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

On the rational homology of high dimensional analogues of spaces of long knots