Comultiplication Structures for a Wedge of Spheres

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1 Filomat 30:13 (2016), DOI /FIL L Published by Faculty of Sciences Mathematics, University of Niš, Serbia Available at: Comultiplication Structures for a Wedge of Spheres Dae-Woong Lee a a Department of Mathematics, Institute of Pure Applied Mathematics, Chonbuk National University, 567 Baekje-daero, Deokjin-gu, Jeonju-si, Jeollabuk-do 54896, Republic of Korea Abstract.In this paper, we consider the various sets of comultiplications of a wedge of spheres provide some methods to calculate many kinds of comultiplications with different properties. In particular, we concentrate on studying to compute the number of comultiplications, associative comultiplications, commutative comultiplications, comultiplications which are both associative commutative of a wedge of spheres. The more spheres that appear in a wedge, the more complicate the proofs computations become. Our methods involve the basic Whitehead products in a wedge of spheres the Hopf-Hilton invariants. 1. Introduction In this paper we assume that all spaces are based have the based homotopy type of based connected CW-complexes. All maps homotopies preserve the base point. Unless otherwise stated, we do not distinguish notationally between a map its homotopy class. Thus equality of maps means equality of homotopy classes of maps; that is, we use = for the homotopy. A pair (Y, ϕ) consisting of a space Y a function ϕ : Y Y Y is called a co-h-space if q 1 ϕ = 1 q 2 ϕ = 1, where q 1 q 2 are the projections Y Y Y onto the first second summs of the wedge product 1 is the identity map of Y. In this case the map ϕ : Y Y Y is called a comultiplication. Equivalently, (Y, ϕ) is a co-h-space if jϕ = : Y Y Y, where is the diagonal map j : Y Y Y Y is the inclusion. A comultiplication ϕ is called associative if (ϕ 1)ϕ = (1 ϕ)ϕ : Y Y Y Y, ϕ is called commutative if Tϕ = ϕ, where T : Y Y Y Y is the switching map. A left inverse for ϕ is a map L : Y Y such that (L 1)ϕ = 0 a right inverse is a map R : Y Y such that (1 R)ϕ = 0, where : Y Y Y is the folding map 0 is the constant map. Co-H-spaces, dual of H-spaces, play a fundamental role in homotopy theory. If the comultiplication ϕ : Y Y Y is homotopy associative has a right left inverse, then for every space Z the set [Y, Z] of homotopy classes from Y to Z becomes a group with the group operation depending on the comultiplication of a space. The important examples of co-h-spaces are all n-spheres, n 1 suspensions of a based space. It is easily seen that the wedge of two co-h-spaces is a co-h-space, therefore it is natural to 2010 Mathematics Subject Classification. Primary 55Q25; Secondary 55Q15, 55P40, 52C45, 68U05 Keywords. Co-H-spaces; comultiplications; homotopy associativity; homotopy commutativity; basic Whitehead products; the Hopf-Hilton invariants; the Hilton s theorem Received: 22 October 2014; Accepted: 03 March 2015 Communicated by Ljubiša D.R. Kočinac This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science Technology (NRF-2015R1D1A1A ). address: dwlee@jbnu.ac.kr (Dae-Woong Lee)

2 D.-W. Lee / Filomat 30:13 (2016), ask about the comultiplications on a wedge of spheres. It turns out that this set of comultiplications is complicated - there are usually many comultiplications (sometimes infinitely many) with many different properties. For example, S 2 S 5 has infinitely many homotopy classes of comultiplications commutative comultiplications. However, it has only 2 homotopy associative comultiplications [5]. Some indication of this complexity appeared in an early paper of Ganea [10, pp ] who gave an intricate argument to show that S 3 S 15 has at least 72 associative comultiplications at most 56 homotopy classes of suspension comultiplications. The study of comultiplications of a co-h-space has been carried out by several authors (see [6], [7], [3], [14], [4] [15]). One can find interesting examples of a co-h-space which is not homotopically equivalent to a suspension [9] of a co-h-space X = S 3 f e 2p+1 which does not admit an associative comultiplication [8], where f π 2p (S 3 ) p is an odd prime. In [11] the set of homotopy classes of comultiplications of a wedge of two Moore spaces was investigated as a natural extension of the earlier work [2]. It is worth while examining the comultiplication structures on the wedge of Moore spaces for further studies. In this paper we study the set of comultiplications on a wedge of k-spheres for k 2. In particular, we focus on the study of the properties cardinalities of comultiplications, associative comultiplications, commutative comultiplications, comultiplications which are both associative commutative of a wedge of spheres which is much more complicate than a wedge of two spheres. More precisely, let S be the number of elements in a set S, let A(Y) CO(Y) be the sets of homotopy classes of associative comultiplications commutative comultiplications, respectively, of a wedge of spheres Y = S n S m S p with m 2n 2 2 n < m < p 4n 4. Then we can show in Theorem 3.15 Theorem 4.7 that if m n are even, then A(Y) = π p (S 2m 1 ) π p (S 2n 1 ) π p (S m+n 1 ) 2 π p (S 2m+n 2 ) π p (S m+2n 2 ) ; CO(Y) = π p (S 2m 1 ) π p (S 2n 1 ) π p (S 3m 2 ) π p (S 3n 2 ) π p (S m+n 1 ) π p (S 2m+n 2 ) 3 π p (S m+2n 2 ) 2. The main results of this paper also show that the cardinalities of A(Y) CO(Y) depend on the homotopy structures of a wedge of spheres the parity of m or n, or both. For examples, A(S 8 S 12 S 27 ) = 32 CO(S 8 S 12 S 27 ) = 16. The paper is organized as follows: In Section 2, we introduce the Hilton s theorem the Hopf-Hilton invariants. We establish basic facts concerning the various types of comultiplications on a wedge of k- spheres give general conditions for comultiplications to be associative or commutative in the case of a wedge of three spheres. In Section 3, we define certain general comultiplications on a wedge of three spheres determine when they are associative. In Section 4.7, we prove the fundamental properties of the commutative comultiplications which are defined in Section 3. The examples will be listed at the end of each section. 2. Comultiplication Structures We first consider a wedge of spheres Y = S n 1 S n k. Let ζj : S n j Y be the inclusion for j = 1,..., k. We define order the basic Whitehead products as follows: Basic products of weight 1 are ζ 1,..., ζ k which are ordered by ζ 1 < < ζ k. Assume that the basic products of weight < n have been defined ordered so that if r < s < n, any basic product of weight r is less than all basic products of weight s. We define inductively a basic product of weight n by a Whitehead product [a, b], where a is a basic product of weight k b is a basic product of weight l, k + l = n, a < b, if b is a Whitehead product [c, d] of basic products c d, then c a. The basic products of weight n are then ordered arbitrarily among themselves are greater than any basic product of weight < n. Suppose ζ j occurs l j times, l j 1 in the basic product w v. Then the height h v of the basic product w v is l j (n j 1) + 1. There is the Hilton s theorem [12] concerning the basic Whitehead products as follows:

3 D.-W. Lee / Filomat 30:13 (2016), Theorem 2.1. Let the basic products of Y = S n 1 S n k be w1, w 2,..., w v,... (in order) with the height of w v being h v. Then for every m, π m (Y) π m (S h v ). The isomorphism θ : v=1 π m(s h v ) π m (Y) is defined by v=1 θ π m (S h v ) = w v : π m (S h v ) π m (Y). Let ϕ : S n S n S n be the stard comultiplication. Then we have the induced homomorphism ϕ : π m (S n ) π m (S n S n ) between homotopy groups by the Hilton s theorem π m (S n S n ) π m (S n ) π m (S n ) π m (S 2n 1 ) π m (S 3n 2 ). Let h v be the height of the vth basic product w v π hv (S n S n ) let pr v : π m (S n S n ) π m (S h v ) be the projection onto the vth summ for v = 1, 2, 3,.... We define the Hopf-Hilton invariant H t : π m (S n ) π m (S h t+3 ) by as in [12]. We rewrite these invariants as follows: Set so on. Thus we have the Hopf-Hilton invariants H t = pr t+3 ϕ, t = 0, 1, 2,... H 1 1 = H 0 : π m (S n ) π m (S 2n 1 ) H 2 1 = H 1, H 2 2 = H 2 : π m (S n ) π m (S 3n 2 ) H l 1, Hl 2,..., Hl t l : π m (S n ) π m (S (l+1)n l ), for l = 1, 2, 3,..., where t l is the number of basic products of length l. We now describe the Hilton s formulas [12, Theorems ] as follows: (1) If Z is any space β, γ π r (Z) α π m (S r ), then (β + γ)α = βα + γα + [β, γ]h 1 1 (α) +[β, [β, γ]]h 2 1 (α) + [γ, [β, γ]]h2 2 (α) (2) Let a π n (S r ), b π r (Z), let m be an integer. Then, if r is odd, { m(b a) for m = 4s or 4s + 1, mb a = m(b a) + [b, b] H 1 (a) for m = 4s 2 or 4s 1. 1 If r is even, mb a = m(b a) + m(m 1) 2 [b, b] H 1 1 (a) + (m+1)m(m 1) 3 [b, [b, b]] H 2 1 (a). From now on, we let X = S n S m Y = S n S m S p with 2 n < m < p unless otherwise stated, we fix the following notations: i 1, i 2 : S n S n S n are the first the second inclusions, respectively. j 1, j 2 : S m S m S m are the first the second inclusions, respectively. α 1, α 2 : S n S n S m S n S m (= X X) are the first the third inclusions, respectively. β 1, β 2 : S m S n S m S n S m (= X X) are the second the fourth inclusions, respectively. r : S n Y, s : S m Y, t : S p Y A : S n S m Y are the inclusions.

4 D.-W. Lee / Filomat 30:13 (2016), pr 1, pr 2 : (S n S m ) (S n S m ) S n S m are the first the second projections, respectively. ι 1, ι 2 : Y Y Y are the first the second inclusions, respectively. q 1, q 2 : Y Y Y are the first the second projections, respectively. I 1, I 2, I 3 : Y Y Y Y are the first, the second the third inclusions, respectively. J 12, J 23 : Y Y Y Y Y are the maps defined by J 12 (z) = (z, ) J 23 (z) = (, z), where z Y Y. T : Y Y Y Yis the switching map. Then the following are straightforward: (r r)i 1 = ι 1 r, (r r)i 2 = ι 2 r, (s s)j 1 = ι 1 s, (s s)j 2 = ι 2 s. (A A)α j = ι j r, (A A)β j = ι j s for j = 1, 2. J 12 ι j = I j for j = 1, 2. (ϕ 1)ι 1 = J 12 ϕ (ϕ 1)ι 2 = I 3. J 23 ι 1 = I 2 J 23 ι 2 = I 3. (1 ϕ)ι 1 = I 1 (1 ϕ)ι 2 = J 23 ϕ. We now order the basic products of weight 1 as follows: α 1 < β 1 < α 2 < β 2 in S n S m S n S m. ι 1 r < ι 1 s < ι 2 r < ι 2 s in Y Y. I 1 r < I 1 s < I 2 r < I 2 s < I 3 r < I 3 s in Y Y Y. For the moment, more generally, let Y = S n 1 S n 2 S n k, X = S n 1 S n 2 S n k 1, 2 n1 < n 2 < < n k let ζ i : S n i Y be the inclusion for each i = 1, 2,..., k pr 1, pr 2 : X X X the first the second projections, respectively. Then every comultiplication ϕ : Y Y Y has the following form ϕ S n 1 = ι 1 ζ 1 + ι 2 ζ 1, ϕ S n 2 = ι 1 ζ 2 + ι 2 ζ 2 + P n2,. ϕ S n k = ι 1 ζ k + ι 2 ζ k + P nk for some P ni π ni (Y Y) such that q 1 P ni = q 2 P ni = 0 for each i = 2, 3,..., k. Conversely, if P ni : S n i Y Y is any map such that q 1 P ni = q 2 P ni = 0 for each i = 2, 3,..., k, then ϕ defined above is a comultiplication. Therefore, we can define a comultiplication as follows: Definition 2.2. Let P ni π ni (Y Y) be any map such that q 1 P ni = 0 = q 2 P ni for each i = 2, 3,..., k. We define a comultiplication ϕ : Y Y Y by ϕ S n 1 = ι 1 ζ 1 + ι 2 ζ 1, ϕ S n 2 = ι 1 ζ 2 + ι 2 ζ 2 + P n2,. ϕ S n k = ι 1 ζ k + ι 2 ζ k + P nk. The element P ni π ni (Y Y) is called the ith perturbation of ϕ for i = 2, 3,..., k. We call P = (P n2, P n3,..., P nk ) the perturbation of ϕ.

