WILLIAM RICHTER. (Communicated by Brooke Shipley)

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1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages S (XX) A CONJECTURE OF GRAY AND THE p-th POWER MAP ON Ω 2 S 2np+1 WILLIAM RICHTER (Communicated by Brooke Shipley) ABSTRACT. For p 2, the p-th power map [p] on Ω 2 S 2np+1 is homotopic to a composite Ω 2 S 2np+1 φn S 2np 1 E2 Ω 2 S 2np+1, where the iber o φ n is BW n. 1. INTRODUCTION Localize spaces and maps at any prime p 2. We prove a conjecture o Gray. Theorem 1.1. For p 2, the p-th power map Ω 2 S 2np+1 a composite Ω 2 S 2np+1 [p] Ω 2 S 2np+1 is homotopic to φn S 2np 1 E2 Ω 2 S 2np+1, where the iber o φ n is BW n. Gray ound a serious gap in Theriault s [14] proo o Gray s conjecture or p 5, which we ill in [11]. Our proo combines [12] with Gray s [7] spectacular construction o BW n. Using ideas o Barratt, Boardman and Steer, we show (Thm. 2.6) that p times the unstable p th James Hop invariant is a cup product. Our actorization Thm. 3.3 applies such cup product inormation to a map ρ similar to a map o Gray s, but deined in a more combinatorial way. This proves Thm. 4.1, the actorization E 2 φ n = [p]. A result o Gray s (Thm. 4.5), which uses only the Serre spectral sequence, shows the iber o φ n is BW n, as φ n kills the Hop invariant Ω 2 S 2n+1 Ω 2 S 2np+1 and is degree p on the bottom cell. Theorem 1.1 relates to [4], where Cohen, Moore and Neisendorer showed π (S 2n+1 ) has exponent p n or p odd, by showing that [p] on Ω 2 S 2n+1 actors, or some π n, as the composite Ω 2 S 2n+1 πn S 2n 1 E2 Ω 2 S 2n+1. Anick [1] solved a conjecture o [4], in 270 pages, constructing a ibration sequence Ω 2 2n+1 αn S S 2n 1 T n ΩS 2n+1, or p 5. Gray and Theriault [9] gave a shorter construction o Anick s ibrations, or p 3, and showed that α n was essentially π n o [4]. Gray noted that Theorem 1.1 gives evidence that φ n = π np, as E 2 φ n = [p] = E 2 π np, and that this would imply that BW n is the loop space ΩT np. See [15] or 2-primary Anick ibration analogues. Theriault [16] constructed the odd-primary Anick ibrations not using [4] (thus reproving the [4] exponent theorem), but using Gray s conjecture. With our proo o Theorem 1.1, [16] now seems correct. 2. COMBINATORIAL JAMES HOPF INVARIANTS AND CUP PRODUCTS We ollow [18], and work in the category TOP o pointed compactly generated weak Hausdor spaces. Whitehead largely ollows Strøm [13], who shows TOP satisies all the axioms o a proper model category (c. [5]), except the limit and colimit axioms, using Received by the editors October 20, 2010, and, in revised orm, October 2, Mathematics Subject Classiication. Primary 55Q40, 55Q25. Key words and phrases. James-Hop invariants, BW n, Gray s conjecture, EHP sequence. 1 c 2011 American Mathematical Society

