For every closed non-orientable surface of genus at least 3, the third bounded cohomology of its measure-preserving homeomorphism group is infinite dimensional.
Volume and Euler classes in bounded cohomology of transformation groups.Glasg
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Bounded cohomology of measure-preserving homeomorphism groups of non-orientable surfaces
For every closed non-orientable surface of genus at least 3, the third bounded cohomology of its measure-preserving homeomorphism group is infinite dimensional.