Loop Space

In topology, a branch of mathematics, the loop space ΩX of a pointed topological space X is the space of based maps from the circle S¹ to X with the compact-open topology. Two elements of a loop space can be naturally concatenated. With this concatenation operation, a loop space is an A_∞-space. The noun adjunct A_∞ describes the manner in which concatenating loops is homotopy coherently associative.

The quotient of the loop space ΩX by the equivalence relation of pointed homotopy is the fundamental group π₁(X).

The iterated loop spaces of X are formed by applying Ω a number of times.

An analogous construction of topological spaces without basepoint is the free loop space. The free loop space of a topological space X is the space of maps from S¹ to X with the compact-open topology. That is to say, the free loop space of a topological space X is the function space ${\displaystyle \mathrm {Map} (S^{1},X)}$. The free loop space of X is denoted by ${\displaystyle {\mathcal {L}}X}$.

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