2024-05-19   69958

Homotopy Sets

If X and Y are topological spaces and f is a map from X to Y, then the homotopy class [f] is the set of all maps from X to Y that are homotopic to f, which roughly means they can be continuously morphed to f. The (free) homotopy set [X,Y] is the set whose elements are homotopy classes of maps from X to Y. The based homotopy set is defined similarly, but using only homotopies that preserve a designated basepoint in X and in Y. The study of homotopy sets is a prominent part of the study of topology. Homotopy groups (article 61813) and cohomology groups (article 28539) may both be expressed as special families of homotopy sets equipped with a natural group structure. This article gathers some results about homotopy sets, with special attention given to the example [T3,SU(k)] where T3 is a 3-torus and SU(k) is a special unitary group.

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www.cphysics.org updated 2024-06-01