The mathematical notion of space includes every shape that exists in our 3-dimensional world such as curves, surfaces, and solid figures.
Since spaces are often presented by formulas, it is usually not possible to visually determine their geometric properties.
This is similar to what is done in high school analytic geometry whereby to certain curves in the plane (geometric objects) are assigned equations (algebraic constructions).
These groups have been studied extensively and give information about the properties of the space X.
There are then operations among these groups (the Whitehead product) which provide additional information about the spaces.
Several generalisations of the Whitehead product appear in (Blakers & Massey 1953) and elsewhere, but the most far-reaching one deals with homotopy sets, that is, homotopy classes of maps from one space to another.
The generalised Whitehead product assigns to an element α in the homotopy set [ΣA, X] and an element β in the homotopy set [ΣB, X], an element [α, β] in the homotopy set [Σ(A ∧ B), X], where A, B, and X are spaces, Σ is the suspension (topology), and ∧ is the smash product.
This was introduced by Cohen (1957) and Hilton (1965) and later studied in detail by Arkowitz (1962), (see also Baues (1989), p. 157).
It is a generalization of the Whitehead product and provides a useful technique in the investigation of homotopy sets.
The relevant MSC code is: 55Q15, Whitehead products and generalizations.
The generalised Whitehead product is then defined as the unique element such that
E[α, β] = 0, where E : [Σ(A ∧ B), X] → [Σ2 (A ∧ B), ΣX] is the suspension homomorphism.
An appropriate Jacobi identity for α and β as above and γ ∈ [ΣC, X], if A, B, and C are suspensions.
The product ΣA × ΣB has the homotopy type of the mapping cone of [ιΣA, ιΣB] ∈ [Σ(A ∧ B), ΣA ∨ ΣB] (Arkowitz (1962)).
Whitehead products for homotopy groups with coefficients are obtained by taking A and B to be Moore spaces (Hilton (1965), pp.
110–114) There is a weak homotopy equivalence between a wedge of suspensions of finitely many spaces and an infinite product of suspensions of various smash products of the spaces according to the Hilton–Milnor theorem.
The map is defined by generalised Whitehead products (Baues & Quintero 2001).
If λU,V : [U, ΩV] → [ΣU, V] is the adjoint isomorphism, where Ω is the loop space functor, then λA∧B,X<σ, τ>= [λA,X (σ), λB,X (τ)] for Y = ΩX.
An Eckmann–Hilton dual of the generalised Whitehead product can be defined as follows.
Let A♭B be the homotopy fiber of the inclusion j : A ∨ B → A × B, that is, the space of paths in A × B which begin in A ∨ B and end at the base point and let γ ∈ [X, ΩA] and δ ∈ [X, ΩB].