A loop based at x0 in a topological spaceMathworldPlanetmath X is simply a continuous map f:[0,1]X with f(0)=f(1)=x0.

The collection of all such loops, modulo homotopy equivalenceMathworldPlanetmathPlanetmath, forms a group known as the fundamental groupMathworldPlanetmathPlanetmath.

More generally, the space of loops in X based at x0 with the compact-open topologyMathworldPlanetmath, represented by Ωx0, is known as the loop spaceMathworldPlanetmath of X. And one has the homotopy groupsMathworldPlanetmath πn(X,x0)=πn-1(Ωx0,ι), where πn represents the higher homotopy groups, and ι is the basepoint in Ωx0 consisting of the constant loop at x0.

Title loop
Canonical name Loop1
Date of creation 2013-03-22 12:16:21
Last modified on 2013-03-22 12:16:21
Owner nerdy2 (62)
Last modified by nerdy2 (62)
Numerical id 5
Author nerdy2 (62)
Entry type Definition
Classification msc 54-00