Obstructions to Smooth Full-Holonomy Cayley Fibrations

Abstract

We study smooth fibrations of compact torsion-free Spin(7)-manifolds by Cayley submanifolds. Using geometric and topological constraints coming from the Spin(7)-structure, we show that only two topological configurations can arise. One of these is excluded by a spinnability criterion for fiber bundles, with the relevant hypothesis verified using gauge-theoretic input, while the remaining case is reduced to an open conjecture in 4-manifold topology. In particular, this rules out smooth Cayley fibrations on all known examples of compact torsion-free Spin(7)-manifolds.