a may be taken to be q , under the assumption that q ≤ q , which implies the result. K -theory and K -homology of ΩΣ X The remainder of this paper proves Theorem 1.4. Sections 5 and 6 assemble necessary background,which we will use in Section 7 to prove the result.When studying the homotopy groups of a suspension Σ X , as in Theorem 1.4, the following approachis natural. Since π ∗ (Σ X ) ∼ = π ∗− (ΩΣ X ), we may instead study ΩΣ X . This is useful because ΩΣ X iswell understood homologically via the Bott-Samelson theorem, which decomposes its homology as thetensor algebra on e H ∗ ( X ). Because we will need to use Adams’ e -invariant, which is defined in termsof K -theory, we wish to replace ordinary homology with K -homology.The purpose of Section 5 is to record the version of the Bott-Samelson theorem which applies to(torsion-free) K -homology, along with a universal coefficient theorem for passing between K -theoryand K -homology. All of the material here is already known (in particular much of it is in [Sel83]) soits summary here is for convenience and clarity.Our conventions on definition of e K ∗ ( X ) are those of [AH61]. In particular, we define e K − ( X ) := e K (Σ X ), and set e K ∗ ( X ) := e K ( X ) ⊕ e K − ( X ). We regard e K ∗ ( X ) and e K ∗ ( X ) as being Z / e K ∗ ( X ) is a Z / K -theory and K -homology modulo the torsion subgroup. For a space X , write e K TF ∗ ( X ) and e K ∗ TF ( X ) for the quotients of the reduced K -homology and K -theory of X bytheir torsion subgroups. The same convention applies in the unreduced case. The universal coefficient theorem for K -theory first appears in some unpublished lecture notes ofAnderson [And], and is first published by Yosimura [Yos75]. Theorem 5.1 (Universal coefficient theorem) . For any CW -complex X and each integer n there is ashort exact sequence 0 → Ext( K n − ( X ) , Z ) → K n ( X ) → Hom( K n ( X ) , Z ) → . In the torsion-free case, the universal coefficient theorem simplifies as follows.
Corollary 5.2.
Let Y be a finite CW -complex. Then we have a natural isomorphism which descendsfrom the second map in the theorem above, K n TF ( Y ) ∼ = −→ Hom( K TF n ( Y ) , Z ) . There is an analogous isomorphism for the reduced theories: e K n TF ( Y ) ∼ = −→ Hom( e K TF n ( Y ) , Z ) . Proof.
Firstly, since Y is assumed to be a finite complex, the group K n − ( Y ) is finitely generatedabelian, so Ext( K n − ( Y ) , Z ) is torsion. Secondly, for any group G , Hom( G, Z ) is torsion-free. Together,these observations show that the exact sequence of Theorem 5.1 has first term torsion and last termtorsion-free. That means that it induces an isomorphism K n TF ( Y ) ∼ = −→ Hom( K n ( Y ) , Z ). Any homo-morphism K n ( Y ) → Z is zero on the torsion subgroup of K n ( Y ), so the injection Hom( K TF n ( Y ) , Z ) ֒ −→ K n ( Y ) , Z ) is an isomorphism, and the unreduced result follows. The reduced statement followsimmediately from the unreduced one.Selick [Sel83] deduces the following from work of Atiyah [Ati62], Mislin [Mis71] and Adams [Ada69]. Theorem 5.3 (K¨unneth theorem for K-homology) . Let X and Y be of the homotopy type of finitecomplexes. Then there is an isomorphism of Z / Z -modules: e K TF ∗ ( X ∧ Y ) ∼ = e K TF ∗ ( X ) ⊗ e K TF ∗ ( Y ) . Remark . It follows immediately from Corollary 5.2 (and knowledge of e K ∗ ( S q )) that e K TF ∗ ( S q ) ∼ = Z .We write ξ q for the generator of e K TF ∗ ( S q ). By the K¨unneth Theorem 5.3, we may choose the ξ q sothat ξ n ⊗ ξ m is identified with ξ n + m under the homeomorphism S n ∧ S m ∼ = S n + m .In the case of K -theory, the analogous result follows directly from [Ada69]. Theorem 5.5 (K¨unneth theorem for K-theory) . Let X and Y be of the homotopy type of finitecomplexes. Then the external product on K -theory defines an isomorphism of Z / e K ∗ TF ( X ) ⊗ e K ∗ TF ( Y ) ∼ = −→ e K ∗ TF ( X ∧ Y ) . For a space X , let X s denote the product of s copies of X . Let ∼ be the relation on X s defined by( x , . . . , x i − , ∗ , x i +1 , x i +2 , . . . x s ) ∼ ( x , . . . , x i − , x i +1 , ∗ , x i +2 , . . . x s ) . Let J s ( X ) be the space X s (cid:30) ∼ . There is a natural inclusion J s ( X ) ֒ −→ J s +1 ( X )( x , . . . , x s ) ( x , . . . , x s , ∗ ) . The
James construction JX is defined to be the colimit of the diagram consisting of the spaces J s ( X ) and the above inclusions. Write i s : J s ( X ) → JX for the map associated to the colimit. Noticethat JX carries a product given by concatenation, which makes it into the free topological monoid on X , and that a topological monoid is in particular an H -space.Let X ∧ i denote the smash product of i copies of X , and let η : X → ΩΣ X be the unit of theadjunction Σ ⊣ Ω. Explicitly, η ( x ) = ( t
7→ h x, t i ∈ Σ X ). Theorem 5.6. [Jam55]1. There is a homotopy equivalence JX ≃ −→ ΩΣ X which respects the H -space structures and iden-tifies i with η .2. There is a homotopy equivalence W ∞ i =1 Σ X ∧ i ≃ −→ Σ JX which restricts to a homotopy equivalence W si =1 Σ X ∧ i ≃ −→ Σ J s ( X ) for each s ∈ N . Lemma 5.7. [Sel83, Lemma 7] Let X have the homotopy type of an ( r − CW -complex.1. ( i s ) ∗ : π N ( J s ( X )) → π N ( JX ) is an isomorphism for N < r ( s + 1) − x ∈ π N ( J s ( X )) for any N . If Σ x is nontrivial then ( i s ) ∗ ( x ) is also nontrivial. Proof.
The first part follows by cellular approximation from the observation that J s ( X ) contains the( r ( s + 1) − JX . The second part follows from the observation that Σ i s has a retractionby Theorem 5.6. 10or spaces X and Y , let X ∗ Y denote the join , which we define to be the homotopy pushout ofthe projections X × Y → X and X × Y → Y . The join is naturally a quotient of X × I × Y , where I denotes the unit interval. Following the treatment in [Ark11], let C denote the subspace of X ∗ Y consisting of points of the form ( x, t, ∗ ), for t ∈ I and x ∈ X , and let C be the subspace consistingof points of the form ( ∗ , t, y ). The subspace C ∪ C ∼ = CX ∪ CY is contractible, so the quotient map q : X ∗ Y → X ∗ Y (cid:30) C ∪ C is a homotopy equivalence. The quotient X ∗ Y (cid:30) C ∪ C is homeomorphicto Σ X ∧ Y . The suspended product Σ( X × Y ) is also a quotient of X × I × Y , and this quotient liesbetween X ∗ Y and X ∗ Y (cid:30) C ∪ C .This gives a factorization of q as X ∗ Y → Σ( X × Y ) → Σ( X ∧ Y ). Let q − denote any choiceof homotopy inverse to q ; all possible choices are homotopic. We may form a new map δ X,Y as thecomposite Σ( X ∧ Y ) q − −−→ X ∗ Y → Σ( X × Y ). It is automatic that δ X,Y splits the quotient map π : Σ( X × Y ) → Σ X ∧ Y . The homotopy class of δ X,Y is well-defined, and we will call δ X,Y the canonical splitting of π . Note that δ X,Y is natural in maps of spaces in the sense that given f : A → X and g : B → Y we obtain a commutative diagramΣ A ∧ B δ A,B / / Σ( f ∧ g ) (cid:15) (cid:15) Σ( A × B ) Σ( f × g ) (cid:15) (cid:15) Σ X ∧ Y δ X,Y / / Σ( X × Y ) . For s ≥
3, consider the quotient map Σ X s → Σ X ∧ s . We define the canonical splitting of thisquotient to be the composite of canonical splittingsΣ X ∧ s → Σ( X × X ) ∧ X ∧ ( s − → Σ(( X × X ) × X ) ∧ X ∧ ( s − → · · · → Σ X s . Of course, we chose an order of multiplication here. This canonical splitting is natural as before.
Definition 5.8.
For a Z -graded (respectively Z / M , let T ( M ) = L ∞ k =1 M ⊗ k denotethe tensor algebra on M . The product is given by concatenation. We refer to M ⊗ k as the weight k component of the tensor algebra T ( M ). We define a Z -grading (respectively Z / T ( M )by setting | x ⊗ x ⊗ · · · ⊗ x k | = P ki =1 | x i | . Definition 5.9.
