  We study high-dimensional analogues of spaces of long knots. These are spaces
of compactly-supported embeddings (modulo immersions) of $\mathbb{R}^m$ into
$\mathbb{R}^n$. We view the space of embeddings as the value of a certain
functor at $\mathbb{R}^m$, and we apply manifold calculus to this functor. Our
first result says that the Taylor tower of this functor can be expressed as the
space of maps between infinitesimal bimodules over the little disks operad. We
then show that the formality of the little disks operad has implications for
the homological behavior of the Taylor tower. Our second result says that when
$2m+1<n$, the singular chain complex of these spaces of embeddings is
rationally equivalent to a direct sum of certain finite chain complexes, which
we describe rather explicitly.
