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VOL. 52 | 2008 On the uniform perfectness of diffeomorphism groups
Takashi Tsuboi

Editor(s) Robert Penner, Dieter Kotschick, Takashi Tsuboi, Nariya Kawazumi, Teruaki Kitano, Yoshihiko Mitsumatsu


We show that any element of the identity component of the group of $C^r$ diffeomorphisms $\mathrm{Diff}_c^r (\boldsymbol{R}^n)_0$ of the $n$-dimensional Euclidean space $\boldsymbol{R}^n$ with compact support $(1 \leqq r \leqq \infty,\ r \ne n+1)$ can be written as a product of two commutators. This statement holds for the interior $M^n$ of a compact $n$-dimensional manifold which has a handle decomposition only with handles of indices not greater than $(n-1)/2$. For the group $\mathrm{Diff}^r (M)$ of $C^r$ diffeomorphisms of a compact manifold $M$, we show the following for its identity component $\mathrm{Diff}^r (M)_0$. For an even-dimensional compact manifold $M^{2m}$ with handle decomposition without handles of the middle index $m$, any element of $\mathrm{Diff}^r (M^{2m})_0$ $(1 \leqq r \leqq \infty,\ r \ne 2m+1)$ can be written as a product of four commutators. For an odd-dimensional compact manifold $M^{2m+1}$, any element of $\mathrm{Diff}^r (M^{2m+1})_0$ $(1 \leqq r \leqq \infty,\ r \ne 2m+2)$ can be written as a product of six commutators.


Published: 1 January 2008
First available in Project Euclid: 28 November 2018

zbMATH: 1183.57024
MathSciNet: MR2509724

Digital Object Identifier: 10.2969/aspm/05210505

Primary: 57R50, 57R52
Secondary: 37C05

Rights: Copyright © 2008 Mathematical Society of Japan


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