For a space Y , let σ : e K TF ∗ ( Y ) ∼ = −→ e K TF ∗ +1 (Σ Y ) be the suspension isomorphism. Let ϕ : e K TF ∗ (Σ Y ) → e K TF ∗ (Σ Y ) be a homomorphism of graded groups, not necessarily induced by a mapof spaces. We call the composite σ − ◦ ϕ ◦ σ the desuspension of θ , denoting it by S − ϕ .Write m s : (ΩΣ X ) s → ΩΣ X for the map given by iteratively performing the standard loop multi-plication on ΩΣ X in any choice of order. Up to homotopy, m s is independent of this choice of order,since ΩΣ X is homotopy associative.Theorem 5.6 gives the existence of a homotopy equivalence Γ : W ∞ i =1 Σ X ∧ i → ΣΩΣ X . There aremany choices of Γ, up to homotopy. The next lemma asserts that Γ can be chosen in a way whichsuits our purpose. Selick [Sel83] describes the composite W ∞ i =1 Σ X ∧ i Γ −→ ΣΩΣ X ≃ −→ Σ JX of Γ with thehomotopy equivalence of Theorem 5.6 (1). This immediately implies the following description of Γ. Lemma 5.10. [Sel83] Let X be a finite CW -complex. The homotopy equivalence Γ : W ∞ i =1 Σ X ∧ i → ΣΩΣ X may be chosen such that:1. S − (Γ ∗ ) : T ( e K TF ∗ ( X )) ∼ = −→ K TF ∗ (ΩΣ X ) is an isomorphism of algebras;2. the restriction of Γ to Σ X ∧ s is homotopic to the compositeΣ X ∧ s → Σ X s Σ( η ) s −−−→ Σ(ΩΣ X ) s Σ m s −−−→ ΣΩΣ X, where the unlabelled arrow is the canonical splitting.11he description of the map Γ in Lemma 5.10 has the following consequence. For a space Y , letev : ΣΩ Y → Y be the evaluation map, which may be described explicitly by ev( h γ, t i ) = γ ( t ) for γ ∈ Ω Y . Lemma 5.11.
Let Γ : W ∞ i =1 Σ X ∧ i → ΣΩΣ X be the homotopy equivalence of Lemma 5.10. Thecomposite ev ◦ Γ is homotopic to the projection onto the first wedge summand.
Proof.
Let ι s : Σ X ∧ s → W ∞ i =1 Σ X ∧ i be the inclusion of the s -th wedge summand. We must show thatev ◦ Γ ◦ ι s ≃ ( Σ X if s = 1, and ∗ otherwise.The following diagram commutes up to homotopyΣ X Σ η / / Σ X $ $ ■■■■■■■■■ ΣΩΣ X ev (cid:15) (cid:15) Σ X. By Lemma 5.10, Σ η = Γ ◦ ι , which implies the s = 1 statement.Now let s ≥
2. Ganea [Gan65, Theorems 1.1 and 1.4] shows that the homotopy fibre of ev is givenby Σ(ΩΣ X ∧ ΩΣ X ) v −→ ΣΩΣ X ev −→ Σ X, where the map v is equal to the compositeΣ(ΩΣ X ∧ ΩΣ X ) → Σ(ΩΣ X × ΩΣ X ) Σ m −−−→ ΣΩΣ X of Σ m with the canonical splitting. We will show that Γ ◦ ι s factors through v , and hence com-poses trivially with ev. Consider the following diagram, where the unlabelled arrows are all canonicalsplittings: Σ(ΩΣ X ∧ ΩΣ X ) / / v , , Σ(ΩΣ X × ΩΣ X ) Σ m / / ΣΩΣ X Σ(ΩΣ X ) ∧ s / / Σ((ΩΣ X ) ( s − ∧ ΩΣ X ) Σ( m ( s − ∧ O O / / Σ(ΩΣ X ) s Σ( m ( s − × O O Σ m s / / ΣΩΣ X Σ X ∧ s Σ η ∧ s O O / / Σ( X ( s − ∧ X ) Σ( η ( s − ∧ η ) O O / / Σ X s . Σ η s O O The composite along the bottom of the diagram is Γ ◦ ι s , so to obtain the desired factorization ofΓ ◦ ι s through v , it suffices to show that the diagram commutes up to homotopy.The top right square commutes because m ◦ m ( s − ≃ m s , by homotopy associativity of the H -space ΩΣ X . The remaining three squares commute by naturality of our canonical splitting. Thiscompletes the proof.Let ρ k be the projection T ( e K TF ∗ ( X )) → e K TF ∗ ( X ) ⊗ k . The next corollary is immediate from Lemma5.11.
Corollary 5.12. S − (ev ∗ ◦ Γ ∗ ) = ρ : T ( e K TF ∗ ( X )) → e K TF ∗ ( X ) . .3 Primitives and commutators It follows from the K¨unneth Theorem (Theorem 5.3), and the fact that Σ( Y × Y ) ≃ Σ Y ∨ Σ Y ∨ Σ( Y ∧ Y ) , that K TF ∗ ( Y × Y ) ∼ = K TF ∗ ( Y ) ⊗ K TF ∗ ( Y ). We may therefore make the following definition. A class y ∈ e K TF ∗ ( Y ) is called primitive if ∆ ∗ ( y ) = y ⊗ ⊗ y , where ∆ : Y → Y × Y is the diagonal, definedby ∆( y ) = ( y, y ).The comultiplication Y → Y ∨ Y on a co- H -space Y is a factorization of ∆ via the inclusion Y ∨ Y ֒ −→ Y × Y . From this point of view, the following lemma is immediate. Lemma 5.13. If Y is a co- H -space, then all elements in e K TF ∗ ( Y ) are primitive.If Y is an H -group, then the multiplication m : Y × Y → Y induces a map e K TF ∗ ( Y ) ⊗ e K TF ∗ ( Y ) → e K TF ∗ ( Y ). We will denote this map by juxtaposition, so that m ∗ ( y ⊗ y ) = y y . Furthermore, thecommutator Y × Y → Y descends to a map c : Y ∧ Y → Y . Expanding the definition of the commutatorin terms of the K -homology K¨unneth Theorem (Theorem 5.3) gives the following lemma. Lemma 5.14.
Let Y be an H -group, and let c : Y ∧ Y → Y be the commutator. If y and y ∈ e K TF ∗ ( Y )are primitive, then c ∗ ( y ⊗ y ) = y y − ( − | y || y | y y . ψ -modules In [Ada66], Adams defines an abelian category which we will follow Selick [Sel83] in calling ψ -modules.The e -invariant, which is our central tool, is defined by Adams in terms of ψ -modules. The purposeof this section is to record results about ψ -modules for later use.A ψ -module consists of an abelian group M , with homomorphisms ψ ℓ : M → M ( ℓ ∈ Z )satisfying the axioms of [Ada66, Section 6]. If X is a space then the group e K ( X ), together with itsAdams operations, is a ψ -module. Since we defined e K − ( X ) by setting e K − ( X ) = e K (Σ X ), it toohas the structure of a ψ -module. Maps of spaces induce maps of ψ -modules. The Adams operation ψ ℓ on e K ( S n ) is multiplication by ℓ n , so in particular Adams operations do not commute with the Bottisomorphism.For graded ψ -modules M and N we will write Hom ψ -Mod ( M, N ) for the abelian group consisting ofgraded ψ -module homomorphisms. The unadorned notation Hom( M, N ) will mean homomorphismsof the underlying graded abelian groups.
Lemma 6.1.
Let M and N be ψ -modules, with N torsion-free. The inclusion of Z -modulesHom ψ -Mod ( M, N ) ֒ −→ Hom(
M, N ) is an injection onto a summand.
Proof.
Let ϕ : M → N be a homomorphism of underlying Z -modules. If, for some k ∈ Z \ { } , k · ϕ isa ψ -module homomorphism, then, since N is torsion-free, ϕ is also a ψ -module homomorphism. Thisimplies that coker(Hom ψ -Mod ( M, N ) ֒ −→ Hom(
M, N )) is torsion-free, which implies the result.For the avoidance of doubt, by the e -invariant we will always mean what Adams calls the complex e -invariant e C [Ada65; Ada66]. Definition 6.2 (Adams’ e -invariant) . Suppose that f : X → Y induces the trivial map on e K ∗ . Thenthe cofibre sequence of f gives a short exact sequence of ψ -modules0 ← e K ( Y ) ← e K ( C f ) ← e K (Σ X ) ← . The e -invariant of f is the element of Ext ψ -Mod ( e K (Σ X ) , e K ( Y )) represented by this exact se-quence. 13he e -invariant does not commute with the Bott isomorphism, but the interaction between the Bottisomorphism and the Adams operations is easy to describe, as follows. Let ψ ℓY be the homomorphism ψ ℓ : e K ( Y ) → e K ( Y ). Then, modulo the Bott isomorphism, we have ψ ℓ Σ X = ℓ · ψ ℓX . That is ‘upondouble suspending, the Adams operations gain a factor ℓ ’. In terms of the e -invariant, all we need toknow is the following. Lemma 6.3. [Ada66, Proposition 3.4b)] There is a homomorphism T : Ext ψ -Mod ( e K (Σ X ) , e K ( Y )) → Ext ψ -Mod ( e K (Σ X ) , e K (Σ Y )), such that T ( e ( f )) = e (Σ f ).We will be concerned only with the e -invariants of maps whose domain is a sphere. One of the two K -groups of a sphere vanishes, in the dimension matching the parity of the sphere, but the e -invariant,as defined above, lives only in K . In order to detect maps regardless of the parity of the sphere onwhich they are defined, we will need to keep track of the e -invariants of f and Σ f , so we will use thefollowing modified e -invariant. Definition 6.4 (Double e -invariant) . LetExt ψ -Mod ( e K ∗ (Σ X ) , e K ∗ ( Y )) := Ext ψ -Mod ( e K (Σ X ) , e K ( Y )) ⊕ Ext ψ -Mod ( e K − (Σ X ) , e K − ( Y )) . Suppose that f : X → Y induces the trivial map on e K ∗ . Then the double e -invariant of f is e ( f ) =( e ( f ) , e (Σ f )) ∈ Ext ψ -Mod ( e K ∗ (Σ X ) , e K ∗ ( Y )) . Pullback of an extension along a homomorphism defines a mapHom ψ -Mod ( M, B ) ⊗ Ext ψ -Mod ( A, B ) → Ext ψ -Mod ( A, M ) . If g : Y → Z then e ( g ◦ f ) is represented by the pullback of e ( f ) and g ∗ : e K ( Z ) → e K ( Y ) [Ada66,Proposition 3.2 b)]. To describe e ( g ◦ f ) we need only apply this result degree-wise, as follows. Forconvenience, we write g ∗ · e ( f ) for the pullback of g ∗ and e ( f ). Define the map θ ( f ) : Hom ψ -Mod ( e K ( Z ) , e K ( Y )) → Ext ψ -Mod ( e K ( Z ) , e K (Σ X )) θ ( f )( x ) = x · e ( f ) . Likewise, define θ − ( f ) : Hom ψ -Mod ( e K − ( Z ) , e K − ( Y )) → Ext ψ -Mod ( e K − ( Z ) , e K − (Σ X )) θ − ( f )( x ) = x · e (Σ f ) . Combining these, let θ ( f ) : Hom ψ -Mod ( e K ∗ ( Z ) , e K ∗ ( Y )) → Ext ψ -Mod ( e K ∗ ( Z ) , e K ∗ (Σ X ))be the direct sum θ ( f ) ⊕ θ − ( f ). These definitions, together with Adams’ above result, give thefollowing lemma. Lemma 6.5.
