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Obstructions to Smooth Full-Holonomy Cayley Fibrations

Viktor F. Majewski, Jacek Rzemieniecki

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.

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.

Paper Structure

This paper contains 5 sections, 10 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Spin(7)-Background
  3. Restricting the Topological Type of Smooth Cayley Fibrations
  4. Excluding the Possible Topological Types
  5. Boundary Dehn Twists and Property (S)

Key Result

Theorem 1.2

Suppose that $X$ is a closed full-holonomy ${\mathrm{Spin}}(7)$-manifold which admits a smooth Cayley fibration. Then the fibration can be written as $\pi: X \to B$ where the base $B$ and the fiber are homeomorphic to $\#^3 \overline{\mathbb{CP}}^2$ and $K3\# K3 \# (S^2 \times S^2)$, respectively. C

Theorems & Definitions (25)

  • Conjecture 1.1
  • Theorem 1.2
  • Conjecture 1.3
  • Corollary 1.4
  • Remark 2.1
  • Theorem 2.2
  • Definition 2.3
  • Remark 3.1
  • Remark 3.2
  • Proposition 3.3
  • ...and 15 more