In the mathematical field of topology, a free loop is a variant of the notion of a loop be a topological space is an equivalence class of continuous functions from the circle if there exists a homeomorphism

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.

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