Title: Smooth perfectness of diffeomorphism groups

Abstract: This is joint work with Josef Teichmann from the Technical University
of Vienna. We show that on a closed smooth manifold equipped with $k$ fiber
bundle structures whose vertical distributions span the tangent bundle,
every smooth diffeomorphism $f$ sufficiently close to the identity can
be written as a product $f=f_1\cdots f_k$, where $f_i$ preserves the
$i$-th fiber. The factors $f_i$ can be chosen to depend smoothly on $f$.
This is an application of the Nash--Moser inverse function theorem.
We apply this result to show that on a certain class of closed smooth
manifolds every diffeomorphism sufficiently close to the identity can be
written as product of commutators and the factors can be chosen to depend
smoothly on $f$. Furthermore we get concrete estimates on how many commutators
are necessary.