Lesson plan
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.
Article summary
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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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
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- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
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