On the uniform perfectness of equivariant diffeomorphism groups for principal G manifolds

Abstract

We proved in [K. Abe, K. Fukui, <i>On commutators of equivariant diffeomorphisms</i>, Proc. Japan Acad. 54 (1978), 52–54] that the identity component $\text{Diff}\,^r_{G,c}(M)_0$ of the group of equivariant $C^r$-diffeomorphisms of a principal $G$ bundle $M$ over a manifold $B$ is perfect for a compact connected Lie group $G$ and $1 \leq r \leq \infty$ ($r \neq \dim B + 1$). In this paper, we study the uniform perfectness of the group of equivariant $C^r$-diffeomorphisms for a principal $G$ bundle $M$ over a manifold $B$ by relating it to the uniform perfectness of the group of $C^r$-diffeomorphisms of $B$ and show that under a certain condition, $\text{Diff}\,^r_{G,c}(M)_0$ is uniformly perfect if $B$ belongs to a certain wide class of manifolds. We characterize the uniform perfectness of the group of equivariant $C^r$-diffeomorphisms for principal $G$ bundles over closed manifolds of dimension less than or equal to 3, and in particular we prove the uniform perfectness of the group for the 3-dimensional case and $r\neq 4$.