Positive sectional curvature on a family of exotic fifteen-sphere

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-García’s bounded-representation transfer theorem. The topological construction and invariant calculation are independent of the curvature input.