Homotopy

Homotopy is a fundamental concept in topology, a branch of mathematics that studies the properties of spaces that are preserved under continuous deformations. Intuitively, two continuous functions between topological spaces are homotopic if one can be continuously deformed into the other.

More formally, let f and g be continuous functions from a topological space X to a topological space Y. A homotopy between f and 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. Here, [0, 1] denotes the closed unit interval, and the product X × [0, 1] is equipped with the product topology. We often write f ≈ g to denote that f and g are homotopic. The variable t in the interval [0, 1] can be thought of as a "time" parameter, and H(x, t) represents the position of f(x) at time t as it continuously deforms into g(x).

Homotopy is an equivalence relation on the set of continuous functions from X to Y. This means that it satisfies the following properties:

• Reflexivity: For any continuous function f: X → Y, f ≈ f.
• Symmetry: If f ≈ g, then g ≈ f.
• Transitivity: If f ≈ g and g ≈ h, then f ≈ h.

The equivalence classes under this relation are called homotopy classes.

Homotopy is a central concept in defining the fundamental group and higher homotopy groups of a topological space, which are important tools for classifying topological spaces. These groups provide algebraic invariants that can distinguish between spaces that are not homeomorphic (i.e., cannot be continuously deformed into each other).

Related concepts include:

• Path Homotopy: A special case of homotopy where X is the unit interval [0, 1]. This is used to define the fundamental group.
• Homotopy Equivalence: Two spaces X and Y are homotopy equivalent if there exist continuous functions f: X → Y and g: Y → X such that g ∘ f is homotopic to the identity map on X and f ∘ g is homotopic to the identity map on Y. Homotopy equivalence is a weaker condition than homeomorphism. Spaces that are homotopy equivalent share many of the same topological properties, such as the same homotopy groups.
• Null-homotopic: A continuous function f: X → Y is null-homotopic if it is homotopic to a constant function (a function that maps every point in X to a single point in Y).
• Isotopy: A stronger condition than homotopy. Two embeddings f and g from X to Y are isotopic if there exists a homotopy H: X × [0, 1] → Y such that H(x, t) is an embedding for all t in [0, 1]. In other words, an isotopy is a homotopy through embeddings.