Bounded cohomology of measure-preserving homeomorphism groups of non-orientable surfaces

TL;DR

This paper proves that the third bounded cohomology of the identity component of measure-preserving homeomorphisms on non-orientable surfaces of genus at least 3 is infinite dimensional, revealing complex algebraic structures.

Contribution

It establishes the infinite dimensionality of the third bounded cohomology for these homeomorphism groups, a novel result in the study of non-orientable surface symmetries.

Findings

Third bounded cohomology is infinite dimensional

Results apply to non-orientable surfaces of genus ≥ 3

Advances understanding of algebraic properties of homeomorphism groups

Abstract

Let $N_g$ be a closed non-orientable surface of genus $g\geq 3$. Let $\operatorname{Homeo}_0(N_g,\mu)$ be the identity component of the group of measure-preserving homeomorphisms of $N_g$. In this work we prove that the third bounded cohomology of $\operatorname{Homeo}_0(N_g,\mu)$ is infinite dimensional.