Opuscula Math. 37, no. 3 (), 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

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.