TY  - JOUR
ID  - Fukui2017
LB  - Fukui2017
AU  - Fukui, Kazuhiko
TI  - On the uniform perfectness of equivariant diffeomorphism groups for principal G manifolds
ST  - On the uniform perfectness of equivariant diffeomorphism groups for principal G manifolds
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2017
VL  - 37
IS  - 3
SP  - 381
EP  - 388
DO  - http://dx.doi.org/10.7494/OpMath.2017.37.3.381
UR  - http://dx.doi.org/10.7494/OpMath.2017.37.3.381
AB  - 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 \(r\neq 4\).
KW  - uniform perfectness
KW  - principal \(G\) manifold
KW  - equivariant diffeomorphism
SN  - 1232-9274
ER  -