homotopische

Homotopische describes aspects related to homotopy in topology. In mathematics, a homotopy is a continuous deformation of one function into another. When two continuous maps f and g from a space X to a space Y can be connected by such a deformation, they are said to be homotopic, written f ~ g. A homotopy is a continuous map H: X × [0,1] → Y with H(x,0) = f(x) and H(x,1) = g(x) for all x in X. The concept provides an equivalence relation on the set of continuous maps, grouping maps into homotopy classes.

Variants include relative homotopy, where the deformation is constrained to keep a subspace A ⊆ X fixed. This leads to relative homotopy groups and relative homotopy classifications, which are useful for studying spaces with boundary or with distinguished subspaces.

Two spaces X and Y are called homotopy equivalent (or of the same homotopy type) if there exist continuous maps f: X → Y and g: Y → X such that the compositions gf and fg are homotopic to the respective identity maps on Y and X. Homotopy equivalence is weaker than homeomorphism but indicates that the spaces share many global topological properties. Invariants such as fundamental groups and higher homotopy groups are preserved under homotopy, as are many homology groups, though the latter are not exclusive to homotopy information.

The homotopy perspective leads to the homotopy category, where objects are spaces and morphisms are homotopy classes of maps. Based variants consider fixed basepoints, yielding a related based homotopy category. Homotopische concepts are central to classifying spaces up to homotopy type, constructing spaces with prescribed homotopy groups, and guiding forms of algebraic topology such as CW-complex theory and rational homotopy theory.