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
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.
- Received: 2016-08-31.
- Accepted: 2016-11-16.
- Published online: 2017-01-30.
