Free loop

For the song, see 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 $X$ be a topological space. Then a free loop in $X$ is an equivalence class of continuous functions from the circle $S^{1}$ to $X$. Two loops are equivalent if they differ by a reparameterization of the circle. That is, $f\sim g$ if there exists a homeomorphism $\psi :S^{1}\rightarrow S^{1}$ such that $g=f\circ \psi .$

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 $LX$ 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: Notes on String Topology