2 2 WILLIAM RICHTER the model structure o homotopy equivalences, Hurewicz ibrations, and NDR pairs. So coibration ( ) will mean NDR pair, and equivalence ( ) will mean homotopy equivalence. Assume all spaces are well-pointed and have the homotopy type o a CW complex. We use the conventions o [12] and [2]. Suspension is given by smashing on the right with S 1 = I/{0, 1}, so ΣX = X S 1. There are adjoint unctors given by the evaluation map σ : ΣΩB B and the suspension map E : B ΩΣB. Given maps : ΣA X and g : A ΩX, we call their adjoints : A ΩX and g : ΣA X. A composite A ΩY ΩE Ω 2 ΣY is adjoint to the suspension o : ΣA Y. Recall the shule shule: Σ n+m (A B) = A B S n S m 1 A T 1 S m A S n B S m = Σ n A Σ m B and the permutation group Σ r action on X [r]. Given : A X and g : A Y, deine the cup product g : A X Y to be the composite A A A g X Y. The cup product is compatible with permutations: given ρ Σ r and maps i : A X, we have (2.1) ρ( 1... r ) = ρ 1 (1)... ρ 1 (r) : A X [r]. Let k = {1,..., k}, and call ( k r) the set o subsets S k o size r. For each S, let (s 1,..., s r ) be its ordered sequence o elements. Order ( k r) using the let-lexicographical (let-lex) order o the (s 1,..., s r ). Given S ( k r), deine the map πs : X k X [r] by π S (x 1,..., x k ) = x s1 x sr, so π S = π s1... π sr is an iterated cup product. Then Lemma 2.1. Let X be a co-h space. Then or any x k and any subset T ( k r 1), the cup product π x π T : X k X [r] is nullhomotopic i x T. Proo. Write π T = π t1 π T, where T is the complement o the smallest element t 1 T. By (2.1), it suices to consider the case x = t 1. Then π x π T = (π x π x ) π T. But π x π x, the composite X k πx X X X, is nullhomotopic, since X is a co-h space. The James construction J(X) (c. [18, VII(2)], [2, 3]) has subspaces J k (X). The identiication map ι k : X k J k (X) induces a coibration ῑ k : X [k] J(X)/J k 1 (X) and a homeomorphism ῑ k : X [k] = J k (X)/J k 1 (X) deined by ῑ k (x 1 x k ) = [x 1,..., x k ], which gives a coibration sequence J k 1 (X) J k (X) X [k]. For X connected, there is an equivalence J(X) ΩΣX. We deine the r th combinatorial James-Hop invariant j r : J(X) J(X [r] ) using the let-lex order, and call H r j r the composite J(X) J(X [r] ) ΩΣX [r]. H 1 : J(X) ΩΣX is the equivalence, as j 1 : J(X) J(X) is the identity. H k : J(X) ΩΣX [k] actors through a map H k : J(X)/J k 1 (X) ΩΣX [k], by construction, or X connected. The composite ΣX k Σι k ΣJ(X) H r ΣX [r] satisies the crucial property ( means the let-lex order) (2.2) H r Σι k = S ( k r) Σπ S [ΣX k, ΣX [r] ]. Note that H 1 Σι k = x k π x [ΣX k, ΣX]. By the proo o [2, Lem. 3.7], we have Lemma 2.2. Given two maps, g : J(X) ΩY, suppose or each k the composites ι k, g ι k : X k ΩY are homotopic. Then and g are homotopic. Deine ΩΣX ΩΣY ΩΣ(X Y ) to be the adjoint o the composite ΣΩΣX ΩΣY σ 1 ΣX ΩΣY 1 σ ΣX Y.