5 D.-W. Lee / Filomat 30:13 (2016), It can be seen that the comultiplication ϕ in Definition 2.2 is commutative if only if TP ni = P ni for each i = 2, 3,..., k; it is associative if only if for each i = 2, 3,..., k. Let A : X Y be the inclusion. Then we have J 12 P ni + (ϕ 1)P ni = J 23 P ni + (1 ϕ)p ni Lemma 2.3. If P nk π nk (Y Y) satisfies q 1 P nk = 0 = q 2 P nk, then there exists a unique W π nk (X X) such that P nk = (A A)W pr 1 W = 0 = pr 2 W. Proof. We consider the following cofibration sequence: X X A A Y Y q q S n k S n k, where q : Y S n k is the projection. Since X X is (n1 1)-connected S n k S n k is (nk 1)-connected, by the Blakers-Massey Theorem (see [13] [17]), the top line in the following commutative diagram is exact π nk (X X) (A A) π nk (Y Y) (q q) π nk (S n k S n k) f π nk (X X) (A A) π nk (Y Y) (q q) e π nk (S n k S n k), where f, e are homomorphisms induced by inclusion maps. We can see that e is an isomorphism. If P nk π nk (Y Y) is such that q 1 P nk = 0 = q 2 P nk, then (P nk ) = 0 so e (q q) P nk = 0. Therefore (q q) P nk = 0 hence, by exactness, there exists a unique W π nk (X X) such that P nk = (A A) W since (A A) is a monomorphism. Finally, since (A A) f W = 0 (A A) is a monomorphism, it follows that f W = 0. Thus we have pr 1 W = 0 = pr 2 W since π nk (X X) π nk (X) π nk (X). We note that there are various types of wedges of spheres commutative diagrams satisfying the statements in the proof of Lemma 2.3. Fortunately, all kinds of wedges of spheres commutative diagrams are condensed into the above case by using the fact that the inclusion map from a wedge of spheres to another induces a monomorphism between homotopy groups. The computation of comultiplications on a wedge of spheres is very complicated. So we restrict our investigation to a wedge of three spheres which is much more complicate than the wedge of two spheres. From now on, in the case of a wedge of three spheres, we use the notation Y = S n S m S p instead of S n 1 S n 2 S n 3, i.e., n = n 1, m = n 2, p = n 3. And we also use the inclusions r : S n Y, s : S m Y t : S p Y instead of ζ 1, ζ 2 ζ 3, respectively, as previously mentioned. Let W be the set of all basic Whitehead products of S n S m S n S m generated by {α 1, β 1, α 2, β 2 }. Let W 1 denote the set of basic Whitehead products generated by {α 1, β 1 } or by {α 2, β 2 }, let W 2 = W W 1. That is, W 1 is the set of basic Whitehead products like [α 1, β 1 ], [α 2, β 2 ], [α 2, [α 2, β 2 ]], [[α 1, β 1 ], [α 1, [α 1, β 1 ]]], W 2 is the set of basic Whitehead products such as [α 1, α 2 ], [α 1, β 2 ], [α 2, [α 1, β 1 ]], [[α 1, β 1 ], [α 2, [α 2, β 2 ]]],. Indeed, W 2 is the set of basic Whitehead products containing at least a 1 at least a 2 in the subscript of the basic Whitehead products. Lemma 2.4. Let pr 1, pr 2 : π p(s n S m S n S m ) π p (S n S m ) be the homomorphisms induced by the first the second projections pr 1 pr 2, respectively. Then Ker(pr 1 ) Ker(pr 2 ) ω W 2 π p (S hω ), where h ω is the height of the basic Whitehead products ω W 2.

6 D.-W. Lee / Filomat 30:13 (2016), Proof. If x ω W 2 π p (S h ω ), then the Hilton s theorem shows that x can be written as an element of π p (S n S m S n S m ) as follows: x = [α 2, [α 1, β 1 ]]ψ [β 2, [α 1, β 1 ]]ψ [α 2, [α 2, [α 1, β 1 ]]]ψ [α 1, α 2 ]ψ [α 1, [α 1, α 2 ]]ψ [β 1, [α 1, α 2 ]]ψ [α 1, β 2 ]ψ [α 1, [α 1, β 2 ]]ψ [β 1, [α 1, β 2 ]]ψ [β 1, α 2 ]ψ [β 1, [β 1, α 2 ]]ψ [β 2, [β 1, α 2 ]]ψ [β 1, β 2 ]ψ [β 1, [β 1, β 2 ]]ψ [β 2, [β 1, β 2 ]]ψ for ψ j π i p (S h ω ) ω W 2. Here, i j are finite for each p since h ω, the summ π p (S h ω ) is embedded into the sum via composition with ω : S h ω S n S m S n S m. By applying pr 1 pr 2 on both sides, we have pr 1 (x) = 0 = pr 2 (x) because all the basic Whitehead products describe above are the elements of W 2. Indeed, all the basic Whitehead products in W 2 are killed off by pr 1 pr 2. For example, pr 1 ([α 1, [α 1, β 2 ]]) = [pr 1 α 1, [pr 1 α 1, pr 1 β 2 ]] = [pr 1 α 1, [pr 1 α 1, 0]] = 0, similarly for pr 2. We can also check the reverse inclusion. Theorem 2.5. Every comultiplication ϕ : Y Y Y on Y = S n S m S p can be uniquely written in the following form ϕ S n = ι 1 r + ι 2 r, ϕ S m = ι 1 s + ι 2 s + (r r) W P, ϕ S p = ι 1 t + ι 2 t + (A A) W Q. Here, (1) W P = w u γ u π m (S n S n ), w u is the uth basic product in S n S n γ u π m (S h u ) for u = 3, 4,..., u=3 (2) W Q π p (S n S m S n S m ) is W Q = ω W 2 ωψ ω, where ω runs over all the basic Whitehead products in W 2, ψ ω π p (S h ω ), h ω is the height of the basic Whitehead products ω W 2. Proof. We prove the result for W Q. By Lemma 2.3, ϕ has the form stated in this theorem for some W Q π p (S n S m S n S m ) with pr 1 W Q = 0 = pr 2 W Q. Let x ω W 2 π p (S h ω ) as in the proof of Lemma 2.4 y ω W 1 π p (S h ω ). Then, by using the Hilton s theorem, we have W Q = α 1 ρ 1 + β 1 ρ 2 + α 2 ρ 3 + β 2 ρ 4 + x + y for ρ 1, ρ 3 π p (S n ) ρ 2, ρ 4 π p (S m ). By Lemmas , by applying pr 1 ( then pr 2 ) on the above equation, we get 0 = pr 1 W Q = pr 1 (α 1 ρ 1 ) + pr 1 (β 1 ρ 2 ) + pr 1 (α 2 ρ 3 ) + pr 1 (β 2 ρ 4 ) + pr 1 (x) + pr 1 (y) = ξ(n)ρ 1 + ξ(m)ρ pr 1 (y), where ξ(n) : S n S n S m ξ(m) : S m S n S m are the inclusion maps. By the Hilton s theorem, ξ(n)ρ 1 = 0 = ξ(m)ρ 2 pr 1 (y) = 0. Since the inclusions ξ(n) ξ(m) induce monomorphisms between homotopy groups, we have ρ 1 = 0 = ρ 2. Similarly, by applying pr 2 on W Q, we have ρ 3 = 0 = ρ 4.

7 D.-W. Lee / Filomat 30:13 (2016), We now consider the case of basic Whitehead products in W 1, namely, [α 1, β 1 ], [α 1, [α 1, β 1 ]], [α 2, β 2 ], [α 2, [α 2, β 2 ]] so on. By the Hilton s theorem, Lemmas , y ω W 1 π p (S h ω ) must satisfy pr 1 (y) = 0 = pr 2 (y), thus it has to be the following type Therefore [α 1, β 1 ]0 + [α 1, [α 1, β 1 ]] [α 2, β 2 ]0 + [α 2, [α 2, β 2 ]] W Q = ω W 2 ωψ ω, where ψ ω π p (S h ω ), as required. The proof in the case of W P goes to the same way. By [1, p. 1167] we can determine the number of comultiplications of a co-h-space. By using Theorem 2.5, we now get a concrete clue how to calculate the number of comultiplications as follows: Corollary 2.6. The number of comultiplications of Y = S n S m S p is π m (S h u ) π p (S hω ). ω W 2 u=3 By using the above results the homotpy groups of spheres [16], we now provide an example. Example 2.7. The cardinality of the set of comultiplications of S 8 S 12 S 27 is 512. Indeed, we let C(Y) be the number of comultiplications of Y. Then C(S 8 S 12 S 27 ) = π 27 (S 26 ) π 27 (S 15 ) π 27 (S 19 ) 2 π 27 (S 23 ) π 27 (S 22 ) 2 π 27 (S 26 ) 2 π 27 (S 26 ) = = Associativity on a wedge of three spheres In this section, we concentrate our attention on the associative comultiplications on a wedge of three spheres. Definition 3.1. Let Y = S n S m S p. We then define a comultiplication ϕ = ϕ P,Q : Y Y Y by ϕ P,Q S n = ι 1 r + ι 2 r, ϕ P,Q S m = ι 1 s + ι 2 s + P, ϕ P,Q S p = ι 1 t + ι 2 t + Q, where P = (r r)[i 1, i 2 ]γ, γ π m (S 2n 1 ), Q : S p Y Y is a map such that q 1 Q = 0 = q 2 Q. We are especially interested in studying the homotopy classes Q π p (Y Y) in order to get some information for the given comultiplication ϕ : Y Y Y. We first consider the perturbations Q j, j = 1, 2,..., 5 of comultiplications ϕ = ϕ P,Qj, j = 1, 2,..., 5 constructed below, then define a perturbation Q π p (Y Y) of a comultiplication ϕ : Y Y Y by Q = in order to get some information about comultiplications, associative comultiplications, commutative comultiplications, comultiplications which are both associative commutative. 5 j=1 Notation In Sections 3 4, we will make use of the symbols ᾱ 1, ᾱ 2 : S n Y Y, ˆα 1, ˆα 2, ˆα 3 : S n Y Y Y, β 1, β 2 : S m Y Y, ˆβ 1, ˆβ 2, ˆβ 3 : S m Y Y Y to denote ι 1 r, ι 2 r, I 1 r, I 2 r, I 3 r, ι 1 s, ι 2 s, I 1 s, I 2 s, I 3 s, respectively. Q j

