Positive sectional curvature on a family of exotic fifteen-sphere
Abstract
We construct metrics of strictly positive sectional curvature on at least 508 pairwise nondiffeomorphic exotic fifteen-spheres, without prescribing orientations. The construction begins with quaternionic line bundles over ℍP2× S4. Surgery along a Hopf fiber gives homotopy spheres with specified equivariant two-disk presentations. A calculation on the surgery filling distinguishes 1015 nonstandard oriented smooth types. We construct positively curved quotient metrics on both disks, match their boundary data under the prescribed attaching map, and smooth the seam by averaging the inverse tangential metrics. The connection estimates and the smoothing argument are proved directly for every two-plane. Thus the curvature conclusion does not require a compatible-disk theorem from the eleven-dimensional construction or an external curvature-gluing theorem. We give quantitative parameter bounds and specify the auxiliary choices for the first nontrivial member after fixing its smooth gauge family.
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BibTeX
@misc{chen2026positive,
title = {{Positive sectional curvature on a family of exotic fifteen-sphere}},
author = {Zhiqi Chen and Shaoqiang Deng and Hui Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03310},
eprint = {2610.03310},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03310},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Positive sectional curvature on a family of exotic fifteen-sphere
AU - Zhiqi Chen
AU - Shaoqiang Deng
AU - Hui Zhang
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.03310
UR - https://arxiv.org/abs/2610.03310
AB - We construct metrics of strictly positive sectional curvature on at least 508 pairwise nondiffeomorphic exotic fifteen-spheres, without prescribing orientations. The construction begins with quaternionic line bundles over ℍP^2× S^4. Surgery along a Hopf fiber gives homotopy spheres with specified equivariant two-disk presentations. A calculation on the surgery filling distinguishes 1015 nonstandard oriented smooth types. We construct positively curved quotient metrics on both disks, match their boundary data under the prescribed attaching map, and smooth the seam by averaging the inverse tangential metrics. The connection estimates and the smoothing argument are proved directly for every two-plane. Thus the curvature conclusion does not require a compatible-disk theorem from the eleven-dimensional construction or an external curvature-gluing theorem. We give quantitative parameter bounds and specify the auxiliary choices for the first nontrivial member after fixing its smooth gauge family.
ER -