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.