Framed configuration spaces and exotic spheres

Abstract

We determine when an exotic sphere $\hat{I}\pounds$ 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 $\hat{I}\pounds$ 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 ${♯}^g(S^{2k+1}\times S^{2k+1})$.

28 pages, 1 figure, to appear in Forum of Mathematics Sigma