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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 X {\displaystyle X} be a topological space. Then a free loop in X {\displaystyle X} is an equivalence class of continuous functions from the circle S 1 {\displaystyle S^{1}} to X {\displaystyle X}. Two loops are equivalent if they differ by a reparameterization of the circle. That is, f ∼ g {\displaystyle f\sim g} if there exists a homeomorphism ψ : S 1 → S 1 {\displaystyle \psi :S^{1}\rightarrow S^{1}} such that g = f ∘ ψ {\displaystyle 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. Free homotopy classes of free loops correspond to conjugacy classes in the fundamental group.
Recently, interest in the space of all free loops L X {\displaystyle 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.