homotop

Homotop is a term that appears only sporadically in mathematical writing and is not widely recognized as a distinct technical concept. In most contexts, the standard term is homotopy, and “homotop” may be encountered as an informal shorthand, a misspelling, or a reference to a related construct such as a homotopy operator or a particular path in a space of maps. When encountered, its precise meaning should be inferred from the surrounding text.

In topology, a homotopy between two continuous maps f and g from a space X to a space Y is a continuous function H: X × [0,1] → Y with H(x,0) = f(x) and H(x,1) = g(x) for all x in X. If such an H exists, f and g are said to be homotopic. Homotopy captures the notion of deforming one map into another through a continuous family of maps. A map is null-homotopic if it is homotopic to a constant map.

A broader concept is homotopy equivalence: spaces X and Y are homotopy equivalent if there exist maps f: X → Y and g: Y → X such that the compositions gf and fg are each homotopic to the respective identity maps. This relation identifies spaces up to continuous deformation and underpins much of algebraic topology, including the study of fundamental groups and higher homotopy groups.

In practice, “homotop” is not used as a formal object in standard references; when seen, it should be interpreted in context as pertaining to a homotopy or to a related operator within a given discussion.

See also: Homotopy, Homotopy equivalence, Homotopy group, Homotopy extension.