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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 .
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism , though usually most classify up to homotopy equivalence .
In homotopy theory and algebraic topology, the word "space" denotes a topological space.In order to avoid pathologies, one rarely works with arbitrary spaces; instead, one requires spaces to meet extra constraints, such as being compactly generated weak Hausdorff or a CW complex.
Given two directed paths γ and δ, a directed homotopy from γ to δ is a morphism of directed spaces h whose underlying map U(h) is a homotopy –in the usual sense– between the underlying paths U(γ) and U(δ). In algebraic topology, there is a homotopy from α to β if and only if there is a homotopy from β to α. Due to non ...
In mathematics, algebraic homotopy is a research program on homotopy theory proposed by J.H.C. Whitehead in his 1950 ICM talk, where he described it as: [1] [2] The ultimate object of algebraic homotopy is to construct a purely algebraic theory, which is equivalent to homotopy theory in the same sort of way that 'analytic' is equivalent to 'pure' projective geometry.
A homotopy pullback (or homotopy fiber-product) is the dual concept of a homotopy pushout. It satisfies the universal property of a pullback up to homotopy. [ citation needed ] Concretely, given f : X → Z {\displaystyle f:X\to Z} and g : Y → Z {\displaystyle g:Y\to Z} , it can be constructed as
In mathematics, homotopical algebra is a collection of concepts comprising the nonabelian aspects of homological algebra, and possibly the abelian aspects as special cases. . The homotopical nomenclature stems from the fact that a common approach to such generalizations is via abstract homotopy theory, as in nonabelian algebraic topology, and in particular the theory of closed model categor