Framed configuration spaces and exotic spheres

We determine when an exotic sphere $\Sigma$ of dimension $d\not \equiv 1 (4)$ can be detected through the homotopy type of its truncated Disc-presheaf. The latter records the diagram of framed configuration spaces of bounded cardinality in $\Sigma$ with natural point-forgetting and -splitting maps between them, and it gives rise to the finite stages in Goodwillie--Weiss' embedding calculus tower. Our proof involves three ingredients that could be of independent interest: a gluing result for Disc-presheaves of manifolds divided into two codimension zero submanifolds, a version of Atiyah duality in the context ofDisc-presheaves, and a computation of the finite residual of the mapping class group of the connected sums $\sharp^g(S^{2k+1}\times S^{2k+1})$.