一族奇异十五维球面上的正截面曲率
Positive sectional curvature on a family of exotic fifteen-sphere
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中文总结 AI 辅助
本文在至少508个两两非微分同胚的奇异十五维球面上构造严格正截面曲率度量，通过四元数线丛和手术方法，并利用Eells--Kuiper不变量区分光滑型，满足有界表示传递定理假设。
中文摘要 AI 辅助
我们在至少508个两两非微分同胚的奇异十五维球面上构造了严格正截面曲率的黎曼度量，且不区分定向。构造始于四元数射影平面与四维球面之积上的四元数线丛。对十一维球面纤维进行手术，产生具有共轭等变附着映射的同伦十五维球面。对手术填充的计算给出了它们的Eells--Kuiper不变量，并区分了1015个非标准定向光滑型。所需的表示具有至多为一的无穷小作用范数，因此这些例子也满足Galaz-García有界表示传递定理的假设。拓扑构造与不变量计算独立于曲率输入。
英文摘要
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.
