## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Groups of Diffeomorphisms: In honor of Shigeyuki Morita on the occasion of his 60th birthday, R. Penner, D. Kotschick, T. Tsuboi, N. Kawazumi, T. Kitano and Y. Mitsumatsu, eds. (Tokyo: Mathematical Society of Japan, 2008), 505 - 524

### On the uniform perfectness of diffeomorphism groups

#### 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.

#### Article information

**Dates**

Received: 8 January 2008

Revised: 19 April 2008

First available in Project Euclid:
28 November 2018

**Permanent link to this document**

https://projecteuclid.org/
euclid.aspm/1543447497

**Digital Object Identifier**

doi:10.2969/aspm/05210505

**Mathematical Reviews number (MathSciNet)**

MR2509724

**Zentralblatt MATH identifier**

1183.57024

**Subjects**

Primary: 57R52: Isotopy 57R50: Diffeomorphisms

Secondary: 37C05: Smooth mappings and diffeomorphisms

**Keywords**

diffeomorphism group uniformly perfect commutator subgroup

#### Citation

Tsuboi, Takashi. On the uniform perfectness of diffeomorphism groups. Groups of Diffeomorphisms: In honor of Shigeyuki Morita on the occasion of his 60th birthday, 505--524, Mathematical Society of Japan, Tokyo, Japan, 2008. doi:10.2969/aspm/05210505. https://projecteuclid.org/euclid.aspm/1543447497