# Free loop

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{{For|the song|Free Loop (One Night Stand)}}
In the [mathematical](/source/mathematics) field of [topology](/source/topology), a '''free loop''' is a variant of the notion of a [loop](/source/loop_(topology)).  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](/source/topological_space).  Then a free loop in <math>X</math> is an [equivalence class](/source/equivalence_class) of [continuous function](/source/continuous_function)s from the [circle](/source/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](/source/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](/source/fundamental_group), is a map from the circle to the space without the basepoint-preserving restriction.  Assuming the space is [path-connected](/source/path-connected), free [homotopy](/source/homotopy) classes of free loops correspond to [conjugacy class](/source/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](/source/string_topology), i.e. the study of new [algebraic structure](/source/algebraic_structure)s on the [homology](/source/singular_homology) of the free loop space.

==See also==
*[Loop space](/source/Loop_space)
*[Loop (topology)](/source/Loop_(topology))
*[Quasigroup](/source/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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Adapted from the Wikipedia article [Free loop](https://en.wikipedia.org/wiki/Free_loop) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Free_loop?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.