For maps f : X → Y and g : Y → Z , the following diagram commutes:[ Y, Z ] f ∗ / / deg (cid:15) (cid:15) [ X, Z ] e (cid:15) (cid:15) Hom ψ -Mod ( e K ∗ ( Z ) , e K ∗ ( Y )) θ ( f ) / / Ext ψ -Mod ( e K ∗ ( Z ) , e K ∗ (Σ X )) . Following [Sel83], write Z ( n ) for the ψ -module e K ( S n ). Explicitly, Z ( n ) has underlying abeliangroup Z , and ψ ℓ acts by multiplication by ℓ n . It follows that e K − ( S n +1 ) := e K ( S n +2 ) ∼ = Z ( n + 1).14 emma 6.6. [Ada66, Proposition 7.8, 7.9] If n < m then Ext ψ -Mod ( Z ( n ) , Z ( m )) injects into Q(cid:30)Z . The e -invariant of a map f : S m − → S n may therefore be regarded as an element of Q(cid:30)Z . Furthermore,the value e ( f ) in Q(cid:30)Z satisfies e (Σ f ) = e ( f ), so in particular, when f is a map between spheres, e ( f )depends only on the stable homotopy class of f .The following theorem is the main technical component of Selick’s paper [Sel83]. Theorem 6.7. [Sel83, Theorem 6] Let f ′ : S m − → S n be such that p t − e ( f ′ ) = 0 in Q(cid:30)Z , for p prime and some t ∈ N . Let Y have the homotopy type of a finite CW -complex and let g : S n → Y be such that Im( g ∗ : e K ( Y ) → e K ( S n )) contains up s e K ( S n ), for s ∈ N and u prime to p . If s < t ,and there exists some ℓ ∈ N for which ψ ℓ ⊗ Q : e K ( Y ) ⊗ Q → e K ( Y ) ⊗ Q does not have ℓ m as an eigenvalue, then e ( g ◦ f ′ ) = 0.The following theorem of Gray [Gra69] will provide the map f ′ for Theorem 6.7. Specifically, thistheorem provides a linearly spaced family of stems, each of which has a stable p -torsion class which isborn on S and detected by the e -invariant. Theorem 6.8. [Gra69, Corollary of Theorem 6.2] Let p be an odd prime and let j ∈ N . Then thereexists a class f p,j ∈ π j ( p − ( S ) with e ( f p,j ) = − p ∈ Q(cid:30)Z .The corresponding 2-primary result is as follows. Adams [Ada66, Theorem 1.5 and Proposition7.14] shows that, for j >
0, the (8 j +3)-rd stem contains a direct summand whose 2-primary componenthas order 8, and that on this component the e -invariant is a surjection onto Z /
4. The sphere of originof the classes in this component was deduced by Curtis in [Cur69].
Theorem 6.9. [Ada66; Cur69] Let j ∈ N . Then there exists a class f ,j ∈ π j +6 ( S ) of order 4, with e ( f ,j ) = − ∈ Q(cid:30)Z . Having assembled preliminaries in Sections 5 and 6, we can begin to work towards the proof of Theorem1.4. Our approach is as follows. From the data of Theorem 1.4, we will construct a commutativediagram of (roughly) the following form, where B is a set and the other objects are Z -modules. B k (cid:15) (cid:15) / / π ∗ (ΩΣ X ) (cid:15) (cid:15) I k / / Ext ψ -Mod (ΩΣ X ) . We will argue that • The image of the top map consists of classes of order dividing p . • The image of the left vertical map generates a submodule isomorphic to the weight k componentof the free graded Lie algebra over Z /p on two generators. • The bottom map is injective.Together, these facts imply that there is a submodule of π ∗ (ΩΣ X ) ∼ = π ∗ +1 (Σ X ), consisting ofclasses of order dividing p , and surjecting onto a module isomorphic to the weight k component of thefree graded Lie algebra over Z /p on two generators. This submodule (which is necessarily a Z /p -vector15pace) must therefore have dimension at least W ( k ) (Theorem 7.5), which will imply that Σ X is p -hyperbolic (Lemma 3.3).The diagram will be obtained by juxtaposing three squares. Subsections 7.1, 7.2, and 7.3 eachconstruct one of these squares. In Subsection 7.4 we put them together and prove Theorem 1.4.Roughly speaking, the top map of the diagram should be thought of as first taking a family of Samelsonproducts and then pulling them back along some suitable map f coming from Gray’s work (Theorem6.8). The vertical maps should be thought of as passing from maps of spaces to K -theoretic invariants,and the bottom map (therefore) should be thought of as tracking the effect of the top map on theseinvariants.Because of the need to work with a finite CW -complex in Selick’s Theorem (Theorem 6.7) we willrestrict the right hand side of the diagram to instead refer to some finite skeleton J s ( X ) of the Jamesconstruction. K -homology Let R be a commutative ring with unit. We take a graded Lie algebra over R to be defined as in[Nei10]. For a non-negatively graded R -module V , let L ( V ) denote the free graded Lie algebra [Nei10,Section 8.5]. Write L k ( V ) for the submodule of L ( V ) generated by the brackets of length k in theelements of V . We will call L k ( V ) the weight k component of L ( V ). Note that this convention differsfrom Neisendorfer’s - he writes L ( V ) k for the weight k component. Definition 7.1.
Let Y be an H -group, and let c : Y ∧ Y → Y be the commutator of Lemma 5.14.Let α ∈ π N ( Y ), and let β ∈ π M ( Y ). The Samelson product of α and β , written h α, β i ∈ π N + M ( Y ), isthe composite h α, β i : S N + M ∼ = S N ∧ S M α ∧ β −−−→ Y ∧ Y c −→ Y. Samelson products fail to make π ∗ ( Y ) into a graded Lie algebra over Z [Nei13, Section 7], but theydo define the structure of a graded magma. In fact, they define a sort of ‘pseudo-Lie algebra’ structure,since they are graded anticommutative and satisfy the graded Jacobi identity. One could define anappropriate notion of ‘free graded pseudo-Lie algebra’, and proceed as follows with that in place of themagma we will use, but we prefer the more lightweight approach.For a graded R -module V , let U ( V ) denote the graded set of homogeneous elements in V . Let B ( V ) be the free magma on U ( V ), where we write the product as a bracket [ x, y ]. We think of B asthe ‘set of brackets of homogeneous elements in V ’. Elements of B ( V ) are nonassociative words inthe elements of U ( V ), so we may define a grading on B ( V ) which extends the grading on U ( V ) via | [ x, y ] | = | x | + | y | . The weight of an element of B ( V ) is its word length. Write B N ( V ) for the subset ofelements in degree N , B k ( V ) for the subset of elements of weight k , and set B kN ( V ) = B k ( V ) ∩ B N ( V ).Let ν : A → ΩΣ X be a map. By the universal property of the free magma B ( π ∗ ( A )), there existsa map Φ πν : B ( π ∗ ( A )) → π ∗ (ΩΣ X )which extends ν ∗ and satisfies Φ πν ([ x, y ]) = h Φ πν ( x ) , Φ πν ( y ) i for all x, y ∈ B ( π ∗ ( A )).For a Z / Z -module V , we define a non-negatively graded Z -module Hom( e K TF ∗ ( S ∗ ) , V ), bysetting Hom( e K TF ∗ ( S ∗ ) , V ) N = ( Hom( e K TF ∗ ( S N ) , V ) if N > , and0 if N ≤ , where the homomorphisms are understood to respect the Z / e K ∗ and V .In the case that V = L is a Z / Z , Hom( e K TF ∗ ( S ∗ ) , L ) inherits a non-negatively graded Lie algebra structure as follows. Let the generators ξ N of e K TF ∗ ( S N ) be as in Remark5.4. Then the bracket [ f, g ] of f ∈ Hom( e K TF ∗ ( S N ) , L ) and g ∈ Hom( e K TF ∗ ( S M ) , L ) is the homomor-phism e K TF ∗ ( S M ) → L carrying ξ N + M to [ f ( ξ N ) , g ( ξ M )] ∈ L . The squaring operation is defined in the16ame way. Likewise, if V is a Z / Z , then Hom( e K TF ∗ ( S ∗ ) , V ) inheritsthe structure of a non-negatively graded associative algebra.Let ν : A → ΩΣ X be a map. There is a composition L ( e K TF ∗ ( A )) → T ( e K TF ∗ ( A )) → e K TF ∗ (ΩΣ X ) , where the first map is the natural map which is the identity on e K TF ∗ ( A ) and satisfies [ x, y ] xy − ( − | x || y | yx , and the second map is obtained by applying the universal property of the tensor algebrato ν ∗ . Let Φ Kν : Hom( e K TF ∗ ( S ∗ ) , L ( e K TF ∗ ( A ))) → Hom( e K TF ∗ ( S ∗ ) , e K TF ∗ (ΩΣ X ))be the pushforward along the above composite. It is then automatic that Φ Kν is a map of non-negativelygraded Lie algebras over Z , where the structures are defined as above.We write deg : π N ( Y ) → Hom( e K TF ∗ ( S N ) , e K TF ∗ ( Y )) for the map f f ∗ . Let deg ′ : B ( π ∗ ( A )) → Hom( e K TF ∗ ( S ∗ ) , L ( e K TF ∗ ( A ))) be the unique map which restricts to deg : π ∗ ( A ) → Hom( e K TF ∗ ( S ∗ ) , e K TF ∗ ( A )) ⊂ Hom( e K TF ∗ ( S ∗ ) , L ( e K TF ∗ ( A ))) and carries brackets to brackets. The above maps are related as follows. Lemma 7.2.
