Unboundedness of some higher Euler classes

Abstract

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