Path space ($\Omega (p,q)$)

The fixed-endpoint path space $\Omega (p,q)$ consists of paths in a manifold with specified initial and terminal points. Smooth, piecewise smooth and Sobolev $H^1$ versions have the same homotopy type in the usual manifold setting. Joining a path to a fixed path back from $q$ to $p$ gives a homotopy equivalence with the based loop space.