Translation numbers for bundle automorphisms and a cocycle on a group of homeomorphisms

TL;DR

This paper introduces translation numbers for bundle automorphisms to analyze characteristic classes and identify undistorted elements in automorphism groups, advancing understanding of bundle dynamics and topology.

Contribution

It develops a new framework of translation numbers for principal bundle automorphisms, linking them to characteristic classes and group element distortion.

Findings

Vanishing of certain characteristic classes for foliated bundles

Detection of undistorted elements in automorphism groups

Introduction of translation numbers for bundle automorphisms

Abstract

We introduce and study translation numbers for automorphisms of principal $\mathbb{Z}$-bundles and flat principal $\mathbb{R}$-bundles. We use them to show a vanishing result of a characteristic class of foliated bundles and to detect undistortion elements in the group of bundle automorphisms.