Loop space

In topology, a branch of mathematics, the loop space Ω"X" of a pointed topological space "X" is the space of (based) loops in "X", i.e. continuous pointed maps from the pointed circle "S"1 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-coherently associative.
The set of path components of Ω"X", i.e. the set of based-homotopy equivalence classes of based loops in "X", is a group, the fundamental group "π"1("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"1 to "X" with the compact-open topology. The free loop space of "X" is often denoted by formula_1. 
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.
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 formula_3 is the set of homotopy classes of maps formula_4,
and formula_5 is the suspension of A, and formula_6 denotes the natural homeomorphism. This homeomorphism is essentially that of currying, modulo the quotients needed to convert the products to reduced products.
In general, formula_7 does not have a group structure for arbitrary spaces formula_8 and formula_9. However, it can be shown that formula_10 and formula_11 do have natural group structures when formula_12 and formula_13 are pointed, and the aforementioned isomorphism is of those groups. Thus, setting formula_14 (the formula_15 sphere) gives the relationship
This follows since the homotopy group is defined as formula_17 and the spheres can be obtained via suspensions of each-other, i.e. formula_18.