3 A CONJECTURE OF GRAY AND THE p-th POWER MAP ON Ω 2 S 2np+1 3 Given : ΣA ΣX and g : ΣA ΣY, deine g : ΣA ΣX Y as the adjoint o the composite A A A g ΩΣX ΩΣY ΩΣX Y. Then g is the composite (2.3) g : ΣA ΣA A 1 ΣX A 1 g ΣX Y. Note that g is a desuspension o what Boardman and Steer [2] call the cup product g. Lemma 2.3. Given maps i : A X and g j : A Y, let = i Σ i : ΣA ΣX and g = j Σg j : ΣA ΣY. Then g = i,j Σ i g j [ΣA, ΣX Y ]. Proo. Composition is let-distributive, and composition is right-distributive i the right map is a suspension. So (1 g)( 1) = ( i j 1 Σg ) j Σi 1 = Σ i g j. i,j Let θ i = (12... i) be the cyclic permutation o length i. Then Proposition 2.4. For any co-h space X and any r, the diagram homotopy commutes. H r J(X) ΩΣX [r] ΩΣX [r] H 1 H r 1 1+Ωθ 2+ +Ωθ r J(X) [2] ΩΣX ΩΣX [r 1] ΩΣX [r] Ω 2 Σ 2 X [r] Proo. The lower and upper composites o the diagram composed with ι k : X k J(X) are adjoint to maps which we will call L, U [Σ 2 X k, Σ 2 X [r] ]. Note this group is abelian. By Lemma 2.2, it suices to show that L = U. L is the suspension o the composite ΣX k Σι k ΣJ(X) H 1 H r 1 ΣX [r]. By naturality o the product, L is the suspension o (H 1 Σι k ) (H r 1Σι k ). Then (2.4) L = x k, T ( k r 1) Σ2 π x π T [Σ 2 X k, Σ 2 X [r] ], by (2.2) and Lemma 2.3. U is the suspension o the sum r i=1 θ ihr Σι k. By (2.2), this sum equals r i=1 S ( k r) Σθ iπ S. But θ i π S = π x π T, where x is the i th element o S and T = S {x}, by (2.1). We have a bijection r ( k ) ( = r {(x, T ) k k r 1) : x T } given by (i, S) (s i, S {s i }). Hence, ater suspending, U = x k, T ( r 1),x T k Σ2 π x π T [Σ 2 X k, Σ 2 X [r] ]. By Lemma 2.1, the condition x T is unnecessary, as the terms with x T are nullhomotopic. By comparing with (2.4), we have L = U [Σ 2 X k, Σ 2 X [r] ]. We specialize soon to X = S 2n. Let Φ: ΩP Q Ω(P Q) be the adjoint o ΣΩP Q σ 1 P Q, so Φ(α q)(t) = α(t) q, or spaces P and Q. Then Lemma 2.5. For e even, the map Φ: ΩU S e Ω(U S e ) is homotopic to the composite (2.5) ΩU S e Σ e 1 σ U S e 1 E Ω(U S e ), or any space U. Take also a space V and a map : U ΩΣV. Then the composite z : Σ e U = U S e E ΩΣV ΩS e+1 Ω(V S e+1 ) = Ω(Σ e+1 V ) is homotopic to the composite Σ e U Σe 1 Σ e V E Ω(Σ e+1 V ). ΩE ΩE

4 4 WILLIAM RICHTER Proo. The shule τ : S e S 1 S 1 S e is homotopic to the identity on S e+1, as it is deined by a permutation with even sign, since e is even. The adjoint o Φ is the composite ΩU S e+1 = ΩU S e S 1 1 ΩU τ ΩU S 1 S e σ 1 S e U S e. Thus Φ : ΩU S e+1 U S e is homotopic to σ 1 S e = Σ e σ, since τ is homotopic to the identity. As the adjoint o (2.5) is Σ(Σ e 1 σ) = Σ e σ as well, this proves the irst part. The second part is similar. By the smash product version o the cup product ormula (2.3), the adjoint o the map z : Σ e U Ω(Σ e+1 V ) is the composite z : U S e S 1 1 τ U S 1 S e 1 S e V S 1 S e 1 τ 1 V S e S 1, since the adjoint o E : S e ΩS e+1 is the identity map on S e S 1 = S e+1. Since τ is homotopic to the