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Intuitively, homotopy groups record information about the basic shape, or holes, of a topological space. To define the n-th homotopy group, the base-point-preserving maps from an n-dimensional sphere (with base point) into a given space (with base point) are collected into equivalence classes, called homotopy classes.
The groups π n+k (S n) with n > k + 1 are called the stable homotopy groups of spheres, and are denoted π S k: they are finite abelian groups for k ≠ 0, and have been computed in numerous cases, although the general pattern is still elusive. [22] For n ≤ k+1, the groups are called the unstable homotopy groups of spheres. [citation needed]
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 ...
A generator for the fundamental group is the closed curve obtained by projecting any curve connecting antipodal points in down to . The projective n {\displaystyle n} -space is compact, connected, and has a fundamental group isomorphic to the cyclic group of order 2: its universal covering space is given by the antipody ...
The fundamental group is a homotopy invariant—topological spaces that are homotopy equivalent (or the stronger case of homeomorphic) have isomorphic fundamental groups. The fundamental group of a topological space X {\displaystyle X} is denoted by π 1 ( X ) {\displaystyle \pi _{1}(X)} .
In mathematics, the Bott periodicity theorem describes a periodicity in the homotopy groups of classical groups, discovered by Raoul Bott (1957, 1959), which proved to be of foundational significance for much further research, in particular in K-theory of stable complex vector bundles, as well as the stable homotopy groups of spheres.
A simply connected map (1-connected map) is one that is an isomorphism on path components (0th homotopy group) and onto the fundamental group (1st homotopy group). n -connectivity for spaces can in turn be defined in terms of n -connectivity of maps: a space X with basepoint x 0 is an n -connected space if and only if the inclusion of the ...
Every group is the fundamental group of some space. [ 2 ] A map f {\displaystyle f} is called a homotopy equivalence if there is another map g {\displaystyle g} such that f ∘ g {\displaystyle f\circ g} and g ∘ f {\displaystyle g\circ f} are both homotopic to the identities.