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arXiv · 2603.26920

Non-Existence of Smooth Full-Holonomy Cayley Fibrations

Jianfeng Lin·Viktor F. Majewski·Jacek Rzemieniecki
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Abstract

We prove that every Cayley fibration of a compact torsion-free Spin(7)-manifold with full holonomy must have singular fibers. This confirms a long-standing expectation in the study of calibrated fibrations with exceptional holonomy. Our argument first uses the rigidity of the Spin(7)-structure to reduce any hypothetical nonsingular Cayley fibration to two possible topological configurations. We then exclude both by proving a new spinnability theorem for smooth fiber bundles over simply-connected 4-manifolds with fiber homeomorphic to the elliptic surfaces E(2) or E(4). The E(4) case, which constitutes the main new difficulty, is established by combining families Seiberg--Witten theory with parametrized equivariant homotopy theory and equivariant K-theory.

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Jianfeng Lin, Viktor F. Majewski, Jacek Rzemieniecki. 2026-03-27. Non-Existence of Smooth Full-Holonomy Cayley Fibrations. https://arxiv.org/abs/2603.26920

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