On the Uniform Perfectness of the Groups of Diffeomorphisms of Even-dimensional Manifolds

We show that the identity component Diff(M)0 of the group of Cr diffeomorphisms of a compact (2m)-dimensional manifold M2m (1 ≤ r ≤ ∞, r = 2m + 1) is uniformly perfect for 2m ≥ 6, i.e., any element of Diff(M)0 can be written as a product of a bounded number of commutators. It is also shown that for a compact connected manifold M2m (2m ≥ 6), the identity component Diff(M)0 of the group of Cr diffeomorphisms of M2m (1 ≤ r ≤ ∞, r = 2m + 1) is uniformly simple, i.e., for elements f and g of Diff(M)0 \ {id}, f can be written as a product of a bounded number of conjugates of g or g−1.