identity, z : Σ e+1 U Σ e+1 V is homotopic to 1 S e = Σ e. But Σ e is also the adjoint o Σ e U Σe 1 Σ e V E Ω(Σ e+1 V ). The composite J(X) J(X) J(X)/J r 2 (X) H1 Hr 1 ΩΣX ΩΣX [r 1] is homotopic to the composite (H 1 H r 1 ) o Proposition 2.4, and the suspension map E is homotopic to the composite X [k] J(X)/Jk 1 (X) H k ΩΣX [k]. Now we specialize Proposition 2.4 to X = S 2n. Let m = 2n(p 1). Then ῑ k Theorem 2.6. Localize at prime p 2. The diagram is homotopy commutative: J(S 2n ) H 1 H p 1 ΩS 2np+1 J(S 2n ) J(S 2n )/J p 2 (S 2n ) ΩS 2n+1 ΩS m+1 ΩS 2np+1 id ῑ p 1 J(S 2n ) S m S 2np. H p Σ m 1 H 1 Proo. The bottom rectangle homotopy commutes by Lemma 2.5 and the above remarks. For p = 2, the top rectangle homotopy commutes [12, Thm. 2.3]. For p > 2, we can desuspend Proposition 2.4, because ΩS 2np+1 is a retract o Ω 2 S 2np+2, since S 2np+1 is an H-space. Then use the above actorization through (H 1 H r 1 ). 3. CUP PRODUCTS AND A FACTORIZATION RESULT FOR A MAP ρ For a co-h space X, we construct a map ρ: Ω(J(X), J p 1 (X)) ΩJ(X) + X [p 1] analogous to Gray s clutching construction collapse map [7], and prove a actorization result involving ρ (Theorem 3.3) that we combine in 4 with Theorem 2.6. Given a coibration K L, call Ω(L, K) = {λ P L : λ(1) K} its homotopy iber. The natural map v : Ω(L, K) Ω(L/K), adjoint to the natural map rom the homotopy iber to the coiber, sends a path λ to the loop [λ]. Let p 1, p 2 : X X be the actors o the comultiplication map X X X o the co-h space X. The selmaps J(p 1 ) and J(p 2 ) o J(X) are homotopic to the identity. Then Lemma 3.1. Let k = a+b+1 or a, b 0. The map J(p 1 ) J(p 2 ) deines a actorization J k (X) J a (X) J k (X) J k (X) J b (X) up to homotopy o the diagonal map on J k (X). Furthermore J(p 1 ) J(p 2 ) induces a map ρ: Ω(J(X), J k (X)) Ω(J(X)/J a (X)) + J k (X)/J b (X), [p] E

5 A CONJECTURE OF GRAY AND THE p-th POWER MAP ON Ω 2 S 2np+1 5 deined explicitly by the ormula ρ(λ) = J(p 1 )λ + J(p 2 )λ(1), and also the map below which makes the ollowing diagram homotopy commutative. J(X) J(X) J(X) J(X)/J k (X) J(X)/J a (X) J(X)/J b (X) Proo. The map J(p 1 ) J(p 2 ) is homotopic to the diagonal map, as is its restriction J k (X) J k (X) J k (X). Any α J k (X) is represented by some (y 1,..., y k ) X k. Let N 1 be the number o indices i or which p 1 (y i ) =, and N 2 be the number o indices i or which p 2 (y i ) =. For any index i, either p 1 (y i ) = or p 2 (y i ) =, so N 1 + N 2 k. Thus either N 1 > b or N 2 > a, since k > a + b. Use k = a + b + 1 to rewrite this as either N 1 k a or N 2 k b. Thus either J(p 1 )(α) J a (X) or J(p 2 )(α) J b (X). This proves the irst claim, which then implies the second and third claims. Let E = J(X), M = J p 1 (X), and Y = J p 2 (X). Lemma 3.1 in the case k = p 1 and a = 0 gives the maps : E/M E E/Y and ρ: Ω(E, M) ΩE + X [p 1], deined explicitly by ρ(λ) = J(p 1 )λ + J(p 2 )λ(1). Then Proposition 3.2. The composite Ω(E) Ω(E, M) ρ ΩE + X [p 1] is trivial, and