Framed configuration spaces and exotic spheres

We determinewhen an exotic sphere Σ of dimension 𝑑 ⧸≡ 1 (mod⁡ ⁢4) can be detected through the homotopy type of its truncated 𝒟⁢isc-presheaf. The latter records the diagram of framed configuration spaces of bounded cardinality in Σ 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 𝒟⁢isc-presheaves of manifolds divided into two codimension zero submanifolds, a version of Atiyah duality in the context of 𝒟⁢isc-presheaves, and a computation of the finite residual of the mapping class group of the connected sums ♯$^𝑔$⁢(𝑆$^{2⁢𝑘+1}$ ×𝑆$^{2⁢𝑘+1}$).