Loop space

In mathematics, the space of loops or (free) loop space of a topological space X is the space of maps from the unit circle S¹ to X together with the compact-open topology.

\Omega X={\mathcal {C}}(S^{1},X).

That is, a particular function space.

In homotopy theory loop space commonly refers to the same construction applied to pointed spaces, i.e. continuous maps respecting base points. In this setting there is a natural "concatenation operation" by which two elements of the loop space can be combined. With this operation, the loop space can be regarded as a magma, or even as an A_∞-space. Concatenation of loops is not strictly associative, but it is associative up to higher homotopies.

If we consider the quotient of the based loop space ΩX with respect to the equivalence relation of pointed homotopy, then we obtain a group, the well-known fundamental group π₁(X).

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

The free loop space construction is right adjoint to the cartesian product with the circle, and the version for pointed spaces to the reduced suspension. This 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]