Smooth Perfectness through Decomposition of Diffeomorphisms into Fiber Preserving Ones

Smooth Perfectness for the Group of Diffeomorphisms

Given the result of Herman, we provide a new elementary proof of the fact that the group of diffeomorphisms is a perfect and hence simple group. Moreover, we show that every diffeomorphism can be represented as a product of commutators f = [g1, h1] · · · [gm, hm] where the factors gi and hi can be chosen to depend smoothly on f . The elegance of our approach is due to the fact that we prove smo...

Smooth Perfectness through Decomposition Ofdiffeomorphisms

On the Perfectness of Nontransitive Groups of Diffeomorphisms

It is proven that the identity component of the group preserving the leaves of a generalized foliation is perfect. This shows that a well-known simplicity theorem on the diffeomorphism group extends to the nontransitive case.

Approximation of Volume-preserving Homeomorphisms by Volume-preserving Diffeomorphisms

Given a volume-preserving homeomorphism of a smooth manifold of dimension n ≥ 5, we give a necessary and sufficient condition for uniform approximability by (volume-preserving) diffeomorphisms.

Compact Stein Surfaces with Boundary as Branched Covers of B

We prove that Stein surfaces with boundary coincide up to orientation preserving diffeomorphisms with simple branched coverings of B4 whose branch set is a positive braided surface. As a consequence, we have that a smooth oriented 3-manifold is Stein fillable iff it has a positive open-book decomposition.