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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

## Abstract

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.

## Information

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

zbMATH: 1183.57024
MathSciNet: MR2509724

Digital Object Identifier: 10.2969/aspm/05210505

Subjects:
Primary: 57R50, 57R52
Secondary: 37C05

Rights: Copyright © 2008 Mathematical Society of Japan

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