On the higher Whitehead product

The Porter's approach (G.\ J.\ Porter, \textit{Higher order Whitehead products}, Topology \textbf{3} (1965), 123--135.) is used to derive some properties of higher order Whitehead products, similar to those ones for triple products obtained by Hardie in K.\ A.\ Hardie, \textit{On a construction of E.\ C.\ Zeeman}, Journal London Math.\ Soc.\ \textbf{35} (1960), 452--464.


Introduction
Whitehead products play an important role in algebraic topology and its applications. The classical Whitehead product [f, g] of (homotopy classes of) maps f : S m → X and g : S n → X is (the homotopy class) represented by a map h : S m+n−1 → X and defined by means of the so called Whitehead map ω : S m+n−1 → S m ∨ S n , the attaching map for the (m + n)-cell of S m × S n .
Arkowitz [2] constructed a generalization ω : Σ(A ∧ B) → ΣA ∨ ΣB of the Whitehead map to define the generalized Whitehead product [f, g] of maps f : ΣA → X and g : ΣB → X. Then Zeeman [13] and Hardie [8] generalized the Whitehead product in a different context, that is, they defined a triple spherical Whitehead product [f 1 , f 2 , f 3 ] of maps f i : S m i → X for i = 1, 2, 3. The main result from [8] deals with the r th order spherical Whitehead product [f 1 , . . . , f r ] for maps f i : S m i → X with r ≥ 3 defined in [9], and in particular, it states that the triple product [f 1 , f 2 , f 3 ] is a coset of a subgroup of π m−1 (X), where m = m 1 + m 2 + m 3 . Many properties which hold for the classical Whitehead product still hold for the triple one as well.
G. J. Porter's approach [15] was to construct the Whitehead map ω r for more than two spaces (see Equation (5)). Hardie's construction from [9] was generalized in [15], where the r th order generalized Whitehead product of maps f i : ΣA i → X for i = 1, . . . , r, with r ≥ 2 was introduced. Then the 2 nd order Whitehead product coincides with the generalized Whitehead product studied by Arkowitz in [2].
Higher order Whitehead products are secondary, tertiary, etc. analogues of ordinary Whitehead products. They first appeared in the late 1960's as a part of a research theme studying higher products, or simply higher structures, in homotopy theory. They are vital for understanding the homotopy theory of certain basic objects, such as an iterated product of spaces, and their maps into other spaces. Recently, higher Whitehead products have re-emerged as key players in the homotopy theory of polyhedral products. These are important objects in toric topology and are being increasingly used in geometric group theory and graph theory. Porter's construction is very useful in many mathematical constructions. For example, given a simplicial complex K on n vertices, Davis and Januszkiewicz [6] associated two fundamental objects of toric topology: the moment-angle complex Z K and the Davis-Januszkiewicz space DJ K . The homotopy fibration sequence CP ∞ and its generalization have been studied in [7] and [11], respectively to show thatω : Z K → DJ K is a sum of higher and iterated Whitehead products for appropriate complexes K.
Next, let F(R n+1 , m) be the Euclidean ordered configuration space. By Salvatore [17,Theorem 7], the homotopy type of F(R n+1 , m) for n ≥ 2 admits a minimal cellular model * = X 0 ⊆ X n ⊆ X 2n ⊆ · · · ⊆ X mn whose cells are attached via higher order Whitehead products.
In this paper, we work with triple and r th order Whitehead products. The aim of Section 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 Section 3, based on some relations from [20], we prove Proposition 3.2, 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 in [8,Section 5]. The main result of Section 4 is Theorem 4.2 which extends the result [8,Theorem 4.3] to higher order Whitehead products. Finally, Section 5 is devoted to some computations concerning the r th order Whitehead product for spheres and projective spaces. In particular, Proposition 5.2 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.
Acknowledgement. The authors are deeply grateful to the referee for a number of invaluable suggestions to work on the final version, especially on Proposition 3.2. They would like to thank Martin Arkowitz for a very helpful discussion and indicating the reference [1] as well.

