Loop space

Summary

The loop space $\Omega X$ of a pointed space $(X, x_0)$ is the space of all loops in $X$ that start and end at $x_0$. Its path components are the elements of the fundamental group of $X$, and in general $\pi_n(\Omega X) \cong \pi_{n+1}(X)$. Concatenation of loops makes $\Omega X$ an H-space. The loop space is dual to the suspension in the sense of Eckmann–Hilton duality.