Bounded cohomology of diffeomorphism groups of higher dimensional spheres

TL;DR

This paper proves that the bounded cohomology of the group of orientation-preserving diffeomorphisms of higher-dimensional spheres vanishes for dimensions four and above, resolving an open question in the field.

Contribution

It establishes the vanishing of bounded cohomology for $ ext{Diff}^r_+(S^n)$ when $n geq 4$, providing a significant advancement in understanding the cohomological properties of these groups.

Findings

Bounded cohomology vanishes for $n geq 4$

Answers an open question from prior research

Applicable for all $r$ between 1 and infinity

Abstract

In this paper we prove the vanishing of the bounded cohomology of $\text{Diff}^r_+(S^n)$ with real coefficients when $n\geq 4$ and $1\leq r\leq \infty$. This answers the question raised in \cite{FNS24} for $\geq 4$ dimensional spheres.