8 D.-W. Lee / Filomat 30:13 (2016), Definition 3.2. For any homotopy classes a 1 π p (S 2m 1 ), a 2, a 3 π p (S 3m 2 ), b 1 π p (S 2n 1 ), b 2, b 3 π p (S 3n 2 ), c 1 π p (S n+m 1 ), c 2, c 3 π p (S m+2n 2 ) c 4, c 5 π p (S 2m+n 2 ), we define Q 1 π p (Y Y) by Q 1 = [ β 1, β 2 ]a 1 + [ β 1, [ β 1, β 2 ]]a 2 + [ β 2, [ β 1, β 2 ]]a 3 +[ᾱ 1, ᾱ 2 ]b 1 + [ᾱ 1, [ᾱ 1, ᾱ 2 ]]b 2 + [ᾱ 2, [ᾱ 1, ᾱ 2 ]]b 3 +[ᾱ 1, β 2 ]c 1 + [ᾱ 1, [ᾱ 1, β 2 ]]c 2 + [ᾱ 2, [ᾱ 1, β 2 ]]c 3 +[ β 1, [ᾱ 1, β 2 ]]c 4 + [ β 2, [ᾱ 1, β 2 ]]c 5. We then define a comultiplication ϕ = ϕ P,Q1 : Y Y Y by substituting Q 1 for Q in Definition 3.1. Proposition 3.3. Let m 2n 2 n < m < p 8n 8. If the Hopf-Hilton invariants H 1 1 (x) = 0 = H2 (y), where 1 x = a i, b i, c j for i = 1, 2, 3, j = 1, 2,..., 5 y = a 1, b 1, c 1, then ϕ = ϕ P,Q1 is associative if only if (1) m n are even, a 2 = a 3 = b 2 = b 3 = c 2 = c 3 = c 4 = c 5 = 0, or (2) m is even n is odd, a 2 = a 3 = c 2 = c 3 = c 4 = c 5 = 0, b 2 = b 3 3b 2 = 0, or (3) m is odd n is even, b 2 = b 3 = c 2 = c 3 = c 4 = c 5 = 0, a 2 = a 3 3a 2 = 0, or (4) m n are odd, c 2 = c 3 = c 4 = c 5 = 0, a 2 = a 3, b 2 = b 3 3a 2 = 0 = 3b 2. Proof. We recall that a necessary sufficient condition for the comultiplication ϕ = ϕ P,Q1 to be associative is that J 12 P + (ϕ 1)P = J 23 P + (1 ϕ)p J 12 Q 1 + (ϕ 1)Q 1 = J 23 Q 1 + (1 ϕ)q 1. It suffices to prove the second equation because P = [ι 1 r, ι 2 r]γ = 0. We compute J 12 Q 1 : We now compute (ϕ 1)Q 1 : J 12 Q 1 = J 12 ( [ β 1, β 2 ]a 1 + [ β 1, [ β 1, β 2 ]]a 2 + [ β 2, [ β 1, β 2 ]]a 3 +[ᾱ 1, ᾱ 2 ]b 1 + [ᾱ 1, [ᾱ 1, ᾱ 2 ]]b 2 + [ᾱ 2, [ᾱ 1, ᾱ 2 ]]b 3 +[ᾱ 1, β 2 ]c 1 + [ᾱ 1, [ᾱ 1, β 2 ]]c 2 + [ᾱ 2, [ᾱ 1, β 2 ]]c 3 +[ β 1, [ᾱ 1, β 2 ]]c 4 + [ β 2, [ᾱ 1, β 2 ]]c 5 ) = [ ˆβ 1, ˆβ 2 ]a 1 + [ ˆβ 1, [ ˆβ 1, ˆβ 2 ]]a 2 + [ ˆβ 2, [ ˆβ 1, ˆβ 2 ]]a 3 +[ ˆα 1, ˆα 2 ]b 1 + [ ˆα 1, [ ˆα 1, ˆα 2 ]]b 2 + [ ˆα 2, [ ˆα 1, ˆα 2 ]]b 3 +[ ˆα 1, ˆβ 2 ]c 1 + [ ˆα 1, [ ˆα 1, ˆβ 2 ]]c 2 + [ ˆα 2, [ ˆα 1, ˆβ 2 ]]c 3 +[ ˆβ 1, [ ˆα 1, ˆβ 2 ]]c 4 + [ ˆβ 2, [ ˆα 1, ˆβ 2 ]]c 5. (ϕ 1)Q 1 = [J 12 ϕs, ˆβ 3 ]a 1 + [J 12 ϕs, [J 12 ϕs, ˆβ 3 ]]a 2 + [ ˆβ 3, [J 12 ϕs, ˆβ 3 ]]a 3 +[J 12 ϕr, ˆα 3 ]b 1 + [J 12 ϕr, [J 12 ϕr, ˆα 3 ]]b 2 + [ ˆα 3, [J 12 ϕr, ˆα 3 ]]b 3 +[J 12 ϕr, ˆβ 3 ]c 1 + [J 12 ϕr, [J 12 ϕr, ˆβ 3 ]]c 2 + [ ˆα 3, [J 12 ϕr, ˆβ 3 ]]c 3 +[J 12 ϕs, [J 12 ϕr, ˆβ 3 ]]c 4 + [ ˆβ 3, [J 12 ϕr, ˆβ 3 ]]c 5 = [ ˆβ 1 + ˆβ 2, ˆβ 3 ]a 1 + [ ˆβ 1 + ˆβ 2, [ ˆβ 1 + ˆβ 2, ˆβ 3 ]]a 2 +[ ˆβ 3, [ ˆβ 1 + ˆβ 2, ˆβ 3 ]]a 3 + [ ˆα 1 + ˆα 2, ˆα 3 ]b 1 +[ ˆα 1 + ˆα 2, [ ˆα 1 + ˆα 2, ˆα 3 ]]b 2 + [ ˆα 3, [ ˆα 1 + ˆα 2, ˆα 3 ]]b 3 +[ ˆα 1 + ˆα 2, ˆβ 3 ]c 1 + [ ˆα 1 + ˆα 2, [ ˆα 1 + ˆα 2, ˆβ 3 ]]c 2 +[ ˆα 3, [ ˆα 1 + ˆα 2, ˆβ 3 ]]c 3 + [ ˆβ 1 + ˆβ 2, [ ˆα 1 + ˆα 2, ˆβ 3 ]]c 4 +[ ˆβ 3, [ ˆα 1 + ˆα 2, ˆβ 3 ]]c 5 = ( [ ˆβ 1, ˆβ 3 ] + [ ˆβ 2, ˆβ 3 ] ) a 1 + ( [ ˆβ 1, [ ˆβ 1, ˆβ 3 ]] + [ ˆβ 1, [ ˆβ 2, ˆβ 3 ]] +[ ˆβ 2, [ ˆβ 1, ˆβ 3 ]] + [ ˆβ 2, [ ˆβ 2, ˆβ 3 ]] ) a 2 + ( [ ˆβ 3, [ ˆβ 1, ˆβ 3 ]] +[ ˆβ 3, [ ˆβ 2, ˆβ 3 ]] ) a 3 + ( [ ˆα 1, ˆα 3 ] + [ ˆα 2, ˆα 3 ] ) b 1 ( [ ˆα1, [ ˆα 1, ˆα 3 ]] + [ ˆα 1, [ ˆα 2, ˆα 3 ]] + [ ˆα 2, [ ˆα 1, ˆα 3 ]]

9 D.-W. Lee / Filomat 30:13 (2016), [ ˆα 2, [ ˆα 2, ˆα 3 ]] ) b 2 + ( [ ˆα 3, [ ˆα 1, ˆα 3 ]] + [ ˆα 3, [ ˆα 2, ˆα 3 ]] ) b 3 ( [ ˆα1, ˆβ 3 ] + [ ˆα 2, ˆβ 3 ] ) c 1 + ( [ ˆα 1, [ ˆα 1, ˆβ 3 ]] + [ ˆα 1, [ ˆα 2, ˆβ 3 ]] +[ ˆα 2, [ ˆα 1, ˆβ 3 ]] + [ ˆα 2, [ ˆα 2, ˆβ 3 ]] ) c 2 + ( [ ˆα 3, [ ˆα 1, ˆβ 3 ]] +[ ˆα 3, [ ˆα 2, ˆβ 3 ]] ) c 3 + ( [ ˆβ 1, [ ˆα 1, ˆβ 3 ]] + [ ˆβ 1, [ ˆα 2, ˆβ 3 ]] +[ ˆβ 2, [ ˆα 1, ˆβ 3 ]] + [ ˆβ 2, [ ˆα 2, ˆβ 3 ]] ) c 4 + ( [ ˆβ 3, [ ˆα 1, ˆβ 3 ]] +[ ˆβ 3, [ ˆα 2, ˆβ 3 ]] ) c 5. Since the Whitehead products such as [ ˆβ 1, [ ˆβ 2, ˆβ 3 ]], [ ˆα 1, [ ˆα 2, ˆα 3 ]], [ ˆα 1, [ ˆα 2, ˆβ 3 ]] [ ˆβ 1, [ ˆα 2, ˆβ 3 ]] in the above equations are not basic products, we need to change them into the basic products by using the Jacobi identity [17, p. 478] as follows: 0 = ( 1) m(m 1) [ ˆβ 1, [ ˆβ 2, ˆβ 3 ]] + ( 1) m(m 1)+m2 [ ˆβ 2, [ ˆβ 1, ˆβ 3 ]] +( 1) m(m 1) [ ˆβ 3, [ ˆβ 1, ˆβ 2 ]], 0 = ( 1) n(m 1) [ ˆα 1, [ ˆα 2, ˆβ 3 ]] + ( 1) n(n 1)+mn [ ˆα 2, [ ˆα 1, ˆβ 3 ]] +( 1) m(n 1) [ ˆβ 3, [ ˆα 1, ˆα 2 ]]. Similarly, for [ ˆα 1, [ ˆα 2, ˆα 3 ]] [ ˆβ 1, [ ˆα 2, ˆβ 3 ]]. More precisely, [ ˆβ 1, [ ˆβ 2, ˆβ 3 ]] = [ ˆα 1, [ ˆα 2, ˆα 3 ]] = [ ˆα 1, [ ˆα 2, ˆβ 3 ]] = { +[ ˆβ 2, [ ˆβ 1, ˆβ 3 ]] [ ˆβ 3, [ ˆβ 1, ˆβ 2 ]] for m odd, [ ˆβ 2, [ ˆβ 1, ˆβ 3 ]] [ ˆβ 3, [ ˆβ 1, ˆβ 2 ]] for m even, { +[ ˆα2, [ ˆα 1, ˆα 3 ]] [ ˆα 3, [ ˆα 1, ˆα 2 ]] for n odd, [ ˆα 2, [ ˆα 1, ˆα 3 ]] [ ˆα 3, [ ˆα 1, ˆα 2 ]] for n even, [ ˆα 2, [ ˆα 1, ˆβ 3 ]] [ ˆβ 3, [ ˆα 1, ˆα 2 ]] for m, n even, +[ ˆα 2, [ ˆα 1, ˆβ 3 ]] + [ ˆβ 3, [ ˆα 1, ˆα 2 ]] for m even, n odd, [ ˆα 2, [ ˆα 1, ˆβ 3 ]] + [ ˆβ 3, [ ˆα 1, ˆα 2 ]] for m odd, n even, +[ ˆα 2, [ ˆα 1, ˆβ 3 ]] [ ˆβ 3, [ ˆα 1, ˆα 2 ]] for m, n odd, [ ˆβ 1, [ ˆα 2, ˆβ 3 ]] = [ ˆα 2, [ ˆβ 1, ˆβ 3 ]] [ ˆβ 3, [ ˆβ 1, ˆα 2 ]] for m, n even, +[ ˆα 2, [ ˆβ 1, ˆβ 3 ]] [ ˆβ 3, [ ˆβ 1, ˆα 2 ]] for m even, n odd, +[ ˆα 2, [ ˆβ 1, ˆβ 3 ]] + [ ˆβ 3, [ ˆβ 1, ˆα 2 ]] for m odd, n even, +[ ˆα 2, [ ˆβ 1, ˆβ 3 ]] [ ˆβ 3, [ ˆβ 1, ˆα 2 ]] for m, n odd. We now compute J 23 Q 1 (1 ϕ)q 1 : J 23 Q 1 = [ ˆβ 2, ˆβ 3 ]a 1 + [ ˆβ 2, [ ˆβ 2, ˆβ 3 ]]a 2 + [ ˆβ 3, [ ˆβ 2, ˆβ 3 ]]a 3 +[ ˆα 2, ˆα 3 ]b 1 + [ ˆα 2, [ ˆα 2, ˆα 3 ]]b 2 + [ ˆα 3, [ ˆα 2, ˆα 3 ]]b 3 +[ ˆα 2, ˆβ 3 ]c 1 + [ ˆα 2, [ ˆα 2, ˆβ 3 ]]c 2 + [ ˆα 3, [ ˆα 2, ˆβ 3 ]]c 3 +[ ˆβ 2, [ ˆα 2, ˆβ 3 ]]c 4 + [ ˆβ 3, [ ˆα 2, ˆβ 3 ]]c 5 (1 ϕ)q 1 = [ ˆβ 1, J 23 ϕs]a 1 + [ ˆβ 1, [ ˆβ 1, J 23 ϕs]]a 2 + [J 23 ϕs, [ ˆβ 1, J 23 ϕs]]a 3 +[ ˆα 1, J 23 ϕr]b 1 + [ ˆα 1, [ ˆα 1, J 23 ϕr]]b 2 + [J 23 ϕr, [ ˆα 1, J 23 ϕr]]b 3 +[ ˆα 1, J 23 ϕs]c 1 + [ ˆα 1, [ ˆα 1, J 23 ϕs]]c 2 + [J 23 ϕr, [ ˆα 1, J 23 ϕs]]c 3 +[ ˆβ 1, [ ˆα 1, J 23 ϕs]]c 4 + [J 23 ϕs, [ ˆα 1, J 23 ϕs]]c 5