Let ν : A → ΩΣ X , for spaces A and X having the homotopy type of finite CW -complexes. The following diagram commutes: B ( π ∗ ( A )) Φ πν / / deg ′ (cid:15) (cid:15) π ∗ (ΩΣ X ) deg (cid:15) (cid:15) Hom( e K TF ∗ ( S ∗ ) , L ( e K TF ∗ ( A ))) Φ Kν / / Hom( e K TF ∗ ( S ∗ ) , e K TF ∗ (ΩΣ X )) . Proof.
By the universal property of the free magma B ( π ∗ ( A )), it suffices to show that the restrictionof the diagram to the weight 1 component B ( π ∗ ( A )) = π ∗ ( A ) commutes, and that all maps respectthe bracket operations. By definition, L ( e K TF ∗ ( A )) = e K TF ∗ ( A ). It then follows immediately from thedefinitions of Φ πν and Φ Kν that restricting the left hand side of the diagram to weight 1 componentsgives the diagram π ∗ ( A ) ν ∗ / / deg (cid:15) (cid:15) π ∗ (ΩΣ X ) deg (cid:15) (cid:15) Hom( e K TF ∗ ( S ∗ ) , e K TF ∗ ( A )) ν ∗ / / Hom( e K TF ∗ ( S ∗ ) , e K TF ∗ (ΩΣ X )) , which commutes, since it just expresses naturality of deg.It remains to show that all maps respect bracket operations. The maps Φ πν and deg ′ respect thebracket operations by definition, and Φ Kν respects bracket operations by construction. We thereforeonly need show that deg respects brackets. Let f ∈ π N (ΩΣ X ), and let g ∈ π M (ΩΣ X ). We must showthat deg( h f, g i ) is the commutator deg( f )deg( g ) − ( − NM deg( g )deg( f ) with respect to the algebraoperation on Hom( e K TF ∗ ( S ∗ ) , e K TF ∗ (ΩΣ X )).Since e K TF ∗ ( S N + M ) ∼ = Z , it suffices to show that the two homomorphisms agree on the generator ξ N + M (Remark 5.4). By Definition 7.1 and the K¨unneth Theorem (Theorem 5.3),deg( h f, g i )( ξ N + M ) = c ∗ ◦ ( f ∗ ⊗ g ∗ )( ξ N ⊗ ξ M ) = c ∗ ◦ ( f ∗ ( ξ N ) ⊗ g ∗ ( ξ M )) . Spheres of dimension at least 1 are co- H spaces, so by Lemma 5.13, ξ N and ξ M are primitive. Bynaturality of the diagonal f ∗ ( ξ N ) and g ∗ ( ξ M ) are still primitive, so by Lemma 5.14, c ∗ ◦ ( f ∗ ( ξ N ) ⊗ g ∗ ( ξ M )) = f ∗ ( ξ N ) g ∗ ( ξ M ) − ( − NM g ∗ ( ξ M ) f ∗ ( ξ N ) , which by definition of the multiplication on Hom( e K TF ∗ ( S ∗ ) , e K TF ∗ (ΩΣ X )) is the result of evaluatingdeg( f )deg( g ) − ( − NM deg( g )deg( f ) on ξ N + M , as required.17e now lift the previous result to J s ( X ), thereby producing the first square of the diagram promisedat the start of this section. Recall that we write i s : J s ( X ) → JX for the inclusion, and that byTheorem 5.6 we have a homotopy equivalence JX ≃ −→ ΩΣ X . We will abuse notation and also write i s for the composite J s ( X ) → JX ≃ −→ ΩΣ X . Corollary 7.3.
Let ν : A → ΩΣ X , for spaces A and X having the homotopy type of finite CW -complexes, with X ( r − r ≥
1. If
N, s ∈ N satisfy N < r ( s + 1) −
1, then ( i s ) ∗ : π N ( J s X ) → π N (ΩΣ X ) is an isomorphism and for each k ≤ s there exists a commutative diagram: B kN ( π ∗ ( A )) f Φ πν / / deg ′ (cid:15) (cid:15) π N ( J s X ) deg (cid:15) (cid:15) Hom( e K TF ∗ ( S N ) , L k ( e K TF ∗ ( A ))) g Φ Kν / / Hom( e K TF ∗ ( S N ) , e K TF ∗ ( J s X )) , with ( i s ) ∗ ◦ f Φ πν = Φ πν and Hom( e K TF ∗ ( S N ) , ( i s ) ∗ ) ◦ g Φ Kν = Φ Kν . Proof.
Consider the diagram of Lemma 7.2. Lemma 5.7 shows that ( i s ) ∗ is an isomorphism on π N ,so let f Φ πν be the unique map such that the condition ( i s ) ∗ ◦ f Φ πν = Φ πν holds. By Theorem 5.6 (2)and Lemma 5.10, the map ( i s ) ∗ : e K TF N ( J s ( X )) → e K TF N (ΩΣ X ) is the inclusion of the tensors of lengthat most s . Since k ≤ s , we may therefore define g Φ Kν to be the unique map such that the conditionHom( e K TF ∗ ( S N ) , ( i s ) ∗ ) ◦ g Φ Kν = Φ Kν holds. Commutativity then follows from Lemma 7.2 by naturalityof deg, since Hom( e K TF ∗ ( S N ) , ( i s ) ∗ ) is injective. Lemma 7.4.
Let V be a non-negatively- or Z / Z -module which is free and finitely generatedin each dimension. Then • L ( V ) and T ( V ) are free Z -modules in every dimension. • The natural map L ( V ) → T ( V ), [ x, y ] xy − ( − | x || y | yx is an injection onto a summand. Proof.
The non-negatively graded case is immediate from [Nei10], Proposition 8.3.1 and p282. For the Z / U from Z -graded modules to Z / Z -graded (Lie) algebras to Z / C from Z / Z -modules which puts V in any even dimension and V in any odd dimension. Both C and U respect freeness and split injections, and there are natural isomorphisms U T ( CV ) ∼ = T ( V ) and U L ( CV ) ∼ = L ( V ). This implies the Z / Theorem 7.5. [Hil55, Theorem 3.2, 3.3] Let V be a torsion-free Z - or Z / Z -module of totaldimension n . Then the total dimension of L k ( V ) is W n ( k ).Let R be a commutative ring with unit. Let M be an R -module, and as usual let T ( M ) denotethe tensor algebra on M . Let ι k : M ⊗ k → T ( M ) be the inclusion, and let ρ k : T ( M ) → M ⊗ k bethe projection. Let τ : T ( M ) → T ( M ) be the composite ι ◦ ρ . Given an R -algebra A , and a map ϕ : M → A , we write e ϕ for the induced map T ( M ) → A , that is, the unique map of algebras such that e ϕ ◦ ι = ϕ .Now, let M and N be R -modules, and let ϕ : M → T ( N ) be a map. In the proof of Theorem 7.7,we will wish to make a ‘leading terms’ style argument. This is made precise in the next Lemma, whichcompares e ϕ with ] τ ◦ ϕ . Informally, we think of ] τ ◦ ϕ as the ‘leading terms part’ of e ϕ . Lemma 7.6.
