On the higher Whitehead product

Porter’s approach is used to derive some properties of higher order Whitehead products, similar to those ones for triple products obtained by Hardie. Computations concerning the higher order Whitehead product for spheres and projective spaces are presented as well.


Introduction
In this paper, we work with triple and r th order Whitehead products. The aim of Sect. 1 is to fix some notations, recall definitions and necessary results from [1,2] and present properties on separation elements, and the relative generalized Whitehead product as well. Section 2 expounds the main facts from [15] on r th Whitehead products which are used to discuss the main result from [8]. In Sect. 3, based on some relations from [20, (5.5), (5.8) and Proposition 5.11], we prove Proposition 4, the main result of that section, which deals with the non-triviality problem of the triple product [η 4 , η 2 4 , 2ι 4 ] ⊆ π 14 (S 4 ), stated by Hardie [8,Section 5]. The main result of Sect. 4 is Theorem 4 which extends the result [8,Theorem 4.3] to higher order Whitehead products. Finally, Sect. 5 is devoted to some computations concerning the r th order Whitehead product for spheres and projective spaces. In particular, Proposition 5 shows that there are spaces X such that the r th order generalized Whitehead product [0 1 , . . . , 0 r ] contains a non-trivial element provided 0 i : S m i → X are trivial for i = 1, . . . , r with r ≥ 4.

Preliminaries
Denote by π(X, Y ) the set of (based) homotopy classes of (based) maps X → Y . In the sequel we do not distinguish between a map and its homotopy class. Write C(−) and Σ(−) for the reduced cone and the reduced suspension functors, respectively. As in Toda [20, p. 16], ι n : S n → S n denotes the identity map of the n-sphere S n , and η n = Σ n−2 η 2 : S n+1 → S n for n ≥ 2, is the iterated suspension of the Hopf map η 2 : S 3 → S 2 . We also use freely other symbols from [20, pp. 39-50 and pp. 172-185] and write ι X for the identity map on a space X .

Relative generalized Whitehead product
Let A and B be spaces. Given maps f : Σ A → X and g : Σ B → X , Arkowitz constructed in [2] the Whitehead map to define the generalized Whitehead product [ f, g] as the map ∇( f ∨ g)ω : Σ(A ∧ B) → X , where ∇ : X ∨ X → X is the folding map and A ∧ B is the smash product. Also, from [2,Proposition 3.3], the product is anti-commutative, that is, In [1], Ando defined the relative generalized Whitehead product, briefly described as follows. Let i Z : Z → C Z be the inclusion map. Given k : X → Y , we say that (h 1 , h 2 ) : i Z → k is a pair-map provided the diagram Y is commutative. Write π 1 (Z , k) for the set of homotopy classes of pair-maps (h 1 , h 2 ) : Note that if Z = Σ Z then the standard co-H -structure ν : which determines a group structure on π 1 (Z , k). Moreover, this group is abelian provided Z = Σ Z . In particular, if k : X 0 → X is an inclusion map and Z = S n , we obtain the ordinary relative homotopy groups π 1 (S n , k) = π n+1 (X, Further, the map k : X → Y implies the dual Puppe long exact sequence where the vertical maps κ i for i = 1, 2 are induced by the inclusions of the wedge into the product, and the top row is exact. By [15,Theorem (2.3)], (i Σ A × ι Σ B ) * κ 1 (ω) = 0 and by the commutativity of the diagram above, and the fact that κ 2 is isomorphism, is an epimorphism, and so δ * is a monomorphism. Thus, by the exactness of the top row, there is a unique Let k : X → Y and Z = A∧B in the above. Then, given f = ( f 1 , f 2 ) ∈ π 1 (Σ A, k) and g ∈ π(Σ B, X ), the element ω determines a pair-map (ω , ω) commuting the diagram As in [1], the relative generalized Whitehead product [ f, g] R ∈ π 1 (Σ(A ∧ B), k) is defined by the pair-map Also therein, Ando compares his construction to the one introduced by Hardie in [8]. We describe that here to obtain Proposition 2.
First, since the generalized Whitehead product [0, g] is trivial and the product Thus, by [1,Theorem (4.6)] we obtain Further, for ϕ : Y → Z and

