Free loop

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.