Unboundedness of some higher Euler classes

TL;DR

This paper demonstrates that certain higher Euler classes associated with homeomorphism groups of Seifert fibered 3-manifolds are unbounded, contrasting with classical bounded Euler classes for circle homeomorphisms.

Contribution

It establishes the unboundedness of higher Euler classes for homeomorphism groups of specific 3-manifolds, providing explicit examples of flat bundles with arbitrary Euler class values.

Findings

Higher Euler classes for these groups are unbounded.

Explicit examples of flat bundles with arbitrary Euler class values.

Contrast with classical bounded Euler classes for circle homeomorphisms.

Abstract

In this paper, we study Euler classes in groups of homeomorphisms of Seifert fibered 3-manifolds. We show that, in contrast to the familiar Euler class for $\mathrm{Homeo}_0(S^1)^\delta$, these Euler classes for $\mathrm{Homeo}_0(M^3)^\delta$ are unbounded classes. In fact, we give examples of flat topological M bundles over a genus 3 surface whose Euler class takes arbitrary values.