{{For|the song|Free Loop (One Night Stand)}} In the mathematical field of topology, a '''free loop''' is a variant of the notion of a 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 functions 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 classes 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 structures on the 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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