Let R be a commutative ring with unit. Let M and N be Z - or Z / R -modules.Let ι k : M ⊗ k → T ( M ) be the inclusion, let ρ k : T ( N ) → N ⊗ k be the projection, and let τ : T ( N ) → T ( N ) be as above. Let ϕ : M → T ( N ) be a map. Then ρ k ◦ e ϕ ◦ ι k = ρ k ◦ ] τ ◦ ϕ ◦ ι k .18 roof. It suffices to check equality on basic tensors. Let v ∈ M ⊗ k be a basic tensor, so that v = v ⊗ v ⊗ · · · ⊗ v k , for v i ∈ M . Then e ϕ ◦ ι k ( v ) = e ϕ ( v ⊗ v ⊗ · · · ⊗ v k ) = ϕ ( v ) ⊗ ϕ ( v ) ⊗ · · · ⊗ ϕ ( v k )= τ ( ϕ ( v )) ⊗ τ ( ϕ ( v )) ⊗ · · · ⊗ τ ( ϕ ( v k )) + terms of weight > k. Applying ρ k to both sides yields the result. Theorem 7.7.
Let F = Q or Z /p for p prime. Let ν : A → ΩΣ X , for spaces A and X having thehomotopy type of finite CW -complexes. Let ν : Σ A → Σ X be the adjoint of ν . If ν ∗ ⊗ F : e K TF ∗ (Σ A ) ⊗ F → e K TF ∗ (Σ X ) ⊗ F is an injection, thenΦ Kν ⊗ F : Hom( e K TF ∗ ( S ∗ ) , L ( e K TF ∗ ( A ))) ⊗ F → Hom( e K TF ∗ ( S ∗ ) , e K TF ∗ (ΩΣ X )) ⊗ F is also an injection. Remark . In the case where ν is a suspension Σ ζ , we have a diagramΩΣ A Ω ν / / ΩΣ XA η O O ν : : ✈✈✈✈✈✈✈✈✈✈ ζ / / X, η O O so in particular ν ∗ factors through the weight 1 component e K TF ∗ ( X ) of the tensor algebra decompositionof e K TF ∗ (ΩΣ X ). This dramatically simplifies the proof, removing the need for Lemma 7.6. In practicethis is not a reasonable assumption - for example, the map µ : S ∨ S → Σ C P of Example 2.5 (whichplays the role of ν ) does not desuspend. Proof.
In this proof, for a space Y , we will identify the algebras T ( e K TF ∗ ( Y )) and e K TF ∗ (ΩΣ Y ), omittingthe isomorphism S − Γ ∗ of Lemma 5.10. We defined Φ Kν to be the pushforward along a certain map L ( e K TF ∗ ( A )) → e K TF ∗ (ΩΣ X ). Call this map Φ Kν ′ . It suffices to prove that Φ Kν ′ ⊗ F is an injection.The triangle identities for the adjunction Σ ⊣ Ω give a commutative diagramΩΣ A Ω ν / / ΩΣ XA. η O O ν : : ✉✉✉✉✉✉✉✉✉ Since Φ Kν ′ is the unique map of Lie algebras extending ν , we have a commuting diagram T ( e K TF ∗ ( A )) ∼ = e K TF ∗ (ΩΣ A ) (Ω ν ) ∗ / / e K TF ∗ (ΩΣ X ) L ( e K TF ∗ ( A )) , ?(cid:31) O O Φ Kν ′ ❦❦❦❦❦❦❦❦❦❦❦❦❦❦ where we note that that by Lemma 7.4, the natural map L ( e K TF ∗ ( A )) → T ( e K TF ∗ ( A )) is an injectiononto a summand. It therefore suffices to show that (Ω ν ) ∗ ⊗ F is an injection.Let g ( ν ∗ ) denote the extension of ν ∗ to T ( e K TF ∗ ( A )), so that g ( ν ∗ ) = (Ω ν ) ∗ (modulo the isomorphism S − Γ ∗ , as above). Since ( ρ i ◦ g ( ν ∗ ) ◦ ι k ) = 0 for i < k , it further suffices to show that ( ρ k ◦ g ( ν ∗ ) ◦ ι k ) ⊗ F is an injection for each k . By Lemma 7.6, with M = e K TF ∗ ( A ) and N ′ = e K TF ∗ ( X ), we have that ρ k ◦ g ( ν ∗ ) ◦ ι k = ρ k ◦ ^ ( τ ◦ ν ∗ ) ◦ ι k . As previously, let ev : ΣΩ Y → Y denote the evaluation map. The following diagram commutes:19 A ν $ $ ■■■■■■■■■ Σ ν / / ΣΩΣ X ev (cid:15) (cid:15) Σ X. The hypothesis therefore implies that the composite (ev ∗ ◦ Σ ν ∗ ) ⊗ F is an injection. Desuspending,we have that ( S − ev ∗ ◦ ν ∗ ) ⊗ F is an injection. By Lemma 5.12,( ρ ◦ ν ∗ ) ⊗ F : e K TF ∗ ( A ) ⊗ F → e K TF ∗ ( X ) ⊗ F is an injection of F -vector spaces. Thus, the image ( ρ ◦ ν ∗ )( e K TF ∗ ( A )) ⊗ F ⊂ e K TF ∗ ( X ) ⊗ F is a directsummand. Thus, the extension ^ ( τ ◦ ν ∗ ) ⊗ F is an injection, and ^ ( τ ◦ ν ∗ ) e K TF ∗ ( A ) ⊗ k ⊂ e K TF ∗ ( X ) ⊗ k foreach k . This implies that ρ k ◦ ^ ( τ ◦ ν ∗ ) ◦ ι k is an injection for each k , as required.The following corollary, which lifts the injectivity back to J s ( X ), is immediate from Theorem 7.7and Lemma 7.3. Corollary 7.9.
Let F = Q or Z /p for p prime. Let ν : A → ΩΣ X , for spaces A and X having thehomotopy type of finite CW -complexes, with X ( r − r ≥
1. Suppose that
N, s, k ∈ N satisfy k ≤ s , so that the map g Φ Kν is as in Corollary 7.3. If ν ∗ ⊗ F : e K TF ∗ (Σ A ) ⊗ F → e K TF ∗ (Σ X ) ⊗ F is an injection, then g Φ Kν ⊗ F : Hom( e K TF ∗ ( S N ) , L k ( e K TF ∗ ( A ))) ⊗ F → Hom( e K TF ∗ ( S N ) , e K TF ∗ ( J s X )) ⊗ F is also an injection.We have now established all that we will need to know about this ‘first square’. Before we move on,we will prove that a space satisfying the hypotheses of Theorem 1.4 at any prime must be rationallyhyperbolic. We will do so as Corollary 7.12, but we first require two lemmas. The first is needed becausethe hypotheses of Theorem 1.4 and Corollary 7.12 are given in terms of surjections on K -theory, butthe machinery we have built so far deals with injections on K -homology. Lemma 7.10.
Let X be a space, and let F = Q or Z /p for p prime. Let µ : S q +1 ∨ S q +1 → Σ X bea map with q i ≥
1, such that the map e K ∗ (Σ X ) ⊗ F µ ∗ ⊗ F −−−→ e K ∗ ( S q +1 ∨ S q +1 ) ⊗ F ∼ = F ⊕ F is a surjection. Then µ ∗ ⊗ F : e K TF ∗ ( S q +1 ∨ S q +1 ) ⊗ F → e K TF ∗ (Σ X ) ⊗ F is an injection. Proof.
Because e K ∗ ( S q +1 ∨ S q +1 ) is torsion-free, the hypothesis implies that the map e K ∗ TF (Σ X ) ⊗ F µ ∗ ⊗ F −−−→ e K ∗ TF ( S q +1 ∨ S q +1 ) ⊗ F ∼ = F ⊕ F is also a surjection. Thus, there exist elements x and y in e K ∗ TF (Σ X ) ⊗ F with ( µ ∗ ⊗ F )( x ) and ( µ ∗ ⊗ F )( y )linearly independent in e K ∗ TF ( S q +1 ∨ S q +1 ) ⊗ F . Via the Universal Coefficient Theorem (Corollary5.2) we may regard x and y as elements of Hom( e K TF ∗ (Σ X ) , Z ) ⊗ F ∼ = Hom( e K TF ∗ (Σ X ) ⊗ F , F ), with x ◦ ( µ ∗ ⊗ F ) and y ◦ ( µ ∗ ⊗ F ) linearly independent. This implies that Im( µ ∗ ⊗ F ) has dimension at least2. Since e K TF ∗ ( S q +1 ∨ S q +1 ) (the domain of µ ∗ ) has dimension 2, it follows that µ ∗ ⊗ F is injective,as required. 20ome preamble to the second lemma is necessary. Let h : π ∗ ( A ) → e K TF ∗ ( A ) be the K -homologicalHurewicz map, which sends f ∈ π N ( A ) to f ∗ ( ξ N ) ∈ e K TF ∗ ( A ). As with deg ′ , let h ′ : B ( π ∗ ( A )) → L ( e K TF ∗ ( A )) be the unique map which restricts to h : π ∗ ( A ) → e K TF ∗ ( A ) ⊂ L ( e K TF ∗ ( A )) and respectsbrackets.Let M be a Z / Z -module. Let χ : Hom( e K TF ∗ ( S ∗ ) , M ) → M be the map which carries ϕ ∈ Hom( e K TF ∗ ( S N ) , M ) to ϕ ( ξ N ) ∈ M (Remark 5.4). If M = L is a Z / e K TF ∗ ( S ∗ ) , L ) that χ is a map of Liealgebras. Lemma 7.11.
