Differential Geometry Working Seminar

Viktor Majewski, University of Waterloo

Obstructions to Smooth Full Holonomy Cayley Fibrations

We study smooth fibrations of compact torsion-free $\textrm{Spin}(7)$-manifolds by Cayley submanifolds.Using geometric and topological constraints coming from the $\textrm{Spin}(7)$-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 $\textrm{Spin}(7)$-manifolds.

MC 5417