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

We proved in [K. Abe, K. Fukui, On commutators of equivariant diffeomorphisms, Proc. Japan Acad. 54 (1978), 52–54] that the identity component [formula] of the group of equivariant Cr-diffeomorphisms of a principal G bundle M over a manifold B is perfect for a compact connected Lie group G and [formula] In this paper, we study the uniform perfectness of the group of equivariant Cr-diffeomorphisms for a principal G bundle M over a manifold B by relating it to the uniform perfectness of the group of Cr-diffeomorphisms of B and show that under a certain condition, [formula] is uniformly perfect if B belongs to a certain wide class of manifolds. We characterize the uniform perfectness of the group of equivariant Cr-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 ≠ 4.