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 adjective 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 $\mathrm{Map}(S^1,X)$ . The free loop space of X is denoted by $\mathcal{L}X$ .

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 accounts for much of the importance of loop spaces in stable homotopy theory.

Relation between homotopy groups of a space and those of its loop space

The basic relation between the homotopy groups is $\pi_k(X) \approxeq \pi_{k-1}(\Omega X)$ .^[1]

More generally,

[\Sigma Z,X] \approxeq [Z, \Omega X]

where, $[A,B]$ is the set of homotopy classes of maps $A \rightarrow B$ , and $\Sigma A$ is the suspension of A. In general $[A, B]$ does not have a group structure for arbitrary spaces $A$ and $B$ . However, it can be shown that $[\Sigma Z,X]$ and $[Z, \Omega X]$ do have natural group structures when $Z$ and $X$ are pointed, and the aforesaid isomorphism is of those groups. ^[2]