10 D.-W. Lee / Filomat 30:13 (2016), = [ ˆβ 1, ˆβ 2 + ˆβ 3 ]a 1 + [ ˆβ 1, [ ˆβ 1, [ ˆβ 2 + ˆβ 3 ]]a 2 +[ ˆβ 2 + ˆβ 3, [ ˆβ 1, ˆβ 2 + ˆβ 3 ]]a 3 + [ ˆα 1, ˆα 2 + ˆα 3 ]b 1 +[ ˆα 1, [ ˆα 1, ˆα 2 + ˆα 3 ]]b 2 + [ ˆα 2 + ˆα 3, [ ˆα 1, ˆα 2 + ˆα 3 ]]b 3 +[ ˆα 1, ˆβ 2 + ˆβ 3 ]c 1 + [ ˆα 1, [ ˆα 1, ˆβ 2 + ˆβ 3 ]]c 2 +[ ˆα 2 + ˆα 3, [ ˆα 1, ˆβ 2 + ˆβ 3 ]]c 3 + [ ˆβ 1, [ ˆα 1, ˆβ 2 + ˆβ 3 ]]c 4 +[ ˆβ 2 + ˆβ 3, [ ˆα 1, ˆβ 2 + ˆβ 3 ]]c 5 = ( [ ˆβ 1, ˆβ 2 ] + [ ˆβ 1, ˆβ 3 ] ) a 1 + ( [ ˆβ 1, [ ˆβ 1, ˆβ 2 ]] + [ ˆβ 1, [ ˆβ 1, ˆβ 3 ]] ) a 2 + ( [ ˆβ 2, [ ˆβ 1, ˆβ 2 ]] + [ ˆβ 2, [ ˆβ 1, ˆβ 3 ]] + [ ˆβ 3, [ ˆβ 1, ˆβ 2 ]] + [ ˆβ 3, [ ˆβ 1, ˆβ 3 ]] ) a 3 + ( [ ˆα 1, ˆα 2 ] + [ ˆα 1, ˆα 3 ] ) b 1 + ( [ ˆα 1, [ ˆα 1, ˆα 2 ]] + [ ˆα 1, [ ˆα 1, ˆα 3 ]] ) b 2 + ( [ ˆα 2, [ ˆα 1, ˆα 2 ]] + [ ˆα 2, [ ˆα 1, ˆα 3 ]] + [ ˆα 3, [ ˆα 1, ˆα 2 ]] + [ ˆα 3, [ ˆα 1, ˆα 3 ]] ) b 3 + ( [ ˆα 1, ˆβ 2 ] + [ ˆα 1, ˆβ 3 ] ) c 1 + ( [ ˆα 1, [ ˆα 1, ˆβ 2 ]] + [ ˆα 1, [ ˆα 1, ˆβ 3 ]] ) c 2 + ( [ ˆα 2, [ ˆα 1, ˆβ 2 ]] + [ ˆα 2, [ ˆα 1, ˆβ 3 ]] + [ ˆα 3, [ ˆα 1, ˆβ 2 ]] + [ ˆα 3, [ ˆα 1, ˆβ 3 ]] ) c 3 + ( [ ˆβ 1, [ ˆα 1, ˆβ 2 ]] + [ ˆβ 1, [ ˆα 1, ˆβ 3 ]] ) c 4 + ( [ ˆβ 2, [ ˆα 1, ˆβ 2 ]] + [ ˆβ 2, [ ˆα 1, ˆβ 3 ]] + [ ˆβ 3, [ ˆα 1, ˆβ 2 ]] + [ ˆβ 3, [ ˆα 1, ˆβ 3 ]] ) c 5. We now prove the following two cases since the other two cases are similar to the first two cases. We note that ( 1)x = x + [1, 1]H 1 (x) = x by the Hilton s formula, where 1 is the identity map of a sphere. We also 1 observe that the following Whitehead products in (1) (2) are all the basic products. (1) If m n are even, then the equality J 12 Q 1 + (ϕ 1)Q 1 = J 23 Q 1 + (1 ϕ)q 1 holds if only if 0 = ( [ ˆβ 3, [ ˆβ 1, ˆβ 2 ]])a 2 + ( [ ˆα 3, [ ˆα 1, ˆα 2 ]])b 2 + ( [ ˆβ 3, [ ˆα 1, ˆα 2 ]])c 2 +( [ ˆα 2, [ ˆβ 1, ˆβ 3 ]])c 4 + [ ˆβ 2, [ ˆα 1, ˆβ 3 ]]c 4 + ( [ ˆβ 3, [ ˆβ 1, ˆα 2 ]])c 4 ([ ˆβ 2, [ ˆβ 1, ˆβ 3 ]] + [ ˆβ 3, [ ˆβ 1, ˆβ 2 ]])a 3 ([ ˆα 2, [ ˆα 1, ˆα 3 ]] + [ ˆα 3, [ ˆα 1, ˆα 2 ]])b 3 ([ ˆα 2, [ ˆα 1, ˆβ 3 ]] + [ ˆα 3, [ ˆα 1, ˆβ 2 ]])c 3 ([ ˆβ 2, [ ˆα 1, ˆβ 3 ]] + [ ˆβ 3, [ ˆα 1, ˆβ 2 ]])c 5 = [ ˆβ 3, [ ˆβ 1, ˆβ 2 ]]( a 2 a 3 ) + [ ˆβ 2, [ ˆβ 1, ˆβ 3 ]]( a 3 ) +[ ˆα 3, [ ˆα 1, ˆα 2 ]]( b 2 b 3 ) + [ ˆα 2, [ ˆα 1, ˆα 3 ]]( b 3 ) +[ ˆβ 3, [ ˆα 1, ˆα 2 ]( c 2 ) + [ ˆα 2, [ ˆα 1, ˆβ 3 ]]( c 3 ) + [ ˆα 3, [ ˆα 1, ˆβ 2 ]]( c 3 ) +[ ˆα 2, [ ˆβ 1, ˆβ 3 ]]( c 4 ) + [ ˆβ 3, [ ˆβ 1, ˆα 2 ]]( c 4 ) +[ ˆβ 2, [ ˆα 1, ˆβ 3 ]](c 4 c 5 ) + [ ˆβ 3, [ ˆα 1, ˆβ 2 ]( c 5 ). Thus ϕ is associative if only if a 2 = a 3 = b 2 = b 3 = c 2 = c 3 = c 4 = c 5 = 0. (2) If m is even n is odd, then the associativity of ϕ = ϕ P,Q1 holds if only if 0 = ( [ ˆβ 3, [ ˆβ 1, ˆβ 2 ]])a 2 + 2[ ˆα 2, [ ˆα 1, ˆα 3 ]]b 2 + ( [ ˆα 3, [ ˆα 1, ˆα 2 ]])b 2 +2[ ˆα 2, [ ˆα 1, ˆβ 3 ]]c 2 + [ ˆβ 3, [ ˆα 1, ˆα 2 ]]c 2 + [ ˆα 2, [ ˆβ 1, ˆβ 3 ]]c 4 +( [ ˆβ 3, [ ˆβ 1, ˆα 2 ]])c 4 + [ ˆβ 2, [ ˆα 1, ˆβ 3 ]]c 4 ([ ˆβ 2, [ ˆβ 1, ˆβ 3 ]] + [ ˆβ 3, [ ˆβ 1, ˆβ 2 ]])a 3 ([ ˆα 2, [ ˆα 1, ˆα 3 ]] + [ ˆα 3, [ ˆα 1, ˆα 2 ]])b 3 ([ ˆα 2, [ ˆα 1, ˆβ 3 ]] + [ ˆα 3, [ ˆα 1, ˆβ 2 ]])c 3 ([ ˆβ 2, [ ˆα 1, ˆβ 3 ]] + [ ˆβ 3, [ ˆα 1, ˆβ 2 ]])c 5 = [ ˆβ 3, [ ˆβ 1, ˆβ 2 ]]( a 2 a 3 ) + [ ˆβ 2, [ ˆβ 1, ˆβ 3 ]]( a 3 ) +[ ˆα 3, [ ˆα 1, ˆα 2 ]]( b 2 b 3 ) + [ ˆα 2, [ ˆα 1, ˆα 3 ]](2b 2 b 3 ) +[ ˆα 2, [ ˆα 1, ˆβ 3 ]](2c 2 c 3 ) + [ ˆβ 3, [ ˆα 1, ˆα 2 ]c 2 +[ ˆα 3, [ ˆα 1, ˆβ 2 ]]( c 3 ) + [ ˆα 2, [ ˆβ 1, ˆβ 3 ]]c 4 +[ ˆβ 3, [ ˆβ 1, ˆα 2 ]]( c 4 ) + [ ˆβ 2, [ ˆα 1, ˆβ 3 ]](c 4 c 5 ) +[ ˆβ 3, [ ˆα 1, ˆβ 2 ]( c 5 ). Thus ϕ is associative if only if a 2 = a 3 = c 2 = c 3 = c 4 = c 5 = 0, b 2 = b 3 3b 2 = 0.

