Opuscula Math. 37, no. 3 (2017), 381-388
http://dx.doi.org/10.7494/OpMath.2017.37.3.381

Opuscula Mathematica

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

Kazuhiko Fukui

Abstract. We proved in [K. Abe, K. Fukui, On commutators of equivariant diffeomorphisms, 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 $$r\neq 4$$.

Keywords: uniform perfectness, principal $$G$$ manifold, equivariant diffeomorphism.

Mathematics Subject Classification: 58D05, 57R30.

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• Kazuhiko Fukui
• Kyoto Sangyo University, Department of Mathematics, Kyoto 603-8555, Japan
• Communicated by P.A. Cojuhari.
• Accepted: 2016-11-16.
• Published online: 2017-01-30.