Framed configuration spaces and exotic spheres

Abstract We determine when an exotic sphere Σ $\Sigma $ normal upper Sigma of dimension d ⧸ ≡ 1 ( mod 4 ) $d\not {\equiv }1\ (\mathrm {mod}\ 4)$ d solidus identical to 1 left parenthesis mod 4 right parenthesis can be detected through the homotopy type of its truncated D isc $\mathscr {D}\mathrm {isc}$ script upper D isc -presheaf. The latter records the diagram of framed configuration spaces of bounded cardinality in Σ $\Sigma $ normal upper 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 D isc $\mathscr {D}\mathrm {isc}$ script upper D isc -presheaves of manifolds divided into two codimension zero submanifolds, a version of Atiyah duality in the context of D isc $\mathscr {D}\mathrm {isc}$ script upper 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 +