Non-Existence of Smooth Full-Holonomy Cayley Fibrations

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.

New version and added author; conjecture now fully proven. 33 p., comments welcome!