the ollowing diagram is homotopy commutative. Ω(E, M) v Ω(E/M) ρ id ῑ p 1 ΩE + X [p 1] ΩE X [p 1] ΩE E/Y Ω(E E/Y ) Proo. The irst assertion is immediate rom the deinition o ρ, as λ(1) = i λ ΩE. The composite M E E/Y actors as M X [p 1] ῑ p 1 E/Y. The straight line homotopy o [λ(s)] E/Y to [λ(1)] = ῑ p 1 λ(1) M/Y = X [p 1] gives a homotopy o the diagram, as (Ω v)(λ)(s) = J(p 1 )λ(s) [J(p 2 )λ(s)], or any λ Ω(E, M), and (Φ (id ῑ p 1 ) ρ)(λ)(s) = J(p 1 )λ(s) ῑ p 1 J(p 2 )λ(1). Let p: E B be a map with p(m) =. Let F be the homotopy iber o p, and suppose that the natural lit o M E is an equivalence M F. Call p: E/M B the map induced by p. Then the ollowing composite is an equivalence. (3.1) Ω(E, M) v Ω(E/M) Ω p ΩB Call ρ : ΩB ΩE + X [k] the composite o ρ with the inverse o equivalence (3.1). We prove an analogue o [12, Thm. 4.1]. Theorem 3.3. Let Z = ΩZ 0 be a loop space, and assume that ΣE ΣE/M has a right homotopy inverse. Take maps : B Z and α: E E/Y Z making the diagram (3.2) E E E/Y p α B Z Φ Ω

6 6 WILLIAM RICHTER homotopy commute. Then Ω : ΩB ΩZ is homotopic to the composite ΩB ρ ΩE + X [p 1] Φ Ω(E X [p 1] ) Ω(id ῑp 1) Ω(E E/Y ) Ωα ΩZ. I X = S 2n, let m = 2n(p 1). Then Ω : ΩB ΩZ is homotopic to the composite (3.3) ΩB ρ Σ m ΩE + where ζ is the adjoint o the composite ζ : Σ m E Σ m 1 σ Σ m 1 E ζ ΩZ, id ῑp 1 E E/Y α Z. Proo. The triangle o the ollowing diagram homotopy commutes by Lemma 3.1. E E/M B α E E/Y Z By (3.2), the outer polygon homotopy commutes. We see the square homotopy commutes by adjointing with Z = ΩZ 0 and using the right homotopy inverse. By looping the right square and using Proposition 3.2, we see the diagram v Ω(E, M) Ω(E/M) ΩB ρ id ῑ p 1 ΩE + X [p 1] ΩE E/Y Ω(E E/Y ) ΩZ homotopy commutes. Now invert the top horizontal composite using equivalence (3.1). In the case X = S 2n, we showed that Ω : ΩB ΩZ is homotopic to the composite ΩB ρ ΩE + S m ΩE S m Φ Φ Ω(E S m ) p Ω Ω p Ωα Ω Ω(id ῑp 1) Ω(E E/Y ) Ωa ΩZ. Lemma 2.5 and the act that Ω(ζ) E = ζ : Σ m 1 E ΩZ establishes (3.3). An obvious result ollows rom the irst part o Proposition 3.2, as the composite o equivalence (3.1) with ΩE Ω(E, M) is Ωp: ΩE ΩB. Lemma 3.4. The composite ΩE Ωp ΩB ρ Σ m ΩE + is nullhomotopic. 4. PROOF OF GRAY S CONJECTURE Specialize to X = S 2n, so E = J(S 2n ). Let B = ΩS 2np+1 and let p: E B be the p th James-Hop invariant H p. The EHP sequences o James [10] or p = 2 and Toda [17] or odd primes show the hypothesis o 3 is satisied, so we have the equivalence Ω(E, M) ΩB o (3.1), where M = J p 1 (S 2n ) and Y = J p 2 (S 2n ). Thus we have the map ρ : ΩB ΩE + X [k] = Σ m ΩE +. Let Z = B and = [p]: B B, the p-th power map on B. The loop o [p] on B is [p] on ΩB. Deine Ω 2 S 2np+1 the composite ΩB ρ Σ m ΩE + s S 2np 1, where s is the composite φn S 2np 1 as (4.1) s: Σ m ΩE + Σ m ΩE Σm 1 σ Σ m 1 E Σm 2 H 1 Σ m 1 X = S 2np 1, where as usual m = 2n(p 1). Now we state our main result. Theorem 4.1. E 2 φ n = [p] [Ω 2 S 2np+1, Ω 2 S 2np+1 ].

