  We show that in codimension at least 3, spaces of locally flat topological
embeddings of manifolds are correctly modelled by derived spaces of maps
between their configuration categories (under mild smoothability conditions,
and perhaps with one notable exception: source dimension 1 and target dimension
4). That general claim was reduced in an earlier paper to the special cases
where the manifolds in question are euclidean spaces. We deal with these
special cases by comparing to other special cases where the manifolds have the
form "torus" and "torus times euclidean space" respectively, and by setting up
a torus trick for configuration categories.
