imported>Jayy V at 21:38, 9 March 2024

2024-03-09T21:38:10Z

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{{For|the song|Free Loop (One Night Stand)}}
In the [[mathematics|mathematical]] field of [[topology]], a '''free loop''' is a variant of the notion of a [[loop (topology)|loop]]. Whereas a loop has a distinguished point on it, called its basepoint, a free loop lacks such a distinguished point. Formally, let <math>X</math> be a [[topological space]]. Then a free loop in <math>X</math> is an [[equivalence class]] of [[continuous function]]s from the [[circle]] <math>S^1</math> to <math>X</math>. Two loops are equivalent if they differ by a reparameterization of the circle. That is, <math>f \sim g</math> if there exists a [[homeomorphism]] <math>\psi : S^1 \rightarrow S^1</math> such that <math>g = f\circ\psi.</math>

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. Assuming the space is [[path-connected]], free [[homotopy]] classes of free loops correspond to [[conjugacy class]]es in the fundamental group.

Recently, interest in the space of all free loops <math>LX</math> has grown with the advent of [[string topology]], i.e. the study of new [[algebraic structure]]s on the [[singular homology|homology]] of the free loop space.

==See also==
*[[Loop space]]
*[[Loop (topology)]]
*[[Quasigroup]]

==Further reading==
* Brylinski, Jean-Luc: Loop spaces, characteristic classes and geometric quantization. Reprint of the 1993 edition. Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston, MA, 2008.
* Cohen and Voronov: [https://arxiv.org/abs/math/0503625 Notes on String Topology]

[[Category:Knot theory]]
[[Category:Topology]]

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Free loop - Revision history

imported>Jayy V at 21:38, 9 March 2024