homotopie

Homotopie, known in English as homotopy, is a fundamental concept in topology that captures the idea of continuously deforming one function into another. Let X and Y be topological spaces. Two continuous maps f, g: X → Y are homotopic if there exists a continuous function H: X × [0,1] → Y, called a homotopy from f to g, such that H(x,0) = f(x) and H(x,1) = g(x) for all x in X. The parameter t in [0,1] describes the deformation from f to g. If such an H exists, f and g are said to be homotopic, written f ~ g.

Homotopy defines an equivalence relation on the set of maps X → Y, and the homotopy class of f is denoted [f]. When focusing on maps X → Y up to homotopy, one works with these classes rather than with individual maps. Spaces X and Y are called homotopy equivalent if there exist maps f: X → Y and g: Y → X such that g∘f is homotopic to id_X and f∘g is homotopic to id_Y. In this sense, X and Y have the same homotopy type.

A space that is homotopy equivalent to a point is contractible. The concept leads to a rich collection of invariants, notably the homotopy groups π_n(X), which generalize the fundamental group π1(X) obtained from based loops. Homotopy theory distinguishes spaces up to homotopy type rather than up to homeomorphism, and it plays a central role in algebraic topology, geometry, and related fields. There are variations, such as based versus free homotopy, and concepts like the homotopy extension property that formalize how homotopies extend over subspaces.