For any space A , there is a commuting diagram B ( π ∗ ( A )) deg ′ / / h ′ (cid:15) (cid:15) Hom( e K TF ∗ ( S ∗ ) , L ( e K TF ∗ ( A ))) χ u u ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ L ( e K TF ∗ ( A )) . Proof.
Commutativity of the diagram π ∗ ( A ) deg / / h (cid:15) (cid:15) Hom( e K TF ∗ ( S ∗ ) , e K TF ∗ ( A )) χ v v ❧❧❧❧❧❧❧❧❧❧❧❧❧❧ e K TF ∗ ( A ) . follows from the definitions. Commutativity of the diagram from the lemma statement then followsfrom the universal property of B ( π ∗ ( A )), since χ respects brackets.We are now ready to prove that a space satisfying the hypotheses of Theorem 1.4 at any primemust be rationally hyperbolic. Corollary 7.12.
Let X be a path connected space having the homotopy type of a finite CW -complex.Suppose that there exists a map µ ∨ µ : S q +1 ∨ S q +1 → Σ X with q i ≥
1, such that the map e K ∗ (Σ X ) ⊗ Z /p ( µ ∨ µ ) ∗ ⊗ Z /p −−−−−−−−−→ e K ∗ ( S q +1 ∨ S q +1 ) ⊗ Z /p ∼ = Z /p ⊕ Z /p is a surjection for some prime p (not necessarily odd). Then Σ X is rationally hyperbolic. Proof.
Set µ = µ ∨ µ . By Lemma 7.10 we have that µ ∗ ⊗ Z /p is injective. The codomain of µ ∗ istorsion free, so µ ∗ ⊗ Q is also injective. Theorem 7.7 (with A = S q ∨ S q and ν = µ ) then impliesthat Hom( e K TF ∗ ( S ∗ ) , L ( e K TF ∗ ( S q ∨ S q ))) ⊗ Q injects into Hom( e K TF ∗ ( S ∗ ) , e K TF ∗ (ΩΣ X )) ⊗ Q via the mapΦ Kν ⊗ Q .The Hurewicz map h is a surjection π ∗ ( S q ∨ S q ) → e K TF ∗ ( S q ∨ S q ), so the submodule generatedby the image of h ′ : B ( π ∗ ( S q ∨ S q )) → L ( e K TF ∗ ( S q ∨ S q )) is precisely the submodule generated bythe image of h under the bracket operation. By Theorem 7.5, this submodule is certainly infinitedimensional, so by Lemma 7.11 the image of deg ′ is also infinite dimensional. Thus, the image of(Φ Kν ◦ deg ′ ) ⊗ Q is also infinite dimensional.By Lemma 7.2, the image of deg : π ∗ (ΩΣ X ) → Hom( e K TF ∗ ( S ∗ ) , e K TF ∗ (ΩΣ X )) contains the imageof Φ Kν ◦ deg ′ . Thus, π ∗ (ΩΣ X ) ⊗ Q ∼ = π ∗ +1 (Σ X ) ⊗ Q surjects onto an infinite dimensional rationalvector space, and hence also has infinite rational dimension, and thus Σ X is rationally hyperbolic, asrequired. 21 .2 Maps derived from the universal coefficient isomorphism In this subsection we will build the second square of our diagram. This square is really just the UniversalCoefficient theorem (Corollary 5.2) in a different form. We will write deg for both K -homological and K -theoretic degree. Lemma 7.13.
Let Y be a space having the homotopy type of a finite CW -complex. There exists anisomorphism U making the following diagram commute. π N ( Y ) deg (cid:15) (cid:15) π N ( Y ) deg (cid:15) (cid:15) Hom( e K TF ∗ ( S N ) , e K TF ∗ ( Y )) U / / Hom( e K ∗ TF ( Y ) , e K ∗ TF ( S N )) . Proof.
For β : e K TF ∗ ( S N ) → e K TF ∗ ( Y ), let U ( β ) be the unique map making the following diagramcommute e K ∗ TF ( Y ) U ( β ) (cid:15) (cid:15) ∼ = / / Hom( e K TF ∗ ( Y ) , Z ) Hom( β, Z ) (cid:15) (cid:15) e K ∗ TF ( S N ) ∼ = / / Hom( e K TF ∗ ( S N ) , Z )where the isomorphisms are those of Corollary 5.2. Since e K TF ∗ ( Y ) is a finitely generated free Z -module, β Hom( β, Z ) is an isomorphism, so U is also an isomorphism. Commutativity of the diagram fromthe statement of this lemma is by naturality of Lemma 5.2. Corollary 7.14.
Let Y be a space having the homotopy type of a finite CW -complex. For a Z -module M , let τ p : M → M ⊗ Z /p be the natural map. There exists an injection U ′ making the followingdiagram commute. π N ( Y ) deg (cid:15) (cid:15) % % ❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏ π N ( Y ) deg (cid:15) (cid:15) & & ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ Hom( e K TF ∗ ( S N ) , e K TF ∗ ( Y )) τ p (cid:15) (cid:15) Hom ψ -Mod ( e K ∗ TF ( Y ) , e K ∗ TF ( S N )) τ p (cid:15) (cid:15) Im( τ p ◦ deg) G g t t ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ U ′ / / Im( τ p ◦ deg) F f t t ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ Hom( e K TF ∗ ( S N ) , e K TF ∗ ( Y )) ⊗ Z /p Hom ψ -Mod ( e K ∗ TF ( Y ) , e K ∗ TF ( S N )) ⊗ Z /p. Proof.
By Lemma 7.13, we have a commutative diagram π N ( Y ) deg (cid:15) (cid:15) π N ( Y ) deg (cid:15) (cid:15) Hom( e K TF ∗ ( S N ) , e K TF ∗ ( Y )) U / / Hom( e K ∗ TF ( Y ) , e K ∗ TF ( S N )) . U an isomorphism, so U ⊗ Z /p is also an isomorphism. By Lemma 6.1, the mapHom ψ -Mod ( e K ∗ TF ( Y ) , e K ∗ TF ( S N )) ⊗ Z /p → Hom( e K ∗ TF ( Y ) , e K ∗ TF ( S N )) ⊗ Z /p is an injection. Maps of spaces induce maps of ψ -modules on K -theory, so the image of U ◦ deg iscontained in Hom ψ -Mod ( e K ∗ TF ( Y ) , e K ∗ TF ( S N )), and hence there exists a map U ′ making the followingdiagram commute: Im( τ p ◦ deg) (cid:127) _ (cid:15) (cid:15) U ′ / / Hom ψ -Mod ( e K ∗ TF ( Y ) , e K ∗ TF ( S N )) ⊗ Z /p (cid:127) _ (cid:15) (cid:15) Hom( e K TF ∗ ( S N ) , e K TF ∗ ( Y )) ⊗ Z /p U ⊗ Z /p ∼ = / / Hom( e K ∗ TF ( Y ) , e K ∗ TF ( S N )) ⊗ Z /p. Both vertical maps are injections, so U ′ has the required properties. S Let f ∈ π j ( S ), and let N ≥
3. Then, for ω ∈ π N ( Y ), the composite S N + j − N − f −−−−→ S N ω −→ Y is defined. The class ω ◦ Σ N − f lies in π M − ( Y ), where M − N + j − f ∗ Σ : π ∗ ( Y ) → π ∗ ( Y ) on ω ∈ π N ( Y ) by setting f ∗ Σ ( ω ) =(Σ N − f ) ∗ ω = ω ◦ Σ N − f . In words, f ∗ Σ pulls classes back along the appropriate suspension of f .Strictly speaking, f ∗ Σ is only a partial map, because it is undefined on π N for N ≤
2, but this will beunimportant.Recall the definition of the double e -invariant e (Definition 6.4). On π N ( Y ), we have by definitionthat f ∗ Σ = (Σ N − f ) ∗ . By Lemma 6.5 we have a commuting square: π N ( Y ) f ∗ Σ / / deg (cid:15) (cid:15) π N + j − ( Y ) e (cid:15) (cid:15) Hom ψ -Mod ( e K ∗ TF ( Y ) , e K ∗ TF ( S N )) θ (Σ N − f ) / / Ext ψ -Mod ( e K ∗ TF ( Y ) , e K ∗ TF ( S N + j − )) . Mimicking the convention for f ∗ Σ , let θ Σ ( f ) : Hom ψ -Mod ( e K ∗ TF ( Y ) , e K ∗ TF ( S ∗ )) → Ext ψ -Mod ( e K ∗ TF ( Y ) , e K ∗ TF ( S ∗ + j − ))be the map which is defined to be equal to θ (Σ N − f ) on the degree N component Hom ψ -Mod ( e K ∗ TF ( Y ) , e K ∗ TF ( S N ))of Hom ψ -Mod ( e K ∗ TF ( Y ) , e K ∗ TF ( S ∗ )). Lemma 7.15.
Let p be a prime, and let f ∈ π j ( S ) with e ( f ) defined. If pf = 0, then there exists amap θ p Σ ( f ) making the following diagram commute for all N : π N ( Y ) deg (cid:15) (cid:15) f ∗ Σ / / π N + j − ( Y ) e (cid:15) (cid:15) Hom ψ -Mod ( e K ∗ TF ( Y ) , e K ∗ TF ( S N )) τ p (cid:15) (cid:15) θ Σ ( f ) / / Ext ψ -Mod ( e K ∗ TF ( Y ) , e K ∗ TF ( S N + j − ))Hom ψ -Mod ( e K ∗ TF ( Y ) , e K ∗ TF ( S N )) ⊗ Z /p. θ p Σ ( f ) roof. Since pf = 0, we have that pe (Σ N − f ) = 0 for all N , which implies that θ Σ ( f ) vanishes on p -divisible elements, so there exists a unique map θ p Σ ( f ) making the diagram commute, as required. Lemma 7.16.
