Obstructions to Smooth Full-Holonomy Cayley Fibrations

TL;DR

This paper investigates the topological constraints on smooth Cayley fibrations of compact Spin(7)-manifolds, ultimately ruling out their existence on known examples through geometric, topological, and gauge-theoretic methods.

Contribution

It identifies topological obstructions to smooth Cayley fibrations on Spin(7)-manifolds, linking gauge theory and 4-manifold topology to these geometric structures.

Findings

Only two topological configurations can occur for such fibrations.

One configuration is excluded by a spinnability criterion.

The remaining case relates to an open conjecture in 4-manifold topology.

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.