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 formula_1 be a topological space. Then a free loop in formula_1 is an equivalence class of continuous functions from the circle formula_3 to formula_1. Two loops are equivalent if they differ by a reparameterization of the circle. That is, formula_5 if there exists a homeomorphism formula_6 such that formula_7.
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. Free homotopy classes of free loops correspond to conjugacy classes in the fundamental group.
Recently, interest in the space of all free loops formula_8 has grown with the advent of string topology, i.e. the study of new algebraic structures on the homology of the free loop space.