  We define the notion of higher-order colocally weakly differentiable maps
from a manifold $M$ to a manifold $N$. When $M$ and $N$ are endowed with
Riemannian metrics, $p\ge 1$ and $k\ge 2$, this allows us to define the
intrinsic higher-order homogeneous Sobolev space $\dot{W}^{k,p}(M,N)$. We show
that this new intrinsic definition is not equivalent in general with the
definition by an isometric embedding of $N$ in a Euclidean space; if the
manifolds $M$ and $N$ are compact, the intrinsic space is a larger space than
the one obtained by embedding. We show that a necessary condition for the
density of smooth maps in the intrinsic space $\dot{W}^{k,p}(M,N)$ is that
$\pi_{\lfloor k p \rfloor} (N) \simeq \{0\}$. We investigate the chain rule for
higher-order differentiability in this setting.
