homotopy

Etymology
From + ; earliest known use in print in 1922,, Analysis Situs.

Noun

 * 1)  A continuous deformation of one continuous function or map to another.
 * 2) * 2010,, Homotopy Quantum Field Theory, , page xi,
 * In this monograph we apply the idea of a TQFT to maps from manifolds to topological spaces. This leads us to a notion of a (d + 1)-dimensional homotopy quantum field theory (HQFT) which may be described as a TQFT for closed oriented d-dimensional manifolds and compact oriented (d + 1)-dimensional cobordisms endowed with maps to a given space X.
 * 1)  The relationship between two continuous functions where homotopy from one to the other is evident.
 * 2)   the systematic study of homotopies and their equivalence classes.
 * 3)  A theory associating a system of groups with each topological space.
 * 4)  A system of groups associated with a topological space.
 * 1)   the systematic study of homotopies and their equivalence classes.
 * 2)  A theory associating a system of groups with each topological space.
 * 3)  A system of groups associated with a topological space.

Usage notes

 * Formally, there are two alternative formulations:
 * 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 line A¹ leads to.
 * The adjective 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.

Translations

 * Danish: homotopi
 * French:
 * Greek: ομοτοπία
 * Hungarian:
 * Icelandic: samtogun
 * Japanese: ホモトピー
 * Korean: 연속 변형 함수, 호모토피
 * Persian: هموتوپی
 * Portuguese: homotopia
 * Spanish: homotopía
 * Ukrainian: гомотопія
 * Vietnamese: đồng luân


 * French:
 * Hungarian:
 * Spanish: homotopía


 * French:
 * Hungarian:
 * Spanish: homotopía