11 D.-W. Lee / Filomat 30:13 (2016), Remark 3.4. We note that if p 4n 4, then the conclusion of Proposition 3.3 holds because x y are suspensions by the Freudenthal suspension theorem [17] thus the Hopf-Hilton invariants are trivial. Definition 3.5. Let Q 2 be the sum of the maps S p c 6 S m+2n 2 [α 2,[α 1,β 1 ]] X X A A Y Y S p c 7 S 2m+n 2 [β 2,[α 1,β 1 ]] X X for some c 6 c 7 ; that is, Q 2 = [ᾱ 2, [ᾱ 1, β 1 ]]c 6 + [ β 2, [ᾱ 1, β 1 ]]c 7, define ϕ = ϕ P,Q2 : Y Y Y by Definition 3.1. A A Y Y Proposition 3.6. Let m 2n 2 n < m < p 8n 8. If the Hopf-Hilton invariants H 1 1 (c 6) H 1 1 (c 7) are trivial, then the comultiplication ϕ = ϕ P,Q2 is associative if only if c 6 = c 7 = 0. Proof. We compute J 12 Q 2 J 23 Q 2 : J 12 Q 2 = J 12 ([ᾱ 2, [ᾱ 1, β 1 ]]c 6 + [ β 2, [ᾱ 1, β 1 ]]c 7 ) = [ ˆα 2, [ ˆα 1, ˆβ 1 ]]c 6 + [ ˆβ 2, [ ˆα 1, ˆβ 1 ]]c 7 We now compute (ϕ 1)Q 2 : J 23 Q 2 = [ ˆα 3, [ ˆα 2, ˆβ 2 ]]c 6 + [ ˆβ 3, [ ˆα 2, ˆβ 2 ]]c 7. (ϕ 1)Q 2 = [ ˆα 3, [J 12 ϕr, J 12 ϕs]]c 6 + [ ˆβ 3, [J 12 ϕr, J 12 ϕs]]c 7 = [ ˆα 3, [J 12 (ᾱ 1 + ᾱ 2 ), J 12 ( β 1 + β 2 )]]c 6 +[ ˆβ 3, [J 12 (ᾱ 1 + ᾱ 2 ), J 12 ( β 1 + β 2 )]]c 7 = [ ˆα 3, [ ˆα 1 + ˆα 2, ˆβ 1 + ˆβ 2 ]]c 6 + [ ˆβ 3, [ ˆα 1 + ˆα 2, ˆβ 1 + ˆβ 2 ]]c 7 = ( [ ˆα 3, [ ˆα 1, ˆβ 1 ]] + [ ˆα 3, [ ˆα 1, ˆβ 2 ]] + ( 1) mn [ ˆα 3, [ ˆβ 1, ˆα 2 ]] +[ ˆα 3, [ ˆα 2, ˆβ 2 ]] ) c 6 + ( [ ˆβ 3, [ ˆα 1, ˆβ 1 ]] + [ ˆβ 3, [ ˆα 1, ˆβ 2 ]] +( 1) mn [ ˆβ 3, [ ˆβ 1, ˆα 2 ]] + [ ˆβ 3, [ ˆα 2, ˆβ 2 ]] ) c 7. Similarly, (1 ϕ)q 2 = [ ˆα 2 + ˆα 3, [ ˆα 1, ˆβ 1 ]]c 6 + [ ˆβ 2 + ˆβ 3, [ ˆα 1, ˆβ 1 ]]c 7 = ( [ ˆα 2, [ ˆα 1, ˆβ 1 ]] + [ ˆα 3, [ ˆα 1, ˆβ 1 ]] ) c 6 ( [ ˆβ 2, [ ˆα 1, ˆβ 1 ]] + [ ˆβ 3, [ ˆα 1, ˆβ 1 ]] ) c 7. Thus ϕ is associative if only if 0 = [ ˆα 3, [ ˆα 1, ˆβ 2 ]]c 6 + ( 1) mn [ ˆα 3, [ ˆβ 1, ˆα 2 ]]c 6 +[ ˆβ 3, [ ˆα 1, ˆβ 2 ]]c 7 + ( 1) mn [ ˆβ 3, [ ˆβ 1, ˆα 2 ]]c 7. We note that the above Whitehead products are all basic products. From the Hilton s theorem, we conclude that the associativity of ϕ P,Q2 holds if only if c 6 = c 7 = 0. Definition 3.7. Let Q 3 be the sum of the maps S p c 8 S m+2n 2 [β 1,[α 1,α 2 ]] X X A A Y Y

12 for some c 8 c 9 ; that is, D.-W. Lee / Filomat 30:13 (2016), S p c 9 S m+2n 2 [β 2,[α 1,α 2 ]] define ϕ = ϕ P,Q3 : Y Y Y by Definition 3.1. X X Q 3 = [ β 1, [ᾱ 1, ᾱ 2 ]]c 8 + [ β 2, [ᾱ 1, ᾱ 2 ]]c 9, A A Y Y Proposition 3.8. Let m 2n 2 n < m < p 8n 8. If the Hopf-Hilton invariants H 1 1 (c 8) H 1 1 (c 9) are trivial, then the comultiplication ϕ = ϕ P,Q3 is associative if only if c 8 = c 9 = 0. Proof. We compute J 12 Q 3 : Similarly, We now compute (ϕ 1)Q 3 : J 12 Q 3 = J 12 ([ β 1, [ᾱ 1, ᾱ 2 ]]c 8 + [ β 2, [ᾱ 1, ᾱ 2 ]]c 9 ) = [ ˆβ 1, [ ˆα 1, ˆα 2 ]]c 8 + [ ˆβ 2, [ ˆα 1, ˆα 2 ]]c 9. J 23 Q 3 = [ ˆβ 2, [ ˆα 2, ˆα 3 ]]c 8 + [ ˆβ 3, [ ˆα 2, ˆα 3 ]]c 9. (ϕ 1)Q 3 = [J 12 ϕs, [J 12 ϕr, ˆα 3 ]]c 8 + [ ˆβ 3, [J 12 ϕr, ˆα 3 ]]c 9 = [J 12 ( β 1 + β 2 ), [J 12 (ᾱ 1 + ᾱ 2 ), ˆα 3 ]]c 8 +[ ˆβ 3, [J 12 (ᾱ 1 + ᾱ 2 ), ˆα 3 ]]c 9 = [ ˆβ 1 + ˆβ 2, [ ˆα 1 + ˆα 2, ˆα 3 ]]c 8 + [ ˆβ 3, [ ˆα 1 + ˆα 2, ˆα 3 ]]c 9 = ( [ ˆβ 1, [ ˆα 1, ˆα 3 ]] + [ ˆβ 1, [ ˆα 2, ˆα 3 ]] + [ ˆβ 2, [ ˆα 1, ˆα 3 ]] +[ ˆβ 2, [ ˆα 2, ˆα 3 ]] ) c 8 + ( [ ˆβ 3, [ ˆα 1, ˆα 3 ]] + [ ˆβ 3, [ ˆα 2, ˆα 3 ]] ) c 9. Similarly, (1 ϕ)q 3 = [ ˆβ 1, [ ˆα 1, ˆα 2 + ˆα 3 ]]c 8 + [ ˆβ 2 + ˆβ 3, [ ˆα 1, ˆα 2 + ˆα 3 ]]c 9 = ( [ ˆβ 1, [ ˆα 1, ˆα 2 ]] + [ ˆβ 1, [ ˆα 1, ˆα 3 ]] ) c 8 + ( [ ˆβ 2, [ ˆα 1, ˆα 2 ]] +[ ˆβ 2, [ ˆα 1, ˆα 3 ]] + [ ˆβ 3, [ ˆα 1, ˆα 2 ]] + [ ˆβ 3, [ ˆα 1, ˆα 3 ]] ) c 9. Since [ ˆβ 1, [ ˆα 2, ˆα 3 ]] is not a basic product, by using the Jacobi identity, we write it as a sum of basic products as follows: [ ˆα 2, [ ˆβ 1, ˆα 3 ]] [ ˆα 3, [ ˆβ 1, ˆα 2 ]] for m, n even, +[ ˆα [ ˆβ 1, [ ˆα 2, ˆα 3 ]] = 2, [ ˆβ 1, ˆα 3 ]] [ ˆα 3, [ ˆβ 1, ˆα 2 ]] for m even, n odd, +[ ˆα 2, [ ˆβ 1, ˆα 3 ]] + [ ˆα 3, [ ˆβ 1, ˆα 2 ]] for m odd, n even, +[ ˆα 2, [ ˆβ 1, ˆα 3 ]] [ ˆα 3, [ ˆβ 1, ˆα 2 ]] for m, n odd. (1) If m n are even, then ϕ P,Q3 is associative if only if 0 = ( [ ˆα 2, [ ˆβ 1, ˆα 3 ]] [ ˆα 3, [ ˆβ 1, ˆα 2 ]] + [ ˆβ 2, [ ˆα 1, ˆα 3 ]] ) c 8 ( [ ˆβ 2, [ ˆα 1, ˆα 3 ]] + [ ˆβ 3, [ ˆα 1, ˆα 2 ]] ) c 9 = [ ˆα 2, [ ˆβ 1, ˆα 3 ]]( c 8 ) + [ ˆα 3, [ ˆβ 1, ˆα 2 ]]( c 8 ) + [ ˆβ 2, [ ˆα 1, ˆα 3 ]](c 8 c 9 ) +[ ˆβ 3 [ ˆα 1, ˆα 2 ]]( c 9 ). (2) If m is even n is odd, then the associativity holds if only if 0 = ( [ ˆα 2, [ ˆβ 1, ˆα 3 ]] [ ˆα 3, [ ˆβ 1, ˆα 2 ]] + [ ˆβ 2, [ ˆα 1, ˆα 3 ]] ) c 8 ( [ ˆβ 2, [ ˆα 1, ˆα 3 ]] + [ ˆβ 3, [ ˆα 1, ˆα 2 ]] ) c 9 = [ ˆα 2, [ ˆβ 1, ˆα 3 ]](c 8 ) + [ ˆα 3, [ ˆβ 1, ˆα 2 ]]( c 8 ) + [ ˆβ 2, [ ˆα 1, ˆα 3 ]](c 8 c 9 ) +[ ˆβ 3 [ ˆα 1, ˆα 2 ]]( c 9 ). Since the above Whitehead products in (1) (2) are all basic products, ϕ is associative if only if c 8 = c 9 = 0. Similarly, the other cases could be treated by using the same method.

13 Definition 3.9. Let Q 4 be the map of compositions D.-W. Lee / Filomat 30:13 (2016), S p c 10 S 2m+n 2 [α 2,[β 1,β 2 ]] X X A A Y Y for some c 10 ; that is, define ϕ = ϕ P,Q4 : Y Y Y by Definition 3.1. Q 4 = [ᾱ 2, [ β 1, β 2 ]]c 10, Proposition Let m 2n 2 n < m < p 8n 8. If the Hopf-Hilton invariant H 1 1 (c 10) is trivial, then the comultiplication ϕ = ϕ P,Q4 is associative if only if c 10 = 0. Proof. We compute J 12 Q 4 (ϕ 1)Q 4 : Similarly, we have Thus the associativity holds if only if J 12 Q 4 = J 12 [ᾱ 2, [ β 1, β 2 ]]c 10 = [ ˆα 2, [ ˆβ 1, ˆβ 2 ]]c 10 (ϕ 1)Q 4 = (ϕ 1)[ᾱ 2, [ β 1, β 2 ]]c 10 = [ ˆα 3, [J 12 ϕs, ˆβ 3 ]]c 10 = [ ˆα 3, [J 12 ( β 1 + β 2 ), ˆβ 3 ]]c 10 = [ ˆα 3, [ ˆβ 1 + ˆβ 2, ˆβ 3 ]]c 10 = [ ˆα 3, [ ˆβ 1, ˆβ 3 ]]c 10 + [ ˆα 3, [ ˆβ 2, ˆβ 3 ]]c 10. J 23 Q 4 = J 23 [ᾱ 2, [ β 1, β 2 ]]c 10 = [ ˆα 3, [ ˆβ 2, ˆβ 3 ]]c 10 (1 ϕ)q 4 = (1 ϕ)[ᾱ 2, [ β 1, β 2 ]]c 10 = [J 23 ϕr, [ ˆβ 1, J 23 ϕs]]c 10 = [J 23 (ᾱ 1 + ᾱ 2 ), [ ˆβ 1, J 23 ( β 1 + β 2 )]]c 10 = [ ˆα 2 + ˆα 3, [ ˆβ 1, ˆβ 2 + ˆβ 3 ]]c 10 = [ ˆα 2, [ ˆβ 1, ˆβ 2 ]]c 10 + [ ˆα 2, [ ˆβ 1, ˆβ 3 ]]c 10 +[ ˆα 3, [ ˆβ 1, ˆβ 2 ]]c 10 + [ ˆα 3, [ ˆβ 1, ˆβ 3 ]]c = [ ˆα 2, [ ˆβ 1, ˆβ 3 ]]( c 10 ) + [ ˆα 3, [ ˆβ 1, ˆβ 2 ]]( c 10 ). We note that the above Whitehead products are basic products. Therefore, ϕ is associative if only if c 10 = 0. Definition Let Q 5 be the sum of the maps S p d 1 S m+n 1 [β 1,α 2 ] X X A A Y Y, S p d 2 S 2m+n 2 [β 1,[β 1,α 2 ]] X X A A Y Y, S p d 3 S m+2n 2 [α 2,[β 1,α 2 ]] X X A A Y Y S p d 4 S 2m+n 2 [β 2,[β 1,α 2 ]] X X A A Y Y for some d 1, d 2, d 3 d 4 ; that is, define ϕ = ϕ P,Q5 : Y Y Y by Definition 3.1. Q 5 = [ β 1, ᾱ 2 ]d 1 + [ β 1, [ β 1, ᾱ 2 ]]d 2 +[ᾱ 2, [ β 1, ᾱ 2 ]]d 3 + [ β 2, [ β 1, ᾱ 2 ]]d 4,