Let X be a finite CW -complex. Let λ Xℓ be the largest eigenvalue of the rational Adamsoperation ψ ℓ ⊗ Q : e K ( X ) ⊗ Q → e K ( X ) ⊗ Q . Then, for i ≥ • the largest eigenvalue of ψ ℓ ⊗ Q on e K (Σ i J s ( X )) ⊗ Q is ℓ i ( λ Xℓ ) s , and • the largest eigenvalue of ψ ℓ ⊗ Q on e K (Σ i +1 J s ( X )) ⊗ Q is ℓ i λ Σ Xℓ ( λ Xℓ ) s − . Proof.
When i ≥
1, Theorem 5.6 gives that Σ J s ( X ) ≃ Σ W st =1 X ∧ t , so Σ i J s ( X ) ≃ S i ∧ W st =1 X ∧ t ,and Σ i +1 J s ( X ) ≃ S i ∧ Σ X ∧ W s − t =1 X ∧ t . By the K¨unneth theorem (Theorem 5.5), this impliesisomorphisms of rings e K (Σ i J s ( X )) ∼ = s M t =1 e K ( S i ) ⊗ e K ( X ) ⊗ t , for i ≥ , and e K (Σ i +1 J s ( X )) ∼ = s − M t =0 e K ( S i ) ⊗ e K (Σ X ) ⊗ e K ( X ) ⊗ t for i ≥ . The K¨unneth isomorphism of Theorem 5.5 is given by the external product of K -theory. Sincethe Adams operations are ring homomorphisms, the above isomorphisms are also isomorphisms of ψ -modules. In particular, the Adams operations on the left are the tensor product of the correspondingoperations on the right.These decompositions hold for e K , so they also hold for Q ⊗ e K , and the remaining problemis to determine the largest eigenvalue of the relevant tensor products of Adams operations. Theeigenvalues of a tensor product of linear endomorphisms are precisely the products of the eigenvalues.The operation ψ ℓ acts on S i by multiplication by ℓ i . Together, these observations imply the result. Lemma 7.17.
Let p be an odd prime. Let X be an ( r − CW -complex. Let N, s ∈ N . Consider the diagram of Lemma 7.15 for Y = J s ( X ) and f = f p,j ∈ π j ( p − ( S ), themap of Theorem 6.8: π N ( J s ( X )) deg (cid:15) (cid:15) f ∗ Σ / / π N +2 j ( p − − ( J s ( X )) e (cid:15) (cid:15) Hom ψ -Mod ( e K ∗ TF ( J s ( X )) , e K ∗ TF ( S N )) τ p (cid:15) (cid:15) θ Σ ( f ) / / Ext ψ -Mod ( e K ∗ TF ( J s ( X )) , e K ∗ TF ( S N +2 j ( p − ))Hom ψ -Mod ( e K ∗ TF ( J s ( X )) , e K ∗ TF ( S N )) ⊗ Z /p. θ p Σ ( f ) ❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡ For ℓ ∈ N , let λ Yℓ be the largest eigenvalue of ψ ℓ ⊗ Q on e K ( Y ) ⊗ Q , and let λ ℓ = max( λ Xℓ , λ Σ Xℓ ).If there exists ℓ ∈ N such that ℓ j ( p − N − > λ sℓ then Ker( e ◦ f ∗ Σ ) ⊂ Ker( τ p ◦ deg), and hence therestriction of θ p Σ ( f ) to Im( τ p ◦ deg) is an injection. Proof.
First note that pf = 0 by Theorem 6.8, so θ p Σ ( f ) is well-defined by Lemma 7.15. Let ω ∈ π N ( J s ( X )). Suppose that ω ∈ Ker( e ◦ f ∗ Σ ), that is, that the e -invariant of the composite S N +2 j ( p − − N − f −−−−→ S N ω −→ J s ( X )24s trivial. By Lemma 6.3, this implies that Σ i ( ω ◦ Σ N − f ) has trivial e -invariant for all i . In particular, e (Σ ω ◦ Σ N − f ) and e ( ω ◦ Σ N − f ) are both 0.By Lemma 7.16, the largest eigenvalue of ψ ℓ ⊗ Q on e K ( J s ( X )) ⊗ Q is at most λ sℓ , and the largesteigenvalue of ψ ℓ ⊗ Q on e K (Σ J s ( X )) ⊗ Q is also at most λ sℓ . We now divide into cases, based on theparity of N .CASE 1 ( N even): Write N = 2 n . Let f ′ = Σ N − f and g = ω in Theorem 6.7. The domain of ω ◦ Σ N − f is S M − , where M − N + 2 j ( p − −
1, so M is even, as is required. To check theeigenvalue hypothesis of Theorem 6.7, write M = 2 m . By Lemma 7.16, the largest eigenvalue of ψ ℓ ⊗ Q on e K ( J s ( X )) ⊗ Q is at most λ sℓ , and ℓ m = ℓ j ( p − n > ℓ j ( p − N − , which we assumed was greaterthan λ sℓ . This means that ℓ m cannot be an eigenvalue of ψ ℓ ⊗ Q on e K ( J s ( X )) ⊗ Q . Now, e ( f ) = 0 byconstruction (Theorem 6.8), so e (Σ N − f ) = 0 by stability (Lemma 6.6). Since e ( ω ◦ Σ N − f ) = 0, thecontrapositive of Theorem 6.7 gives that ω ∗ has p -divisible image in e K ( S N ). Since N is even, thisimplies that τ p ◦ deg( ω ) = 0, as required.CASE 2 ( N odd): Write n = 2 n + 1. Let f ′ = Σ N − f and g = Σ ω in Theorem 6.7, and proceedsimilarly to case 1. The domain of Σ ω ◦ Σ N − f is S M − , where M − N + 2 j ( p − M is even,as is required. To check the eigenvalue hypothesis of Theorem 6.7, write M = 2 m . By Lemma 7.16,the largest eigenvalue of ψ ℓ ⊗ Q on e K (Σ J s ( X )) ⊗ Q is at most λ sℓ , and ℓ m = ℓ j ( p − n = ℓ j ( p − N − ,which we assumed was greater than λ sℓ . This means that ℓ m cannot be an eigenvalue of ψ ℓ ⊗ Q on e K (Σ J s ( X )) ⊗ Q . As in the previous case, e (Σ N − f ) = 0. Since e (Σ ω ◦ Σ N − f ) = 0, the contrapositiveof Theorem 6.7 gives that (Σ ω ) ∗ has p -divisible image in e K ( S N +1 ). Since N is odd, this implies that τ p ◦ deg( ω ) = 0, as required. This completes the case, and hence the proof. Construction . Let p be an odd prime. Let ν : A → ΩΣ X , for spaces A and X having thehomotopy type of finite CW -complexes, with X ( r − r ≥
1. Let f ∈ π i ( S ) with e ( f )defined. Suppose that N, k, s ∈ N satisfy N < r ( s + 1) − k ≤ s . The diagrams of the precedingsubsections may be combined as follows.Recall the definition of deg ′ from the preamble to Lemma 7.2. Let I ( A ) be the submodule ofHom( e K TF ∗ ( S ∗ ) , L ( e K TF ∗ ( A ))) ⊗ Z /p generated by Im( τ p ◦ deg ′ ). The same grading conventions asusual apply: we write I k ( A ) for the weight k part, we write I N ( A ) for the degree N part, and let I kN ( A ) = I k ( A ) ∩ I N ( A ).From Corollary 7.3, using the assumptions that N < r ( s + 1) − k ≤ s (which make f Φ πν and g Φ Kν well-defined) we obtain the following diagram, where the images of the vertical maps have been‘popped out’ to their right. B kN ( π ∗ ( A )) f Φ πν / / ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘ τ p ◦ deg ′ (cid:15) (cid:15) π N ( J s ( X )) τ p ◦ deg (cid:15) (cid:15) ) ) ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ I kN ( A ) / / H h u u ❧❧❧❧❧❧❧❧❧❧❧❧❧❧ Im( τ p ◦ deg) G g u u ❥❥❥❥❥❥❥❥❥❥❥❥❥❥ Hom( e K TF ∗ ( S N ) , L k ( e K TF ∗ ( A ))) ⊗ Z /p g Φ Kν ⊗ Z /p / / Hom( e K TF ∗ ( S N ) , e K TF ∗ ( J s ( X ))) ⊗ Z /p. Next, from Corollary 7.14 (with Y = J s ( X )) we have a diagram25 N ( J s ( X )) ) ) ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ τ p ◦ deg (cid:15) (cid:15) π N ( J s ( X )) * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ τ p ◦ deg (cid:15) (cid:15) Im( τ p ◦ deg) H h u u ❦❦❦❦❦❦❦❦❦❦❦❦❦❦ U ′ / / Im( τ p ◦ deg) G g t t ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ Hom( e K TF ∗ ( S N ) , e K TF ∗ ( J s ( X ))) ⊗ Z /p Hom ψ -Mod ( e K ∗ TF ( J s ( X )) , e K ∗ TF ( S N )) ⊗ Z /p. Lastly, we obtain the following diagram from Lemma 7.15: π N ( J s ( X )) deg (cid:15) (cid:15) f ∗ Σ / / π N + i − ( J s ( X )) e (cid:15) (cid:15) Hom ψ -Mod ( e K ∗ TF ( J s ( X )) , e K ∗ TF ( S N )) τ p (cid:15) (cid:15) θ Σ ( f ) / / Ext ψ -Mod ( e K ∗ TF ( J s ( X )) , e K ∗ TF ( S N + i − ))Hom ψ -Mod ( e K ∗ TF ( J s ( X )) , e K ∗ TF ( S N )) ⊗ Z /p. θ p Σ ( f ) ❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢ Concatenating these diagrams gives a diagram as follows: B kN ( π ∗ ( A )) f ∗ Σ ◦ f Φ πν / / τ p ◦ deg ′ (cid:15) (cid:15) π N + i − ( J s ( X )) e (cid:15) (cid:15) I kN ( A ) θ p Σ ( f ) ◦ U ′ ◦ ( g Φ Kν ⊗ Z /p ) / / Ext ψ -Mod ( e K ∗ TF ( J s ( X )) , e K ∗ TF ( S N + i − )) . In this subsection, we combine the results of the previous subsections to produce results about thisdiagram.
