Path space
= Path space
{title2=$\Omega(p,q)$}
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>.