Finiteness of the topological rank of diffeomorphism groups

TL;DR

This paper proves that the topological rank of the identity component of diffeomorphism groups of compact smooth manifolds is finite for all differentiability classes, extending previous results from one-dimensional cases to higher dimensions.

Contribution

It generalizes the finiteness of the topological rank of diffeomorphism groups from one-dimensional manifolds to all compact smooth manifolds of any dimension.

Findings

Topological rank of Diff$_0^k(M)$ is finite for all $k \\geq 1$.

Extension of previous one-dimensional results to higher dimensions.

Provides a broader understanding of the structure of diffeomorphism groups.

Abstract

For a compact smooth manifold $M$ (with boundary) we prove that the topological rank of the diffeomorphism group Diff$_0^k(M)$ is finite for all $k\geq 1$. This extends a result from [2] where the same claim is proved in the special case of dim M = k = 1.