14 D.-W. Lee / Filomat 30:13 (2016), Proposition Let m 2n 2 n < m < p 8n 8. If the Hopf-Hilton invariants H 1 1 (d 1), H 2 1 (d 1), H 1 1 (d 2), H 1 1 (d 3) H 1 1 (d 4) are trivial, then the comultiplication ϕ = ϕ P,Q5 is associative if only if d 2 = d 3 = d 4 = 0. Proof. We compute J 12 Q 5 : Similarly, We now compute (ϕ 1)Q 5 : J 12 Q 5 = J 12 ( [ β 1, ᾱ 2 ]d 1 + [ β 1, [ β 1, ᾱ 2 ]]d 2 +[ᾱ 2, [ β 1, ᾱ 2 ]]d 3 + [ β 2, [ β 1, ᾱ 2 ]]d 4 ) = [ ˆβ 1, ˆα 2 ]d 1 + [ ˆβ 1, [ ˆβ 1, ˆα 2 ]]d 2 + [ ˆα 2, [ ˆβ 1, ˆα 2 ]]d 3 +[ ˆβ 2, [ ˆβ 1, ˆα 2 ]]d 4. J 23 Q 5 = [ ˆβ 2, ˆα 3 ]d 1 + [ ˆβ 2, [ ˆβ 2, ˆα 3 ]]d 2 + [ ˆα 3, [ ˆβ 2, ˆα 3 ]]d 3 +[ ˆβ 3, [ ˆβ 2, ˆα 3 ]]d 4. (ϕ 1)Q 5 = [J 12 ϕs, ˆα 3 ]d 1 + [J 12 ϕs, [J 12 ϕs, ˆα 3 ]]d 2 +[ ˆα 3, [J 12 ϕs, ˆα 3 ]]d 3 + [ ˆβ 3, [J 12 ϕs, ˆα 3 ]]d 4 = [J 12 ( β 1 + β 2 ), ˆα 3 ]d 1 +[J 12 ( β 1 + β 2 ), [J 12 ( β 1 + β 2 ), ˆα 3 ]]d 2 +[ ˆα 3, [J 12 ( β 1 + β 2 ), ˆα 3 ]]d 3 + [ ˆβ 3, [J 12 ( β 1 + β 2 ), ˆα 3 ]]d 4 = [ ˆβ 1 + ˆβ 2, ˆα 3 ]d 1 + [ ˆβ 1 + ˆβ 2, [ ˆβ 1 + ˆβ 2, ˆα 3 ]]d 2 +[ ˆα 3, [ ˆβ 1 + ˆβ 2, ˆα 3 ]]d 3 + [ ˆβ 3, [ ˆβ 1 + ˆβ 2, ˆα 3 ]]d 4 = ( [ ˆβ 1, ˆα 3 ] + [ ˆβ 2, ˆα 3 ] ) d 1 + ( [ ˆβ 1, [ ˆβ 1, ˆα 3 ]] +[ ˆβ 1, [ ˆβ 2, ˆα 3 ]] + [ ˆβ 2, [ ˆβ 1, ˆα 3 ]] + [ ˆβ 2, [ ˆβ 2, ˆα 3 ]] ) d 2 + ( [ ˆα 3, [ ˆβ 1, ˆα 3 ]] + [ ˆα 3, [ ˆβ 2, ˆα 3 ]] ) d 3 + ( [ ˆβ 3, [ ˆβ 1, ˆα 3 ]] + [ ˆβ 3, [ ˆβ 2, ˆα 3 ]] ) d 4. We need to change the non-basic product [ ˆβ 1, [ ˆβ 2, ˆα 3 ]] into basic products by using the Jacobi identity as follows: ( 1) m(n 1) [ ˆβ 1, [ ˆβ 2, ˆα 3 ]] = ( 1) m(m 1)+mn [ ˆβ 2, [ ˆβ 1, ˆα 3 ]] ( 1) n(m 1) [ ˆα 3, [ ˆβ 1, ˆβ 2 ]]. We note that the Whitehead products on the right-h side in the above equation are both basic products. Similarly, (1 ϕ)q 5 = [ ˆβ 1, J 23 ϕr]d 1 + [ ˆβ 1, [ ˆβ 1, J 23 ϕr]]d 2 +[J 23 ϕr, [ ˆβ 1, J 23 ϕr]]d 3 + [J 23 ϕs, [ ˆβ 1, J 23 ϕr]]d 4 = [ ˆβ 1, J 23 (ᾱ 1 + ᾱ 2 )]d 1 + [ ˆβ 1, [ ˆβ 1, J 23 (ᾱ 1 + ᾱ 2 )]]d 2 +[J 23 (ᾱ 1 + ᾱ 2 ), [ ˆβ 1, J 23 (ᾱ 1 + ᾱ 2 )]]d 3 +[J 23 ( β 1 + β 2 ), [ ˆβ 1, J 23 (ᾱ 1 + ᾱ 2 )]]d 4 = [ ˆβ 1, ˆα 2 + ˆα 3 ]d 1 + [ ˆβ 1, [ ˆβ 1, ˆα 2 + ˆα 3 ]]d 2 +[ ˆα 2 + ˆα 3, [ ˆβ 1, ˆα 2 + ˆα 3 ]]d 3 +[ ˆβ 2 + ˆβ 3, [ ˆβ 1, ˆα 2 + ˆα 3 ]]d 4 = ( [ ˆβ 1, ˆα 2 ] + [ ˆβ 1, ˆα 3 ] ) d 1 + ( [ ˆβ 1, [ ˆβ 1, ˆα 2 ]] + [ ˆβ 1, [ ˆβ 1, ˆα 3 ]] ) d 2 + ( [ ˆα 2, [ ˆβ 1, ˆα 2 ]] + [ ˆα 2, [ ˆβ 1, ˆα 3 ]] + [ ˆα 3, [ ˆβ 1, ˆα 2 ]] +[ ˆα 3, [ ˆβ 1, ˆα 3 ]] ) d 3 + ( [ ˆβ 2, [ ˆβ 1, ˆα 2 ]] + [ ˆβ 2, [ ˆβ 1, ˆα 3 ]] +[ ˆβ 3, [ ˆβ 1, ˆα 2 ]] + [ ˆβ 3, [ ˆβ 1, ˆα 3 ]] ) d 4.

15 We now find the conditions for the equation D.-W. Lee / Filomat 30:13 (2016), J 12 Q 5 + (ϕ 1)Q 5 = J 23 Q 5 + (1 ϕ)q 5 to be satisfied. We also note that the following Whitehead products in (1) (2) are all basic products. (1) If m is odd n is even, then ϕ P,Q5 is associative if only if 0 = [ ˆβ 2, [ ˆβ 1, ˆα 3 ]]d 2 + [ ˆα 3, [ ˆβ 1, ˆβ 2 ]]d 2 + [ ˆβ 2, [ ˆβ 1, ˆα 3 ]]d 2 ( [ ˆα 2, [ ˆβ 1, ˆα 3 ]] + [ ˆα 3, [ ˆβ 1, ˆα 2 ]] ) d 3 ( [ ˆβ 2, [ ˆβ 1, ˆα 3 ]] + [ ˆβ 3, [ ˆβ 1, ˆα 2 ]] ) d 4 = [ ˆα 3, [ ˆβ 1, ˆβ 2 ]]d 2 + [ ˆα 2, [ ˆβ 1, ˆα 3 ]]( d 3 ) + [ ˆα 3, [ ˆβ 1, ˆα 2 ]]( d 3 ) [ ˆβ 2, [ ˆβ 1, ˆα 3 ]](2d 2 d 4 ) + [ ˆβ 3, [ ˆβ 1, ˆα 2 ]]( d 4 ). (2) If m n are odd, then the associativity holds if only if 0 = [ ˆβ 2, [ ˆβ 1, ˆα 3 ]]d 2 + ( [ ˆα 3, [ ˆβ 1, ˆβ 2 ]])d 2 + [ ˆβ 2, [ ˆβ 1, ˆα 3 ]]d 2 ( [ ˆα 2, [ ˆβ 1, ˆα 3 ]] + [ ˆα 3, [ ˆβ 1, ˆα 2 ]] ) d 3 ( [ ˆβ 2, [ ˆβ 1, ˆα 3 ]] + [ ˆβ 3, [ ˆβ 1, ˆα 2 ]] ) d 4 = [ ˆα 3, [ ˆβ 1, ˆβ 2 ]]( d 2 ) + [ ˆα 2, [ ˆβ 1, ˆα 3 ]]( d 3 ) + [ ˆα 3, [ ˆβ 1, ˆα 2 ]]( d 3 ) [ ˆβ 2, [ ˆβ 1, ˆα 3 ]](2d 2 d 4 ) + [ ˆβ 3, [ ˆβ 1, ˆα 2 ]]( d 4 ). Thus ϕ is associative if only if d 2 = d 3 = d 4 = 0. Similarly, the other cases could be proved by the same way. We now consider the sum of the perturbations Q j, j = 1, 2,..., 5 defined above in order to investigate all the comultiplications on a wedge of three spheres completely in a certain range as follows: Definition Let Q be the sum of the perturbations Q 1, Q 2,..., Q 5 ; that is, Q = 5 i=1 Q i = [ β 1, β 2 ]a 1 + [ β 1, [ β 1, β 2 ]]a 2 + [ β 2, [ β 1, β 2 ]]a 3 +[ᾱ 1, ᾱ 2 ]b 1 + [ᾱ 1, [ᾱ 1, ᾱ 2 ]]b 2 + [ᾱ 2, [ᾱ 1, ᾱ 2 ]]b 3 +[ᾱ 1, β 2 ]c 1 + [ᾱ 1, [ᾱ 1, β 2 ]]c 2 + [ᾱ 2, [ᾱ 1, β 2 ]]c 3 +[ β 1, [ᾱ 1, β 2 ]]c 4 + [ β 2, [ᾱ 1, β 2 ]]c 5 + [ᾱ 2, [ᾱ 1, β 1 ]]c 6 +[ β 2, [ᾱ 1, β 1 ]]c 7 + [ β 1, [ᾱ 1, ᾱ 2 ]]c 8 + [ β 2, [ᾱ 1, ᾱ 2 ]]c 9 +[ᾱ 2, [ β 1, β 2 ]]c 10 + [ β 1, ᾱ 2 ]d 1 + [ β 1, [ β 1, ᾱ 2 ]]d 2 +[ᾱ 2, [ β 1, ᾱ 2 ]]d 3 + [ β 2, [ β 1, ᾱ 2 ]]d 4, define ϕ = ϕ P,Q : Y Y Y by Definition 3.1. The following is one of the technical results in this paper. Theorem Let m 2n 2, n < m < p 8n 8 let the Hopf-Hilton invariants H 1 1 (x) H2 1 (y) be trivial, where x = a i, b i, c j, d k for i = 1, 2, 3, j = 1, 2,..., 10, k = 1, 2, 3, 4 y = a 1, b 1, c 1, d 1. Then the comultiplication ϕ = ϕ P,Q is associative if only if (1) m n are even, a 2 = a 3 = b 2 = b 3 = c 3 = c 6 = d 4 = 0, c 4 = c 5 = c 7 = c 10 = d 2 c 2 = c 8 = c 9 = d 3, or (2) m is even n is odd, a 2 = a 3 = d 4 = 0, c 4 = c 5 = c 7 = c 10 = d 2, b 2 = b 3, 3b 2 = 0, c 3 = c 6 = 2d 3 c 2 = c 8 = c 9 = d 3, or (3) m is odd n is even, b 2 = b 3 = c 3 = c 6 = 0, a 2 = a 3, 3a 2 = 0, c 2 = c 8 = c 9 = d 3, c 4 = c 5 = c 7 = c 10 = d 2 d 4 = 2c 4, or (4) m n are odd, a 2 = a 3, b 2 = b 3, 3a 2 = 0 = 3b 2, c 2 = c 8 = c 9 = d 3, 2c 2 = c 3 = c 6, c 4 = c 5 = c 7 = c 10 = d 2 2d 2 = d 4. Proof. To prove this theorem, we will make use of the previous results from Propositions 3.3, 3.6, 3.8, We also note that following Whitehead products in (1), (2), (3) (4) are all the basic products.

