Path space fibration

Summary

Let $(X, x_0)$ be a pointed space and $PX$ the space of paths that start at $x_0$. The path space fibration is the map $PX \to X$ that sends each path to its end point. It is a fibration, its total space $PX$ is contractible, and its fiber over $x_0$ is the loop space $\Omega X$. From its long exact sequence of homotopy groups one gets $\pi_n(\Omega X) \cong \pi_{n+1}(X)$. It plays the role of a universal bundle for loops and is the basic input of the Serre spectral sequence computations of loop spaces.