Positive sectional curvature on a family of exotic fifteen-sphere

Zhiqi Chen; Shaoqiang Deng; Hui Zhang

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arXiv · 2610.03310

Positive sectional curvature on a family of exotic fifteen-sphere

Zhiqi Chen·Shaoqiang Deng·Hui Zhang

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

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Zhiqi Chen, Shaoqiang Deng, Hui Zhang. 2026-10-02. Positive sectional curvature on a family of exotic fifteen-sphere. https://arxiv.org/abs/2610.03310

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