16 D.-W. Lee / Filomat 30:13 (2016), (1) If m n are even, then the equality J 12 Q + (ϕ 1)Q = J 23 Q + (1 ϕ)q holds if only if 0 = [ ˆβ 3, [ ˆβ 1, ˆβ 2 ]]( a 2 a 3 ) + [ ˆβ 2, [ ˆβ 1, ˆβ 3 ]]( a 3 ) +[ ˆα 3, [ ˆα 1, ˆα 2 ]]( b 2 b 3 ) + [ ˆα 2, [ ˆα 1, ˆα 3 ]]( b 3 ) +[ ˆβ 3, [ ˆα 1, ˆα 2 ]]( c 2 c 9 ) + [ ˆα 2, [ ˆα 1, ˆβ 3 ]]( c 3 ) +[ ˆα 3, [ ˆα 1, ˆβ 2 ]]( c 3 + c 6 ) + [ ˆα 2, [ ˆβ 1, ˆβ 3 ]]( c 4 c 10 ) +[ ˆβ 3, [ ˆβ 1, ˆα 2 ]]( c 4 + c 7 d 4 ) + [ ˆβ 2, [ ˆα 1, ˆβ 3 ]](c 4 c 5 ) +[ ˆβ 3, [ ˆα 1, ˆβ 2 ]]( c 5 + c 7 ) + [ ˆα 3, [ ˆβ 1, ˆα 2 ]](c 6 c 8 d 3 ) +[ ˆα 2, [ ˆβ 1, ˆα 3 ]]( c 8 d 3 ) + [ ˆβ 2, [ ˆα 1, ˆα 3 ]](c 8 c 9 ) +[ ˆα 3, [ ˆβ 1, ˆβ 2 ]]( d 2 c 10 ) + [ ˆβ 2, [ ˆβ 1, ˆα 3 ]]( d 4 ). Thus ϕ is associative if only if a 2 = a 3 = b 2 = b 3 = c 3 = c 6 = d 4 = 0, c 4 = c 5 = c 7 = c 10 = d 2 c 2 = c 8 = c 9 = d 3. (2) If m is even n is odd, then the associativity of the comultiplication ϕ = ϕ P,Q : Y Y Y holds if only if 0 = [ ˆβ 3, [ ˆβ 1, ˆβ 2 ]]( a 2 a 3 ) + [ ˆβ 2, [ ˆβ 1, ˆβ 3 ]]( a 3 ) +[ ˆα 3, [ ˆα 1, ˆα 2 ]]( b 2 b 3 ) + [ ˆα 2, [ ˆα 1, ˆα 3 ]](2b 2 b 3 ) +[ ˆα 2, [ ˆα 1, ˆβ 3 ]](2c 2 c 3 ) + [ ˆβ 3, [ ˆα 1, ˆα 2 ](c 2 c 9 ) +[ ˆα 3, [ ˆα 1, ˆβ 2 ]]( c 3 + c 6 ) + [ ˆα 2, [ ˆβ 1, ˆβ 3 ]](c 4 c 10 ) +[ ˆβ 3, [ ˆβ 1, ˆα 2 ]]( c 4 + c 7 d 4 ) + [ ˆβ 2, [ ˆα 1, ˆβ 3 ]](c 4 c 5 ) +[ ˆβ 3, [ ˆα 1, ˆβ 2 ]]( c 5 + c 7 ) + [ ˆα 3, [ ˆβ 1, ˆα 2 ]](c 6 c 8 d 3 ) +[ ˆα 2, [ ˆβ 1, ˆα 3 ]](c 8 d 3 ) + [ ˆβ 2, [ ˆα 1, ˆα 3 ]](c 8 c 9 ) +[ ˆα 3, [ ˆβ 1, ˆβ 2 ]](d 2 c 10 ) + [ ˆβ 2, [ ˆβ 1, ˆα 3 ]]( d 4 ). Thus ϕ is associative if only if a 2 = a 3 = d 4 = 0, c 4 = c 5 = c 7 = c 10 = d 2, b 2 = b 3, 3b 2 = 0, c 3 = c 6 = 2d 3 c 2 = c 8 = c 9 = d 3. (3) If m is odd n is even, then the comultiplication ϕ = ϕ P,Q is associative if only if 0 = [ ˆβ 3, [ ˆβ 1, ˆβ 2 ]]( a 2 a 3 ) + [ ˆβ 2, [ ˆβ 1, ˆβ 3 ]](2a 2 a 3 ) +[ ˆα 3, [ ˆα 1, ˆα 2 ]]( b 2 b 3 ) + [ ˆα 2, [ ˆα 1, ˆα 3 ]]( b 3 ) +[ ˆα 2, [ ˆα 1, ˆβ 3 ]]( c 3 ) + [ ˆβ 3, [ ˆα 1, ˆα 2 ]](c 2 c 9 ) +[ ˆα 3, [ ˆα 1, ˆβ 2 ]]( c 3 + c 6 ) + [ ˆα 2, [ ˆβ 1, ˆβ 3 ]](c 4 c 10 ) +[ ˆβ 3, [ ˆβ 1, ˆα 2 ]](c 4 + c 7 d 4 ) + [ ˆβ 2, [ ˆα 1, ˆβ 3 ]](c 4 c 5 ) +[ ˆβ 3, [ ˆα 1, ˆβ 2 ]]( c 5 + c 7 ) + [ ˆα 3, [ ˆβ 1, ˆα 2 ]](c 6 + c 8 d 3 ) +[ ˆα 2, [ ˆβ 1, ˆα 3 ]](c 8 d 3 ) + [ ˆβ 2, [ ˆα 1, ˆα 3 ]](c 8 c 9 ) +[ ˆβ 2, [ ˆβ 1, ˆα 3 ]](2d 2 d 4 ) + [ ˆα 3, [ ˆβ 1, ˆβ 2 ]](d 2 c 10 ). Thus ϕ is associative if only if b 2 = b 3 = c 3 = c 6 = 0, a 2 = a 3, 3a 2 = 0, c 2 = c 8 = c 9 = d 3, c 4 = c 5 = c 7 = c 10 = d 2 d 4 = 2c 4. (4) If m n are odd, then the given comultiplication is associative if only if 0 = [ ˆβ 3, [ ˆβ 1, ˆβ 2 ]]( a 2 a 3 ) + [ ˆβ 2, [ ˆβ 1, ˆβ 3 ]](2a 2 a 3 ) +[ ˆα 3, [ ˆα 1, ˆα 2 ]]( b 2 b 3 ) + [ ˆα 2, [ ˆα 1, ˆα 3 ]](2b 2 b 3 ) +[ ˆα 2, [ ˆα 1, ˆβ 3 ]](2c 2 c 3 ) + [ ˆβ 3, [ ˆα 1, ˆα 2 ]]( c 2 c 9 ) +[ ˆα 3, [ ˆα 1, ˆβ 2 ]]( c 3 + c 6 ) + [ ˆα 2, [ ˆβ 1, ˆβ 3 ]](c 4 c 10 ) +[ ˆβ 3, [ ˆβ 1, ˆα 2 ]]( c 4 c 7 d 4 ) + [ ˆβ 2, [ ˆα 1, ˆβ 3 ]](c 4 c 5 ) +[ ˆβ 3, [ ˆα 1, ˆβ 2 ]]( c 5 + c 7 ) + [ ˆα 3, [ ˆβ 1, ˆα 2 ]]( c 6 c 8 d 3 ) +[ ˆα 2, [ ˆβ 1, ˆα 3 ]](c 8 d 3 ) + [ ˆβ 2, [ ˆα 1, ˆα 3 ]](c 8 c 9 ) +[ ˆβ 2, [ ˆβ 1, ˆα 3 ]](2d 2 d 4 ) + [ ˆα 3, [ ˆβ 1, ˆβ 2 ]]( d 2 c 10 ). Thus ϕ is associative if only if a 2 = a 3, 3a 2 = 0, b 2 = b 3, 3b 2 = 0, c 2 = c 8 = c 9 = d 3, 2c 2 = c 3 = c 6, c 4 = c 5 = c 7 = c 10 = d 2 2d 2 = d 4.

17 D.-W. Lee / Filomat 30:13 (2016), Let A(Y) be the set of homotopy classes of associative comultiplications on a co-h-space Y let S be the number of elements in a set S. Then we have the one of the main theorems in this paper as follows: Theorem Let m 2n 2, n < m < p 4n 4 let Y = S n S m S p. (1) If m n are even, then (2) If m is even n is odd, then (3) If m is odd n is even, then (4) If m n are odd, then A(Y) = π p (S 2m 1 ) π p (S 2n 1 ) π p (S m+n 1 ) 2 π p (S 2m+n 2 ) π p (S m+2n 2 ). A(Y) = π p (S 2m 1 ) π p (S 2n 1 ) {b 2 π p (S 3n 2 ) 3b 2 = 0} π p (S m+n 1 ) 2 π p (S m+2n 2 ) π p (S 2m+n 2 ). A(Y) = π p (S 2m 1 ) π p (S 2n 1 ) {a 2 π p (S 3m 2 ) 3a 2 = 0} π p (S m+n 1 ) 2 π p (S m+2n 2 ) π p (S 2m+n 2 ). A(Y) = π p (S 2m 1 ) π p (S 2n 1 ) {a 2 π p (S 3m 2 ) 3a 2 = 0} {b 2 π p (S 3n 2 ) 3b 2 = 0} π p (S m+n 1 ) 2 π p (S m+2n 2 ) π p (S 2m+n 2 ). Proof. Since every comultiplication has the form in Definition 3.13, by Theorem 3.14, the result follows. Example The set A(S 4 S 5 S 12 ) is infinite, A(S 8 S 12 S 27 ) = 32. Indeed, A(S 8 S 12 S 27 ) = π 27 (S 23 ) π 27 (S 15 ) π 27 (S 19 ) 2 π 27 (S 30 ) π 27 (S 26 ) = = 32, similarly for the other case. 4. Commutativity We recall that ϕ = ϕ P,Q : Y Y Y in Definition 3.1 is commutative if only if TP = P TQ = Q. We also note that Tι 1 = ι 2 Tι 2 = ι 1, where T : Y Y Y Y is the switching map. Proposition 4.1. Let ϕ = ϕ P,Q1 : Y Y Y be the comultiplication in Definition 3.2. Then, under the same hypotheses of Proposition 3.3, ϕ is commutative if only if (1) m n are even, a 2 = a 3, b 2 = b 3 c i = 0 for i = 1, 2,..., 5, or (2) m is even n is odd, a 2 = a 3, b 2 = b 3, 2b 1 = 0 c i = 0 for i = 1, 2,..., 5, or (3) m is odd n is even, a 2 = a 3, b 2 = b 3, 2a 1 = 0 c i = 0 for i = 1, 2,..., 5, or (4) m n are odd, a 2 = a 3, b 2 = b 3, 2a 1 = 0 = 2b 1 c i = 0 for i = 1, 2,..., 5. Proof. Since we have Q 1 = [ β 1, β 2 ]a 1 + [ β 1, [ β 1, β 2 ]]a 2 + [ β 2, [ β 1, β 2 ]]a 3 +[ᾱ 1, ᾱ 2 ]b 1 + [ᾱ 1, [ᾱ 1, ᾱ 2 ]]b 2 + [ᾱ 2, [ᾱ 1, ᾱ 2 ]]b 3 +[ᾱ 1, β 2 ]c 1 + [ᾱ 1, [ᾱ 1, β 2 ]]c 2 + [ᾱ 2, [ᾱ 1, β 2 ]]c 3 +[ β 1, [ᾱ 1, β 2 ]]c 4 + [ β 2, [ᾱ 1, β 2 ]]c 5, TQ 1 = [ β 2, β 1 ]a 1 + [ β 2, [ β 2, β 1 ]]a 2 + [ β 1, [ β 2, β 1 ]]a 3 +[ᾱ 2, ᾱ 1 ]b 1 + [ᾱ 2, [ᾱ 2, ᾱ 1 ]]b 2 + [ᾱ 1, [ᾱ 2, ᾱ 1 ]]b 3 +[ᾱ 2, β 1 ]c 1 + [ᾱ 2, [ᾱ 2, β 1 ]]c 2 + [ᾱ 1, [ᾱ 2, β 1 ]]c 3 +[ β 2, [ᾱ 2, β 1 ]]c 4 + [ β 1, [ᾱ 2, β 1 ]]c 5 = ( 1) m2 [ β 1, β 2 ]a 1 + ( 1) m2 [ β 2, [ β 1, β 2 ]]a 2 +( 1) m2 [ β 1, [ β 1, β 2 ]]a 3 + ( 1) n2 [ᾱ 1, ᾱ 2 ]b 1 +( 1) n2 [ᾱ 2, [ᾱ 1, ᾱ 2 ]]b 2 + ( 1) n2 [ᾱ 1, [ᾱ 1, ᾱ 2 ]]b 3 +( 1) mn [ β 1, ᾱ 2 ]c 1 + ( 1) mn [ᾱ 2, [ β 1, ᾱ 2 ]]c 2 +( 1) mn [ᾱ 1, [ β 1, ᾱ 2 ]]c 3 + ( 1) mn [ β 2, [ β 1, ᾱ 2 ]]c 4 +( 1) mn [ β 1, [ β 1, ᾱ 2 ]]c 5.

