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The two dashed paths shown above are homotopic relative to their endpoints. The animation represents one possible homotopy. In topology, two continuous functions from one topological space to another are called homotopic (from Ancient Greek: ὁμός homós "same, similar" and τόπος tópos "place") if one can be "continuously deformed" into the other, such a deformation being called a ...
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. Although algebraic topology primarily uses algebra to study topological ...
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 .
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).}
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
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 the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants , which reflect, in algebraic terms, the structure of spheres viewed as topological spaces , forgetting about their precise geometry.