homotopias

Homotopies are a fundamental concept in algebraic topology, a branch of mathematics that studies the properties of spaces that are preserved under continuous deformations. A homotopy is a continuous transformation of one continuous function into another. More formally, given two continuous functions f and g from a topological space X to a topological space Y, a homotopy H from f to g is a continuous function H: X × [0, 1] → Y such that H(x, 0) = f(x) and H(x, 1) = g(x) for all x in X. The function H can be thought of as a "continuous deformation" of f into g, where the parameter t in [0, 1] represents the "time" of the deformation.

Homotopies are used to define various notions of equivalence between topological spaces. For example, two spaces X and Y are said to be homotopy equivalent if there exist continuous functions f: X → Y and g: Y → X such that the compositions g ∘ f and f ∘ g are homotopic to the identity functions on X and Y, respectively. Homotopy equivalence is a weaker notion of equivalence than homeomorphism, as it allows for continuous deformations of the spaces.

Homotopies are also used to define homotopy groups, which are important invariants in algebraic topology. The nth homotopy group of a space X, denoted πn(X), is the set of homotopy classes of continuous functions from the n-dimensional sphere Sn to X, where the homotopies are required to fix the basepoint of Sn. Homotopy groups are used to study the shape of topological spaces and to distinguish between different spaces.

In summary, homotopies are a powerful tool in algebraic topology, used to define notions of equivalence between spaces and to study the shape of spaces through homotopy groups. They provide a way to understand the properties of spaces that are preserved under continuous deformations.