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
The free loop space construction is right adjoint to the cartesian product with the circle, while the loop space construction is right adjoint to the reduced suspension. This adjunction can be understood as currying, where the cartesian product is adjoint to the hom functor. This adjunction accounts for much of the importance of loop spaces in stable homotopy theory. Informally, this duality, as well as currying, is referred to as Eckmann–Hilton duality.
Eckmann–Hilton duality
The loop space is dual to the suspension of the same space; this duality is sometimes called Eckmann–Hilton duality. The basic observation is that
where
In general,
This follows, since the homotopy group is defined as