Theorem 7.19.
Let p be an odd prime. Let ν : A → ΩΣ X , for spaces A and X having the homotopytype of finite CW -complexes, with X ( r − r ≥
1. Let
N, k, s ∈ N with N < r ( s + 1) − k ≤ s . Let f = f p,j ∈ π j ( p − ( S ), the map of Theorem 6.8.For ℓ ∈ N , let λ Yℓ be the largest eigenvalue of ψ ℓ ⊗ Q on e K ( Y ) ⊗ Q , and let λ ℓ = max( λ Xℓ , λ Σ Xℓ ).If • ν ∗ ⊗ Z /p : e K TF ∗ (Σ A ) ⊗ Z /p → e K TF ∗ (Σ X ) ⊗ Z /p is an injection, and • there exists ℓ ∈ N such that ℓ j ( p − N − > λ sℓ ,then θ p Σ ( f ) ◦ U ′ ◦ ( g Φ Kν ⊗ Z /p ) : I kN ( A ) → Ext ψ -Mod ( e K ∗ TF ( J s ( X )) , e K ∗ TF ( S N +2 j ( p − )) is an injection. Proof.
By Corollary 7.9, since ν ∗ ⊗ Z /p is an injection, g Φ Kν ⊗ Z /p is also an injection. By Corollary7.14 U ′ is an injection. By Lemma 7.17 the hypothesis on ℓ implies that the restriction of θ p Σ ( f ) toIm( τ p ◦ deg) is an injection. The map θ p Σ ( f ) ◦ U ′ ◦ ( g Φ Kν ⊗ Z /p ) is thus a composite of injections, hencean injection, as required.In the proof of Theorem 1.4, we will wish to restrict attention to those elements of B ( π ∗ ( A )) whoare brackets of classes in π ∗ ( A ) in some dimensional range q min ≤ n ≤ q max . All such classes lie indimensions kq min ≤ N ≤ kq max . Said more precisely, we have an inclusion B k ( L q max n = q min π n ( A )) ⊂ S kq max N = kq min B kN ( π ∗ ( A )). We will now study the diagram of Construction 7.18 in this dimensional range.26 onstruction . Let p be an odd prime, ν : A → ΩΣ X for finite CW -complexes A and X with X ( r − r ≥
1, and f ∈ π i ( S ) with e ( f ) defined. Let q max > q min be natural numbers.Fix k ∈ N , and let s = kq max + 1. For N ∈ N with kq min ≤ N ≤ kq max , we have that N < r ( s + 1) − k ≤ s . Combining the diagrams obtained from Construction 7.18 for this range of values of N gives the following diagram: S kq max N = kq min B kN ( π ∗ ( A )) f ∗ Σ ◦ f Φ πν / / τ p ◦ deg ′ (cid:15) (cid:15) L kq max N = kq min π N + i − ( J s ( X )) e (cid:15) (cid:15) L kq max N = kq min I kN ( A ) θ p Σ ( f ) ◦ U ′ ◦ ( g Φ Kν ⊗ Z /p ) / / L kq max N = kq min Ext ψ -Mod ( e K ∗ TF ( J s ( X )) , e K ∗ TF ( S N + i − )) . We now show that by choosing a large enough c ∈ N , and setting f = f p,ck , the eigenvaluehypothesis of Theorem 7.19 may be satisfied across the dimensional range of Construction 7.20 for allsufficiently large k . Corollary 7.21.
Let p be an odd prime. Let ν : A → ΩΣ X , for spaces A and X having the homotopytype of finite CW -complexes, with X path-connected. Let q max > q min be natural numbers. Let c, k ∈ N . Let f = f p,ck ∈ π ck ( p − ( S ) be the map of Theorem 6.8. If ν ∗ ⊗ Z /p : e K TF ∗ (Σ A ) ⊗ Z /p → e K TF ∗ (Σ X ) ⊗ Z /p is an injection then there exists c ∈ N such that for large enough k ∈ N , θ p Σ ( f ) ◦ U ′ ◦ ( g Φ Kν ⊗ Z /p ) : kq max M N = kq min I kN ( A ) → kq max M N = kq min Ext ψ -Mod ( e K ∗ TF ( J s ( X )) , e K ∗ TF ( S N +2 ck ( p − ))is an injection. Proof.
By Theorem 7.19, it suffices to show that for each N with kq min ≤ N ≤ kq max there exists ℓ ∈ N such that ℓ ck ( p − N − > λ sℓ = λ kq max +1 ℓ . Take any ℓ ≥
2. Since N ≥ kq min , it suffices to find c such that for large enough k we have ℓ ck ( p − kq min − > λ kq max ℓ . Taking logs on both sides, this isequivalent to ( ck ( p −
1) + kq min −
12 ) log( ℓ ) > kq max log( λ ℓ ) . It is now clear that we may choose c large enough that this equation holds for large enough k , inparticular, any c ≥ q max log( λ ℓ ) − ( q min2 log( ℓ )) p − will do.We are now ready to prove Theorem 1.4. Proof of Theorem 1.4.
Let µ = µ ∨ µ , with adjoint µ : S q ∨ S q → ΩΣ X . Let f = f p,ck ∈ π ck ( p − ( S ). Consider the diagram of Construction 7.20, with A = S q ∨ S q , q max = max( q , q ), q min = min( q , q ), and ν = µ . We have such a diagram for each k ∈ N : S kq max N = kq min B kN ( π ∗ ( S q ∨ S q )) f ∗ Σ ◦ f Φ πµ / / τ p ◦ deg ′ (cid:15) (cid:15) L kq max N = kq min π N +2 ck ( p − − ( J s ( X )) e (cid:15) (cid:15) L kq max N = kq min I kN ( S q ∨ S q ) θ p Σ ( f ) ◦ U ′ ◦ ( g Φ Kµ ⊗ Z /p ) / / L kq max N = kq min Ext ψ -Mod ( e K ∗ TF ( J s ( X )) , e K ∗ TF ( S N +2 ck ( p − )) .
27y assumption, µ ∗ ⊗ Z /p : e K ∗ (Σ X ) ⊗ Z /p → e K ∗ ( S q +1 ∨ S q +1 ) ⊗ Z /p is a surjection. By Lemma7.10 this implies that µ ∗ ⊗ Z /p : e K TF ∗ ( S q +1 ∨ S q +1 ) ⊗ Z /p → e K TF ∗ (Σ X ) ⊗ Z /p is an injection. Thus, by Corollary 7.21, we may fix c such that for large enough k , θ p ( f ) ◦ U ′ ◦ ( g Φ Kµ ⊗ Z /p )is an injection.The Hurewicz map h is a surjection π ∗ ( S q ∨ S q ) → e K TF ∗ ( S q ∨ S q ), so the submodule generated bythe image of the map h ′ : B ( π ∗ ( S q ∨ S q )) → L ( e K TF ∗ ( S q ∨ S q )) of Lemma 7.11 contains the submodulegenerated by e K TF ∗ ( S q ∨ S q ) under the bracket operation. In particular, it contains the weight k component L k ( e K TF ∗ ( S q ∨ S q )) for each k . By Theorem 3.1, dim Z ( L k ( e K TF ∗ ( S q ∨ S q ))) = W ( k ).Note that L k ( e K TF ∗ ( S q ∨ S q )) = L kq max N = kq min L k ( e K TF ∗ ( S q ∨ S q )).It then follows from Lemma 7.11 that dim Z /p ( L kq max N = kq min I kN ( S q ∨ S q )) ≥ W ( k ). Since θ p Σ ( f ) ◦ U ′ ◦ ( g Φ Kµ ⊗ Z /p ) is an injection for large enough k , it follows that the dimension of e ( L kq max N = kq min π N +2 ck ( p − − ( J s ( X ))) is at least W ( k ). By Corollary 7.3 ( i s ) ∗ is an injection, sothe dimension of ( i s ) ∗ ( L kq max N = kq min π N +2 ck ( p − − ( J s ( X ))) ⊂ L kq max N = kq min π N +2 ck ( p − − (ΩΣ X ) is alsoat least W ( k ). Thus, Σ X satisfies the hypotheses of Lemma 3.3 with a = 2 c ( p −
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