Homotopioita

Homotopioita, or homotopies, are a fundamental concept in topology describing continuous deformations between maps. If X and Y are topological spaces and f, g: X → Y are continuous, a homotopy between f and g is a continuous function H: X × I → Y, where I = [0, 1], such that H(x, 0) = f(x) and H(x, 1) = g(x) for all x in X. Intuitively, H(x, t) traces a path of points in Y from f(x) to g(x) as t goes from 0 to 1.

Homotopic maps are equivalent under the relation of being connected by such a homotopy. This relation is an equivalence relation on the set of continuous maps X → Y, and the equivalence classes are called homotopy classes, denoted [f]. When f and g are homotopic, they induce the same effects on many algebraic invariants, such as homotopy groups.

Two maps f: X → Y and g: X → Y are said to be homotopic if there exists a homotopy H with H(x, 0) = f(x) and H(x, 1) = g(x). A related notion is a relative homotopy, where H(a, t) is fixed for all a in a subspace A ⊆ X. A deformation retract is a particular kind of homotopy that deforms a space X onto a subspace A while keeping A fixed.

If spaces X and Y admit maps f: X → Y and g: Y → X with g ∘ f homotopic to the identity on X and f ∘ g homotopic to the identity on Y, X and Y are homotopy equivalent. This concept underpins the idea of homotopy invariants, such as the fundamental group and higher homotopy groups, which are preserved under homotopy equivalence. Variants include smooth, PL, or relative homotopies, depending on additional structures on the spaces.