Free loop

Free loop

In the mathematical field of topology, a free loop is a variant of the mathematical notion of a loop. Whereas a loop has a distinguished point on it, called a basepoint, a free loop lacks such a distinguished point. Formally, let "X" be a topological space. Then a free loop in "X" is an equivalence class of continuous functions from the circle S1 to "X". Two loops are equivalent if they differ by a reparameterization of the circle. That is, ƒ ≈ "g" if:g = fcircpsifor a homeomorphism ψ : S1 → S1.

Thus, a free loop, as opposed to a based loop used in the definition of the fundamental group, is a map from the circle to the space without the basepoint-preserving restriction. Free homotopy classes of free loops correspond to conjugacy classes in the fundamental group.

ee also

*loop space
*Loop (topology)

References

Brylinski, Jean-Luc:Loop spaces, characteristic classes and geometric quantization.Reprint of the 1993 edition. Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston, MA, 2008.


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