Normal subgroups of diffeomorphism and homeomorphism groups of R^n and other open manifolds

TL;DR

This paper classifies all normal subgroups of diffeomorphism and homeomorphism groups of R^n and other open manifolds, revealing their structure and simplicity properties in various cases.

Contribution

It provides a complete determination of normal subgroups for these groups, except in specific exceptional cases, and studies the simplicity of certain quotient groups.

Findings

Normal subgroups of diffeomorphism groups are classified except for specific dimensions.

The quotient of certain diffeomorphism groups by specific normal subgroups is simple.

The structure of homeomorphism groups of R^n is characterized in detail.

Abstract

We determine all the normal subgroups of the group of C^r diffeomorphisms of R^n, r = 1,2,...,infinity, except when r=n+1 or n=4, and also of the group of homeomorphisms of R^n (r=0). We also study the group A_0 of diffeomorphisms of an open manifold M that are isotopic to the identity. If M is the interior of a compact manifold with nonempty boundary, then the quotient of A_0 by the normal subgroup of diffeomorphisms that coincide with the identity near to a given end e of M is simple.