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 [math]\displaystyle{ X }[/math] be a topological space. Then a free loop in [math]\displaystyle{ X }[/math] is an equivalence class of continuous functions from the circle [math]\displaystyle{ S^1 }[/math] to [math]\displaystyle{ X }[/math]. Two loops are equivalent if they differ by a reparameterization of the circle. That is, [math]\displaystyle{ f \sim g }[/math] if there exists a homeomorphism [math]\displaystyle{ \psi : S^1 \rightarrow S^1 }[/math] such that [math]\displaystyle{ 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]\displaystyle{ 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
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: Notes on String Topology
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