On the homeomorphism groups of manifolds and their universal coverings

TL;DR

This paper investigates the algebraic and topological properties of the homeomorphism groups of manifolds, focusing on their simplicity, conjugation-invariant norms, boundedness, and the structure of their universal coverings.

Contribution

It establishes the simplicity and perfectness of the homeomorphism groups, analyzes conjugation-invariant norms, and explores the structure of their universal coverings.

Findings

$ ext{H}_c(M)$ is perfect and simple under mild conditions

Conjugation-invariant norms on $ ext{H}_c(M)$ are studied for boundedness

The structure of the universal covering group of $ ext{H}_c(M)$ is characterized

Abstract

Let $\mathcal H_c(M)$ stand for the path connected identity component of the group of all compactly supported homeomorphisms of a manifold $M$. It is shown that $\mathcal H_c(M)$ is perfect and simple under mild assumptions on $M$. Next, conjugation-invariant norms on $\H_c(M)$ are considered and the boundedness of $\mathcal H_c(M)$ is investigated. Finally, the structure of the universal covering group of $\mathcal H_c(M)$ is studied.