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], 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] and write ι X for the identity map on a space X.
1.1. Relative generalized Whitehead product. Let A and B be spaces. Given maps f : ΣA → X and g : ΣB → 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- In [1], Ando defined the relative generalized Whitehead product, briefly described as follows. Let i Z : Z ֒→ CZ 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 for the set of homotopy classes of pair-maps Note that if Z = ΣZ ′ then the standard co-H-structure ν : ΣZ ′ → ΣZ ′ ∨ ΣZ ′ leads to a pair-map which determines a group structure on π 1 (Z, k). Moreover, this group is abelian provided Z ′ = ΣZ ′′ . 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, X 0 ) = [(D n+1 , S n ), (X, X 0 )], where D n+1 = CS n is the (n + 1)-disc.
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.
Let k : X → Y and Z = A ∧ B in the above. Then, given f = (f 1 , f 2 ) ∈ π 1 (ΣA, k) and g ∈ π(ΣB, 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 1.3.
First, since the generalized Whitehead product [0, g] is trivial and the product ΣA × ΣB is a push-out, we obtain a map λ A (g) : Further, for ϕ : Y → Z and Next, suppose that A = ΣA ′ and consider [f 2 , g] ′′ : Σ(A∧B) → X given by Remark 1.1. We work with relative Whitehead products for an inclusion map k : X 0 ֒→ X. It means that (homotopy classes of) maps f : (CΣA, ΣA) → (X, X 0 ) and g : ΣB → 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 (ΣA, k) and g, g i ∈ π(ΣB, X) for i = 1, 2. According to [1,Proposition (4.5)], if A and B are suspensions then [f, 1.2. Generalized separation element. Given maps f, g : CA → 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) : ΣA → X given by: Then, following (mutatis mutandis) Hardie [8, Section 1] and James [12, Section 10], we may state: and let k : X → Y be any map. Then: Moreover, let f i , g i : CΣA → X be maps for i = 1, 2 such that f 1|ΣA = f 2|ΣA and g 1|ΣA = g 2|ΣA . 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 : A × I → CA be the quotient map such that the inclusion 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 For (viii), we note that the map j * : π(Σ 2 A, Y ) → π 1 (ΣA, k) is determined by the map (CΣA, ΣA) → (Σ 2 A, * ). Then, we simply mimic the proof of [8, We finish this section by stating a relation between the separation element and the products defined in Section 1.1 (cf. [8, (2 .7)]). Then Then by the definition of the generalized separation element and the discussion above , kg], 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], let T 0 (A) = A 1 × · · · × A r and T s (A) ⊆ T 0 (A) be the subset of points (a 1 , . . . , a r ) with at least s coordinates a i = * for s = 1, . . . , r. Thus, T r−1 (A) = A 1 ∨ · · · ∨ A r is the wedge sum and T 1 (A) = F W (A) is the fat wedge. Also, we write Λ(A) = A 1 ∧ · · · ∧ A r and Σ(A) = (ΣA 1 , . . . , ΣA r ).
Remark 2.1. 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 Ψ In [15], Porter constructed the generalized Whitehead map (cf. (1)) to define the r th order generalized Whitehead product , Remark 2.2. In the sequel we will use the same notation σ for a permutation of the set {1, . . . , r} and for its induced homeomorphism A 1 ∧ · · · ∧ A r → A σ(1) ∧ · · · ∧ A σ(r) . Proof. Denote by σA = (A σ(1) , . . . , A σ(r) ) the image of A under the permutation σ ∈ S r . It is clear that there is a commutative diagram . Finally, note that σω r (σf ) = sgn(σ)(Σ r−1 σ) * (ω r (f )) and the proof follows.
According to [15,Theorem (2.3)] there is a homotopy equivalence So, we have a pair-map (Ω r , ω r ) commuting the diagram Considering the spaces and inclusions as in Remark 2.1, let θ :   and only if f 1 ∨· · ·∨f r : T r−1 Σ(A) → X has an extension to T 0 Σ(A). Further, Σω r (f ) = 0 for f : Proof. In virtue of Proposition 2.3, 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 2.5(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 Lemma 2.7. 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 : Finally, let F r : X × A r → X extending ι X ∨ f r : X ∨ A r → X and then the composite F r (F × ι Ar ) : X × T 0 (A) → X extends ι X ∨ f 1 ∨ · · · ∨ f r , which completes the proof.
Referring to Williams [22], we say that a space X has property P r if for every f i : ΣA i → X with i = 1, . . . , r, we have 0 ∈ [f 1 , . . . , f r ]. Certainly, in view of Theorem 2.5(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 2.5(iii) and Lemma 2.7 it follows: Corollary 2.8. Every G-space X with respect to any ΣA 1 , . . . , ΣA r has property P r . In particular, If f : ΣA → X and X is a G-space with respect to any ΣA 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 : i is a suspension one defines an addition + (i) by From this, it follows: for any integer n and i = 1, . . . , r.
Hardie [8]   that it is non-empty and has order fifteen. In order to prove this, we need to check first that all lower products vanish. We keep below the standard notations from Toda's book [20].

Main result
Theorem 2.4 provides the following commutative diagram which lead to pair-maps for i = 1, . . . , r.
, k) and  i : ΣA i ֒→ T 2 Σ(A) be the canonical inclusion map for i = 1, . . . , r, where k : T 2 Σ(A) ֒→ T 1 Σ(A) is the inclusion map. Then each relative generalized Whitehead product . . , r, and σ i ∈ S r is the permutation inducing the homeomorphism Hence, we got the relative generalized Whitehead products τ 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 where σ i ∈ S r is the permutation inducing the homeomorphism Proof. (i) By Lemma 4.1 and commutativity of the diagram above, h = ∂(Ω r , ω r ) = j * (ω r ) so that ω r ∈ j −1 * (h). In view of Proposition 1.2(viii), there exists g ′ : Since f ′ k = f ′′ k = f , by (4) and Propositions 1.2-1.3, we get (7), we ) → X and then, in view of Proposition 1.2(vii), there exist maps . . , r. Note that the maps g i determine a map Since, by Theorem 2.4, 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.
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: Lemma 5.4. Let h 0 f ∈ π k (S 2n−1 ) be the 0 th Hopf-Hilton invariant for f ∈ π k (S n ). Then: [γ nR f, i nR ] = 0, for odd n,