Homotopy theory | Lie algebras
The Whitehead product is a mathematical construction introduced in . It has been a useful tool in determining the properties of spaces. 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. Some of these properties are connectedness (is the space in one or several pieces), the number of holes the space has, the knottedness of the space, and so on. Spaces are then studied by assigning algebraic constructions to them. 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). The most common algebraic constructions are groups. These are sets such that any two members of the set can be combined to yield a third member of the set (subject to certain restrictions). In homotopy theory, one assigns a group to each space X and positive integer p called the pth homotopy group of X. 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. This has been very important in the study of homotopy groups. Several generalisations of the Whitehead product appear in 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 and and later studied in detail by , (see also , 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. (Wikipedia).
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