Path fibration over a connected manifold.

Question

Let $M$ be a differentiable manifold. We can consider $P(M):=\{\gamma:[0,1] \to M\}$, so we have a natural projection on $M$ $$ P(M) \to M $$ $$ \gamma \mapsto \gamma(1) ,$$ in the fibre of this application what have we the space $\Omega(M;p,q)$ of piecewise $C^\infty$ paths from $p$ to $q$ or the space $\Omega(M;p,p)$ of piecewise $C^\infty$ from $p$ to $p$? I know that $\Omega(M;p,q)$ is homotopically equivalenti to $\Omega(M;p,p)$.