Remark 1
We work with relative Whitehead products for an inclusion map k : X 0 → X . It means that (homotopy classes of) maps f : (CΣ A, Σ A) → (X, X 0 ) and g : Σ B → X 0 determine the map of pairs We finish this section with some properties of the relative product [−,−] R . Let f, f i ∈ π 1 (Σ A, k) and g, g i ∈ π(Σ B, X ) for i = 1, 2. According to [1,Proposition (4.5)], if A and B are suspensions then [ f,

Generalized separation element
Given maps f, g : C A → X such that f |A = g |A , following James [12, Section 10] and Tsuchida [21, Section 3], we define the generalized separation element as (the homotopy class of) the map d( f, g) : Σ A → X given by: Further, if A = Σ A then we consider Then, following (mutatis mutandis) Hardie [8, Section 1] and James [12, Section 10], we may state: Proposition 1 Let f, g, h : C A → X be maps such that f |A = g |A = h |A , and let k : X → Y be any map. Then: Moreover, let f i , g i : CΣ A → X be maps for i = 1, 2 such that f 1|Σ A = f 2|Σ A and g 1|Σ A = g 2|Σ A . Then: Let j * and δ * be the maps as in the exact sequence (2). Then: Proof Since the items (i)-(vi) follow directly from the definition of d, we just prove (vii) and sketch a proof of (viii).
For (vii), let q : After some computations and changes of parameters we can show that: 3 4 ≤ t ≤ 1.
Since the first two parts provide a map homotopic to d( f, f ) = 0 and the third part is homotopic to −ϕ, we obtain −ϕ = d( f, f ). For (viii), we note that the map j * : . Then, we simply mimic the proof of [8, Theorem 1.9].
We finish this section by stating a relation between the separation element and the products defined in Sect. 1.1 (cf. [8, (2.7)]).
Then by the definition of the generalized separation element and the discussion above and the proof follows.

Higher order generalized Whitehead product
Let r ≥ 2 be an integer and denote by A = (A 1 , . . . , A r ) an r -tuple of topological spaces with base points * . The fat wedge of A is the space Following Porter's notation [15], Remark 2 In the sequel, we need to take some coordinate a i 0 = * . To do this, we define . . , r −1 and has a self-explanatory notation. Also, there are canonical embeddings In [15], Porter constructed the generalized Whitehead map (cf. (1)) to define the r th order generalized Whitehead product . . , f r . Hardie [9] gave the definition of [ f 1 , . . . , f r ] when all the A i 's are spheres (called the r th order spherical Whitehead product).

Remark 3
In the sequel we will use the same notation σ for a permutation of the set {1, . . . , r } and for its induced homeomorphism Proposition 3 Let f i : Σ A i → X for i = 1, . . . , r and σ ∈ S r be a permutation of the set {1, . . . , r }. Then and the proof follows.
According to [15,Theorem (2.3)] there is a homotopy equivalence So, we have a pair-map ( r , ω r ) commuting the diagram Considering the spaces and inclusions as in Remark 2, let θ : Proof In virtue of Proposition 3, we can suppose that i 0 = 1. By hypothesis the map 0 ∨ f 2 ∨ · · · ∨ f r : T r −1 Σ(A) → X has an extension F 1 : Then, F is an extension of 0 1 ∨ f 2 ∨ · · · ∨ f r , and the result follows by Theorem 2(iii).
Recall from [19] that a space X is called a G-space if for any map f : S n → X and n ≥ 1 the map ∇(ι X ∨ f ) : X ∨ S n → X extends to F : X × S n → X . According to [19,Theorem 3.2], any r th order spherical Whitehead product of a G-space contains zero. Now, let G(A, X ) ⊆ π(A, X ) be the set of (homotopy classes of) maps f : Lemma 1 If X is a G-space with respect to the spaces A 1 , . . . , A r then any map ι X ∨ f 1 ∨ · · · ∨ f r : X ∨ A 1 ∨ · · · ∨ A r → X extends to F : X × A 1 × · · · × A r → X.
Proof The proof is by induction on r . Let r = 2 and F i : Suppose that the statement is true for r − 1, that is, there exists F : Referring to Williams [22], we say that a space X has property P r if for every Certainly, in view of Theorem 2(ii), any H -space has not only property P r for all r ≥ 2 but 0 is the only element of [ f 1 , . . . , f r ]. (Williams [22]: We note at this point that it is unresolved conjecture as to whether X has property P r implies that 0 is the only element of [ f 1 , . . . , f r ].) Directly from Theorem 2(iii) and Lemma 1 it follows:

