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Free loop

Topological variant of the loop

Wikipedia contributorsAdapted for Fathomly · View authors
1 min read
Source updated July 16, 2026

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.

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This article is adapted from the Wikipedia article “Free loop”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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