The torus trick for configuration categories

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). 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.