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 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 fsim g} if there exists a homeomorphism ψ : S 1 → S 1 {displaystyle psi :S^{1} ightarrow S^{1}} such that g = f ∘ ψ {displaystyle g=fcirc psi } . 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 fsim g} if there exists a homeomorphism ψ : S 1 → S 1 {displaystyle psi :S^{1} ightarrow S^{1}} such that g = f ∘ ψ {displaystyle g=fcirc 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.