7 A CONJECTURE OF GRAY AND THE p-th POWER MAP ON Ω 2 S 2np+1 7 Proo. ΣE ΣE/M has a right homotopy inverse by the James splitting, so we can apply Theorem 3.3 and Theorem 2.6. Thus [p]: ΩB ΩB is the homotopy class o the composite ΩB ρ Σ m ΩE + composite ζ : Σ m E Σ m 1 E ΩB, using the adjoint o the α B. By Theorem 2.6, α is the composite Σm 1 σ id ῑp 1 E E/Y ζ α: J(S 2n ) J(S 2n )/J p 2 (S 2n ) H1 Hp 1 ΩS 2n+1 ΩS m+1 ΩS 2np+1 and ζ is the composite Σ m J(X) Σm 1 H 1 Σ m X E Ω(Σ m+1 X). So ζ is the composite Σ m 1 J(X) Σm 2 H 1 Σ m 1 X E2 Ω 2 (Σ m+1 X). Thus [p] is the composite ΩB ρ Σ m ΩE + s Σ m 1 X E2 Ω 2 (Σ m+1 X), or ΩB φn S 2np 1 X E2 Ω 2 S 2np+1. Thus E 2 φ n = [p] [Ω 2 S 2np+1, Ω 2 S 2np+1 ]. We show the homotopy iber o φ n is BW n using an argument essentially due to Gray and largely contained in [8], an earlier version o [7]. Lemma 3.4 immediately implies Lemma 4.2. The composite Ω 2 S 2n+1 The next result is due to James and Toda [10, 17]. ΩH Ω 2 S 2np+1 φn S 2np 1 is nullhomotopic. Lemma 4.3. Let h: ΩJ p 1 (S 2n ) ΩS 2np 1 be a map which gives an isomorphism in (2np 2)-dimensional integral cohomology. Then the homotopy iber o h is S 2n 1. Remark 4.4. Gray s proo o Lemma 4.3 or odd primes [6, Thm. 1(a)] using the Z cohomology Serre ss also works or p = 2. Letting F be the homotopy iber o h, the cohomology o the total space ΩJ p 1 (S 2n ) is the tensor product o the cohomologies o the base and iber. The other 2-primary EHP sequence S 2n ΩS 2n+1 ΩS 4n+1 has a harder proo [3, Thm. 3.1], using the binomial coeicient identity ( 2n 2k) ( n k) (mod Z/2). Note Ω 2 S 2np+1 Theorem 4.5. A map Ω 2 S 2np+1 φn S 2np 1 is degree p on the bottom cell by Theorem 4.1. We prove the bottom cell and the composite Ω 2 S 2n+1 S 2np 1 has homotopy iber BW n i is degree p on ΩH Ω 2 S 2np+1 S 2np 1 is nullhomotopic. Proo. Looping the EHP ibration [10, 17] gives the homotopy ibration sequence (4.2) ΩJ p 1 (S 2n ) ΩE Ω 2 S 2n+1 ΩH Ω 2 S 2np+1. Take the ibration sequence ΩS 2np 1 π L Ω 2 S 2np+1 S 2np 1. A null-homotopy o the composite ΩH gives a lit ν : Ω 2 S 2n+1 L o H and a map o homotopy ibers h: ΩJ p 1 (S 2n ) ΩS 2np 1, yielding the homotopy commutative diagram ΩS 2np 1 π L Ω 2 S 2np+1 S 2np 1, h ν ΩH ΩJ p 1 (S 2n ΩE ) Ω 2 S 2n+1 F where F is the homotopy iber o both h and ν. The Z cohomology Serre ss o the path ibration proves H 2np 1 (Ω 2 S 2n+1, Z) = Z/p. The Z/p cohomology Serre ss proves

8 8 WILLIAM RICHTER H Ω 2 S 2n+1 is Z/p in dimensions 2np 2 and 2np 1, connected by a Bockstein by the integral result. Since H 2np 1 (ΩJ p 1 (S 2n, Z/p) = 0, the Z/p cohomology Serre exact sequence o (4.2) shows that (ΩH) is an isomorphism in dimension 2np 1 and (ΩE) is an isomorphism in dimension 2np 2. The Z/p cohomology o L is also Z/p in dimensions 2np 2 and 2np 1, connected by a Bockstein. By naturality o the Bockstein, ν is an isomorphism in dimension 2np 2, so h is an isomorphism in dimension 