The bounded cohomology of transformation groups of Euclidean spaces and discs

TL;DR

This paper computes the bounded cohomology of various transformation groups of Euclidean spaces and discs, revealing that many characteristic classes are unbounded, and introduces new methods for studying these groups.

Contribution

It provides the first full computation of the bounded cohomology for non-compactly supported transformation groups, extending understanding of their algebraic and topological properties.

Findings

Groups of orientation-preserving homeomorphisms and diffeomorphisms of R^n are boundedly acyclic.

Many characteristic classes of flat bundles are unbounded.

The bounded cohomology of homeomorphism groups of discs and the Cantor set is computed.

Abstract

We prove that the groups of orientation-preserving homeomorphisms and diffeomorphisms of $\mathbb{R}^n$ are boundedly acyclic, in all regularities. This is the first full computation of the bounded cohomology of a transformation group that is not compactly supported, and it implies that many characteristic classes of flat $\mathbb{R}^n$- and $S^n$-bundles are unbounded. We obtain the same result for the group of homeomorphisms of the disc that restrict to the identity on the boundary, and for the homeomorphism group of the non-compact Cantor set. In the appendix, Alexander Kupers proves a controlled version of the annulus theorem which we use to study the bounded cohomology of the homeomorphism group of the discs.