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

Abstract

Abstract. We proved in [K. Abe, K. Fukui, On commutators of equivariant diffeomorphisms, Proc. Japan Acad. 54 (1978), [52][53][54] that the identity component 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 ≤ r ≤ ∞ (r = 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, 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 = 4.

Boundedness of bundle diffeomorphism groups over a circle

“…The boundedness of the diffeomorphism group Diff r (M ) of a smooth manifold M is studied by D. Burago, S. Ivanov and L. Polterovich [3] and T. Tsuboi [18,19,20], et al The case of equivariant diffeomorphism groups under free Lie group actions is studied by K. Fukui [1,7], J. Lech, I. Michalik and T. Rybicki [13] et al, and the case of leaf-preserving diffeomorphism groups is studied in [6,15,17].…”

On the boundedness of equivariant homeomorphism groups

“…Recently Fukui studied in [10] the uniform perfectness of the identity component of the group of compactly supported equivariant C r -diffeomorphisms under the above assumptions. He showed a relation between the uniform perfectness of this group and the uniform perfectness of the identity component of the group of C r -diffeomorphisms of B.…”

“…Firstly, we would like to extend the main result in [10] to all homeomorphisms, partly by using similar methods. In particular, we observe that the groups studied in [16] are uniformly perfect under some assumptions.…”