2np 2. Thus h is degree r on the bottom cell, where r is prime to p, so the selmap rι o S 2np 1 is an equivalence. Composing h with the homotopy inverse o Ω(rι) and applying Lemma 4.3 shows that the homotopy iber o h is S 2n 1. Thereore F = S 2n 1, and we have a homotopy ibration sequence S 2n 1 E 2 Ω 2 S 2n+1 ν L. Thus L is a delooping o W n, the iber o E 2, so L is BW n. Theorem 1.1 now ollows rom Theorem 4.1, Lemma 4.2 and Theorem ACKNOWLEDGMENTS Thanks to Brayton Gray, whose helpul advice and probing questions led to a much clearer exposition than [12]. Gray suggested the murky geometry o [12] (involving a dual Barratt-Toda theorem and Boardman-Steer s geometric Hop invariant) be replaced by a more algebraic argument, and that the original proo o [12, Lem. 5.1] was inadequate. Gray s simple Theorem 4.5 replaces a much more technical Ganea-theoretic proo that our map φ n is homotopic to the map Gray used. Thanks to Michael Barratt or explaining his unpublished ideas about symmetry ormulas or the unsuspended James Hop invariants used in 2. Thanks to Fred Cohen, Paul Goerss and John Klein or helpul discussions. REFERENCES 1. D. Anick, Dierential algebras in topology, Research Notes in Mathematics, vol. 3, A K Peters, Ltd., J. M. Boardman and B. Steer, On Hop invariants, Comment. Math. Helv. 42 (1968), F. Cohen, A course in some aspects o classical homotopy theory, Alg. Top., Proc. Seattle 1985 (H. Miller and D. Ravenel, eds.), Lec. Notes in Math., vol. 1286, Springer, 1987, pp F. Cohen, J. C. Moore, and J. Neisendorer, Exponents in homotopy groups, Ann. o Math. 110 (1979), W. Dwyer and J. Spalinski, Homotopy theory and model categories, James handbook, B. Gray, On Toda s ibration, Math. Proc. Camb. Phil. Soc. 97 (1985), , On the iterated suspension, Topology 27 (1988), , On the double suspension, Algebraic Topology: Proc. Intern. Con. Arcata (G. Carlsson et al., ed.), Lec. Notes in Math., vol. 1370, Springer, 1989, pp B. Gray and S. Theriault, An elementary construction o Anick s ibration, Geometry and Topology 14 (2010), I. M. James, On the suspension triad, Annals o Math. 63 (1956), W. Richter, A note on the H-space structure o BW n, 6 pages. 12. W. Richter, The H-space squaring map on Ω 3 S 4n+1 actors through the double suspension, Proc. Amer. Math. Soc. 123 (1995), A. Strøm, The homotopy category is a homotopy category, Arch. Math. 23 (1972), S. Theriault, Proos o two conjectures o Gray involving the double suspension., Proc. Amer. Math. Soc. 131 (2003), S. Theriault, 2-primary Anick ibrations, Journal o Topology (2011), S. Theriault, Anick s ibration and the odd primary homotopy exponent o spheres, preprint arxiv: , H. Toda, On the double suspension E 2, J. Inst. Polytechnics, Osaka City Univ. 7 (1956), G. W. Whitehead, Elements o Homotopy Theory, GTM, vol. 61, Springer, DEPARTMENT OF MATHEMATICS, NORTHWESTERN UNIV., EVANSTON, IL address: richter@@math.northwestern.edu