Corollary 2 Every G-space X with respect to any Σ A 1 , . . . , Σ A r has property P r . In particular, if f : Σ A → X and X is a G-space with respect to any Σ A then
0 ∈ [ f, ×r . . . , f ] for any r ≥ 2.
In general, the generalized Whitehead product is not additive. But, in the sequel, we need an addition operation defined in [15] for some particular functions. We say that maps f, g : Then, in view of [15, Theorem (2.13)], we can state: From this, it follows:
Proof First, recall that for f ∈ π k (S m ) and g ∈ π l (S m ) with relatively prime orders, the Whitehead product [ f, g] = 0. Hence, because the orders of η 4 and η 2 4 are two, we can restrict to π 4 9 and π 4 10 only. For the first statement notice that which lead to pair-maps for i = 1, . . . , r .
is the inclusion map. Then each relative generalized Whitehead product simplifies to . . , r , and σ i ∈ S r is the permutation inducing the homeomorphism σ i : Next, consider the commutative diagram where k : T 2 Σ(A) → T 1 Σ(A) and k : T 1 Σ(A) → T 0 Σ(A) are inclusion maps, δ * 1 , δ * 2 are boundaries and j * is the obvious map.
Following mutatis mutandis the result due to Nakaoka-Toda [14, Lemma (1.2)] for spheres and a proof of its generalization [10, Formula (0.1)], we may state for suspensions: Notice that in view of the formula (3), we have Now, we are in a position to generalize the main result of [8,Theorem 4.3].
Theorem 4 Let f : T 2 Σ(A) → X be any map, j i : Σ A i → T 2 Σ(A) be the canonical inclusion maps, and f i = f j i : Σ A i → X for i = 1, . . . , r.
where σ i ∈ S r is the permutation inducing the homeomorphism σ i :

is an extension of f then for any
Proof (i) By Lemma 3 and commutativity of the diagram above, h = ∂( r , ω r ) = j * (ω r ) so that ω r ∈ j −1 * (h). In view of Proposition 1(viii), there exists g : T 1 Σ(A)).
Since, by Theorem 1, the space T 1 Σ(A) is a push-out, the universal property guarantees the existence of f : The same computations as in (i) show that γ = ω r ( f ) − ω r ( f ) and the proof is complete. (iii) This is a consequence of (i) and (ii).
On the other hand, since ρ * and q * are monomorphisms, the elements q * ρ * (α) and q * ρ * (β) are non-trivial as well. Further, in view of (8), the map is an isomorphism with the inverse for n = m 1 + m 2 and n = m 3 + · · · + m r .
Let R and C be the fields of real and complex numbers, respectively and H the skew R-algebra of quaternions. Denote by FP n the n-projective space over F = R, C or H, put d = dim R F and set i nF : S d → FP n for the inclusion map. Let γ nF : S (n+1)d−1 → FP n be the canonical quotient map. In view of [3, Corollary (7.4)] and [4, (4.1-3)], we obtain a key formula: