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A homotopy between two embeddings of the torus into : as "the surface of a doughnut" and as "the surface of a coffee mug".This is also an example of an isotopy.. Formally, a homotopy between two continuous functions f and g from a topological space X to a topological space Y is defined to be a continuous function: [,] from the product of the space X with the unit interval [0, 1] to Y such that ...
In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology , but nowadays is learned as an independent discipline.
The older definition of the homotopy category hTop, called the naive homotopy category [1] for clarity in this article, has the same objects, and a morphism is a homotopy class of continuous maps. That is, two continuous maps f : X → Y are considered the same in the naive homotopy category if one can be continuously deformed to the other.
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group , denoted π 1 ( X ) , {\displaystyle \pi _{1}(X),} which records information about loops in a space .
An equivalent definition of homotopical connectivity is based on the homotopy groups of the space. A space is n-connected (or n-simple connected) if its first n homotopy groups are trivial. Homotopical connectivity is defined for maps, too. A map is n-connected if it is an isomorphism "up to dimension n, in homotopy".
Regular homotopy for immersions is similar to isotopy of embeddings: they are both restricted types of homotopies. Stated another way, two continuous functions f , g : M → N {\displaystyle f,g:M\to N} are homotopic if they represent points in the same path-components of the mapping space C ( M , N ) {\displaystyle C(M,N)} , given the compact ...
In mathematics, especially homotopy theory, the mapping cone is a construction in topology analogous to a quotient space and denoted . Alternatively, it is also called the homotopy cofiber and also notated . Its dual, a fibration, is called the mapping fiber.
The homotopy group functors assign to each path-connected topological space the group () of homotopy classes of continuous maps . Another construction on a space X {\displaystyle X} is the group of all self-homeomorphisms X → X {\displaystyle X\to X} , denoted H o m e o ( X ) . {\displaystyle {\rm {Homeo}}(X).}