Boundedness of Euler classes for volume-preserving diffeomorphism groups

Ascertain whether the Euler classes in H^{2n}(Diff(R^{2n}, vol); R) and H^{2n}(Diff(S^{2n−1}, vol); R) are bounded; specifically, determine if these classes lie in the image of the comparison map from bounded cohomology and therefore admit representatives with uniformly bounded seminorm.

Background

While the paper proves unboundedness of various characteristic classes for flat bundles in broad settings, classical results show vanishing of the Euler class for certain amenable bases and low-regularity contexts. The boundedness of Euler classes for volume-preserving diffeomorphism groups remains unclear and would bridge dynamics, foliation theory, and bounded cohomology.

This question probes whether the Euler classes associated with volume-preserving diffeomorphisms exhibit bounded behavior in bounded cohomology, contrasting with known unboundedness phenomena in related settings.