Definitions For Homotopy

Noun

HOMOTOPY. (countable and uncountable, plural HOMOTOPYs) (topology) A continuous deformation of one continuous function or map to another. : {{ux, en, The concept of homotopy represents a formalisation of the intuitive idea of a smooth deformation of one curve into another.}} (uncountable) The relationship between two continuous functions where homotopy from one to the other is evident. (informal) {{ellipsis of, en, homotopy theory}} (the systematic study of homotopies and their equivalence classes). (topology) A theory associating a system of groups with each topological space. (topology) A system of groups associated with a topological space.

Usage notes

Formally, there are two alternative formulations:{{pedia, Homotopy#Formal definition}} Given topological spaces X, Y and continuous maps f,g: X\rightarrow Y A continuous map H: [0,1]\times X\rightarrow Y such that H(x,0)=f(x) and H(x,1)=g(x) \forall x\in X. A family of continuous maps h_t: X\rightarrow Y, t\in [0,1] such that h_0=f, h_1=g and the map (x,t)\mapsto h_t is continuous from X to Y. (Note that it is not sufficient to require that each map h_t(x) be continuous.) Replacing the unit interval [0,1] with the affine space, affine line A¹ leads to {{w, A¹ homotopy theory}}. The adjective homotopic is used specifically in the sense, with respect to two functions, of "having the relationship of being in homotopy". Being homotopic is an equivalence relation on the class of all continuous functions between given topological spaces. An equivalence class of such a relation is called a homotopy class.

Hyponyms

continuous deformation isotopy, regular homotopy

Derived terms

{{rel-top3, terms derived from "homotopy"}} cohomotopy homotopic homotopy category homotopy class homotopy equivalence homotopy equivalent homotopy fiber, homotopy fibre homotopy group homotopy invariant homotopy relative to a subspace homotopy theory homotopy type linear homotopy regular homotopy straight-line homotopy {{rel-bottom}}

Related terms

homotopic

Translations

Danish: homotopi, c French: homotopie, f Greek: ομοτοπία, f Hungarian: homotópia Icelandic: samtogun, f Japanese: ホモトピー Korean: 연속 변형 함수, 호모토피 Persian: هموتوپی (hemotopi) Portuguese: homotopia, f Spanish: homotopía, f Ukrainian: гомотопія, f Vietnamese: đồng luân French: homotopie, f Hungarian: homotópia Spanish: homotopía, f French: homotopie, f Hungarian: homotópia Spanish: homotopía, f