Loop space

In topology, a branch of mathematics, the loop space ΩX of a pointedtopological spaceX is the space of (based) loops in X, i.e. continuous pointed maps from the pointed circleS to X, equipped with the compact-open topology. Two loops can be multiplied by concatenation. With this operation, the loop space is an A_∞-space. That is, the multiplication is homotopy-coherentlyassociative.

The set of path components of ΩX, i.e. the set of based-homotopy equivalence classes of based loops in X, is a group, the fundamental groupπ₁(X).

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

There is an analogous construction for topological spaces without basepoint. The free loop space of a topological space X is the space of maps from the circle S to X with the compact-open topology. The free loop space of X is often denoted by ${\displaystyle {\mathcal {L}}X}$.

As a functor, the free loop space construction is right adjoint to cartesian product with the circle, while the loop space construction is right adjoint to the reduced suspension. This adjunction accounts for much of the importance of loop spaces in stable homotopy theory. (A related phenomenon in computer science is currying, where the cartesian product is adjoint to the hom functor.) Informally this is referred to as Eckmann–Hilton duality.