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A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology. [3] In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work with compactly generated spaces, CW complexes, or spectra.
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 .
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.
The term () is the second homotopy group of B, which is defined to be the set of homotopy classes of maps from to B, in direct analogy with the definition of . If E happens to be path-connected and simply connected, this sequence reduces to an isomorphism
In this sense, ANRs avoid all the homotopy-theoretic pathologies of arbitrary topological spaces. For example, the Whitehead theorem holds for ANRs: a map of ANRs that induces an isomorphism on homotopy groups (for all choices of base point) is a homotopy equivalence. Since ANRs include topological manifolds, Hilbert cube manifolds, Banach ...
The homotopy pullback of along the identity is nothing but the mapping path space of . The universal property of a homotopy pullback yields the natural map , a special case of a natural map from a limit to a homotopy limit. In the case of a homotopy fiber, this map is an inclusion of a fiber to a homotopy fiber.
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".