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Given two topological spaces X and Y, a homotopy equivalence between X and Y is a pair of continuous maps f : X → Y and g : Y → X, such that g ∘ f is homotopic to the identity map id X and f ∘ g is homotopic to id Y. If such a pair exists, then X and Y are said to be homotopy equivalent, or of the same homotopy type.
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.
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 .
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.
Tables of homotopy groups of spheres are most conveniently organized by showing π n+k (S n). The following table shows many of the groups π n+k (S n). The stable homotopy groups are highlighted in blue, the unstable ones in red. Each homotopy group is the product of the cyclic groups of the orders given in the table, using the following ...
A 1 homotopy theory is founded on a category called the A 1 homotopy category ().Simply put, the A 1 homotopy category, or rather the canonical functor (), is the universal functor from the category of smooth -schemes towards an infinity category which satisfies Nisnevich descent, such that the affine line A 1 becomes contractible.
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 homotopy fiber (sometimes called the mapping fiber) [1] is part of a construction that associates a fibration to an arbitrary continuous function of topological spaces:. It acts as a homotopy theoretic kernel of a mapping of topological spaces due to the fact it yields a long exact sequence of ...
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