18 D.-W. Lee / Filomat 30:13 (2016), We note that the Whitehead product [ᾱ 1, [ β 1, ᾱ 2 ] in the above equations is still not a basic product. We need to change it into the basic product by using the Jacobi identity as follows: ( 1) n(n 1) [ᾱ 1, [ β 1, ᾱ 2 ] = ( 1) m(n 1)+n2 [ β 1, [ᾱ 1, ᾱ 2 ]] ( 1) n(m 1) [ᾱ 2, [ᾱ 1, β 1 ]]. By using the basic products on the right-h side of the above equation, we see that the following Whitehead products in (1) (2) are all basic products. We recall that by the Hilton s formula. (1) If m is even n is odd, then ( [ β 1, β 2 ])a 1 = [ β 1, β 2 ]( 1)a 1 = [ β 1, β 2 ]( a 1 + [1, 1]H 1 1 (a 1)) Q 1 TQ 1 = [ β 1, [ β 1, β 2 ]](a 2 a 3 ) + [ β 2, [ β 1, β 2 ]](a 3 a 2 ) +[ᾱ 1, ᾱ 2 ](2b 1 [1, 1]H 1 1 (b 1)) +[ᾱ 1, [ᾱ 1, ᾱ 2 ]](b 2 + b 3 ) + [ᾱ 2, [ᾱ 1, ᾱ 2 ]](b 3 + b 2 ) +[ᾱ 1, β 2 ]c 1 + [ᾱ 1, [ᾱ 1, β 2 ]]c 2 + [ᾱ 2, [ᾱ 1, β 2 ]]c 3 +[ β 1, [ᾱ 1, β 2 ]]c 4 + [ β 2, [ᾱ 1, β 2 ]]c 5 +[ β 1, ᾱ 2 ]( c 1 ) + [ᾱ 2, [ β 1, ᾱ 2 ]]( c 2 ) +[ β 1, [ᾱ 1, ᾱ 2 ]]( c 3 ) + [ᾱ 2, [ᾱ 1, β 1 ]]( c 3 ) +[ β 2, [ β 1, ᾱ 2 ]]( c 4 ) + [ β 1, [ β 1, ᾱ 2 ]]( c 5 ). Since the Hopf-Hilton invariant H 1 1 (b 1) is trivial, ϕ is commutative if only if a 2 = a 3, b 2 = b 3, 2b 1 = 0 c i = 0 for i = 1, 2,..., 5. (2) If m n are odd, then Q 1 TQ 1 = [ β 1, β 2 ](2a 1 [1, 1]H 1 1 (a 1)) + [ᾱ 1, ᾱ 2 ](2b 1 [1, 1]H 1 1 (b 1)) +[ β 1, [ β 1, β 2 ]](a 2 + a 3 ) + [ β 2, [ β 1, β 2 ]](a 3 + a 2 ) +[ᾱ 1, [ᾱ 1, ᾱ 2 ]](b 2 + b 3 ) + [ᾱ 2, [ᾱ 1, ᾱ 2 ]](b 3 + b 2 ) +[ᾱ 1, β 2 ]c 1 + [ᾱ 1, [ᾱ 1, β 2 ]]c 2 + [ᾱ 2, [ᾱ 1, β 2 ]]c 3 +[ β 1, [ᾱ 1, β 2 ]]c 4 + [ β 2, [ᾱ 1, β 2 ]]c 5 +[ β 1, ᾱ 2 ]c 1 + [ᾱ 2, [ β 1, ᾱ 2 ]]c 2 +[ β 1, [ᾱ 1, ᾱ 2 ]]c 3 + [ᾱ 2, [ᾱ 1, β 1 ]]( c 3 ) +[ β 2, [ β 1, ᾱ 2 ]]c 4 + [ β 1, [ β 1, ᾱ 2 ]]c 5. Since the Hopf-Hilton invariants H 1 1 (a 1) H 1 1 (b 1) are trivial, ϕ is commutative if only if a 2 = a 3, b 2 = b 3, 2a 1 = 0, 2b 1 = 0 c i = 0 for i = 1, 2,..., 5. Similarly, the other cases could be proved by the same method. Proposition 4.2. Let ϕ = ϕ P,Q2 : Y Y Y be the comultiplication in Definition 3.5. Then, under the same hypotheses of Proposition 3.6, ϕ is commutative if only if c 6 = c 7 = 0. Proof. From Q 2 = [ᾱ 2, [ᾱ 1, β 1 ]]c 6 + [ β 2, [ᾱ 1, β 1 ]]c 7, we have TQ 2 = [ᾱ 1, [ᾱ 2, β 2 ]]c 6 + [ β 1, [ᾱ 2, β 2 ]]c 7. Since TQ 2 consists of two non-basic Whitehead products, we need to change them into the basic product by using the Jacobi identity as follows: ( 1) n(m 1) [ᾱ 1, [ᾱ 2, β 2 ]] = ( 1) n(n 1)+mn [ᾱ 2, [ᾱ 1, β 2 ]] ( 1) m(n 1) [ β 2, [ᾱ 1, ᾱ 2 ]] ( 1) m(m 1) [ β 1, [ᾱ 2, β 2 ]] = ( 1) n(m 1)+m2 [ᾱ 2, [ β 1, β 2 ]] ( 1) m(n 1) [ β 2, [ β 1, ᾱ 2 ]].

19 We recall that ( 1)c j = c j + [1, 1]H 1 1 (c j) = c j, for j = 6, 7. (1) If m n are even, then (2) If m is odd n is even, then D.-W. Lee / Filomat 30:13 (2016), Q 2 TQ 2 = [ᾱ 2, [ᾱ 1, β 1 ]]c 6 + [ β 2, [ᾱ 1, β 1 ]]c 7 +[ᾱ 2, [ᾱ 1, β 2 ]]c 6 + [ β 2, [ᾱ 1, ᾱ 2 ]]c 6 +[ᾱ 2, [ β 1, β 2 ]]c 7 + [ β 2, [ β 1, ᾱ 2 ]]c 7. Q 2 TQ 2 = [ᾱ 2, [ᾱ 1, β 1 ]]c 6 + [ β 2, [ᾱ 1, β 1 ]]c 7 +[ᾱ 2, [ᾱ 1, β 2 ]]c 6 + [ β 2, [ᾱ 1, ᾱ 2 ]]( c 6 ) +[ᾱ 2, [ β 1, β 2 ]]( c 7 ) + [ β 2, [ β 1, ᾱ 2 ]]( c 7 ). Since the above Whitehead products in (1) (2) are all the basic products, ϕ is commutative if only if c 6 = c 7 = 0 in any case. Similarly, we obtain the same result for the other cases. Proposition 4.3. Let ϕ = ϕ P,Q3 : Y Y Y be the comultiplication in Definition 3.7. Then, under the same hypotheses of Proposition 3.8, ϕ is commutative if only if (1) n is even c 8 = c 9, or (2) n is odd c 8 = c 9. Proof. Since Q 3 = [ β 1, [ᾱ 1, ᾱ 2 ]]c 8 + [ β 2, [ᾱ 1, ᾱ 2 ]]c 9, we have If n is even, then In case that n is odd, we obtain TQ 3 = [ β 2, [ᾱ 2, ᾱ 1 ]]c 8 + [ β 1, [ᾱ 2, ᾱ 1 ]]c 9 = ( 1) n2 [ β 2, [ᾱ 1, ᾱ 2 ]]c 8 + ( 1) n2 [ β 1, [ᾱ 1, ᾱ 2 ]]c 9. Q 3 TQ 3 = [ β 1, [ᾱ 1, ᾱ 2 ]](c 8 c 9 ) + [ β 2, [ᾱ 1, ᾱ 2 ]](c 9 c 8 ). Q 3 TQ 3 = [ β 1, [ᾱ 1, ᾱ 2 ]](c 8 + c 9 ) + [ β 2, [ᾱ 1, ᾱ 2 ]](c 9 + c 8 ). Since the right-h side of the above equations are basic products, we complete the proof. Proposition 4.4. Let ϕ = ϕ P,Q4 : Y Y Y be the comultiplication in Definition 3.9. Then, under the same hypotheses of Proposition 3.10, ϕ is commutative if only if c 10 = 0. Proof. We note that Q 4 = [ᾱ 2, [ β 1, β 2 ]]c 10 TQ 4 = [ᾱ 1, [ β 2, β 1 ]]c 10 = ( 1) m2 [ᾱ 1, [ β 1, β 2 ]]c 10. Since the last term is not a basic Whitehead product, we change it into the basic products as follows: ( 1) n(m 1) [ᾱ 1, [ β 1, β 2 ]] = ( 1) m(n 1)+mn [ β 1, [ᾱ 1, β 2 ]] ( 1) m(m 1) [ β 2, [ᾱ 1, β 1 ]]. Therefore, by considering the equation Q 4 TQ 4 = 0, we complete the proof. Proposition 4.5. Let ϕ = ϕ P,Q5 : Y Y Y be the comultiplication in Definition Then, under the same hypotheses of Proposition 3.12, ϕ is commutative if only if d 1 = d 2 = d 3 = d 4 = 0. Proof. We compute Q 5 TQ 5 as follows: Q 5 = [ β 1, ᾱ 2 ]d 1 + [ β 1, [ β 1, ᾱ 2 ]]d 2 + [ᾱ 2, [ β 1, ᾱ 2 ]]d 3 + [ β 2, [ β 1, ᾱ 2 ]]d 4 TQ 5 = [ β 2, ᾱ 1 ]d 1 + [ β 2, [ β 2, ᾱ 1 ]]d 2 + [ᾱ 1, [ β 2, ᾱ 1 ]]d 3 +[ β 1, [ β 2, ᾱ 1 ]]d 4 = ( 1) mn [ᾱ 1, β 2 ]d 1 + ( 1) mn [ β 2, [ᾱ 1, β 2 ]]d 2 +( 1) mn [ᾱ 1, [ᾱ 1, β 2 ]]d 3 + ( 1) mn [ β 1, [ᾱ 1, β 2 ]]d 4. By considering the equation Q 5 TQ 5 = 0 again whose terms consist of basic Whitehead products, we complete the proof.