Positive sectional curvature on a family of exotic fifteen-sphere

Abstract

We construct Riemannian metrics of strictly positive sectional curvature on at least 508 pairwise nondiffeomorphic exotic fifteen-spheres, without distinguishing orientations. The construction starts with quaternionic line bundles over the product of the quaternionic projective plane and the four-sphere. Surgery on an eleven-sphere fiber produces homotopy fifteen-spheres with conjugation-equivariant attaching maps. A calculation on the surgery filling gives their Eells--Kuiper invariants and distinguishes 1015 nonstandard oriented smooth types. The required representation has infinitesimal action norm at most one, so the examples also satisfy the hypothesis of Galaz-Garc\'ia's bounded-representation transfer theorem. The topological construction and invariant calculation are independent of the curvature input.