On the Uniform Perfectness of Diffeomorphism Groups

We show that any element of the identity component of the group of Cr diffeomorphisms Diffc(R )0 of the n-dimensional Euclidean space R n with compact support (1 5 r 5 ∞, r = n + 1) can be written as a product of two commutators. This statement holds for the interior Mn 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 Diff(M) of Cr diffeomorphisms of a compact manifold M , we show the following for its identity component Diff(M)0. For an even-dimensional compact manifold M2m with handle decomposition without handles of the middle index m, any element of Diff(M)0 (1 5 r 5 ∞, r = 2m + 1) can be written as a product of four commutators. For an odd-dimensional compact manifold M2m+1, any element of Diff(M)0 (1 5 r 5 ∞, r = 2m + 2) can be written as a product of six commutators.