Smooth perfectness for the group of diffeomorphisms
Stefan Haller
(Universität Wien)
Abstract:
The connected component of the diffeomorphism group of a
closed manifold is simple according to results of Epstein, Herman, Mather,
and Thurston. The difficult part here is to show that the group is
perfect, i.e. every element can be written as a product of commutators
f=[g_1,h_1]...[g_m,h_m]. In this talk we will discuss a new
way to establish such a presentation which permits to
choose the factors g_i and h_i to depend smoothly on f,
and which allows to give estimates on how many commutators are necessary.
For example, every diffeomorphism f of the sphere Sn
which is sufficiently close to the identity can be written as a product
of eighteen commutators f=[g_1,h_1]...[g_{18},h_{18}].
Given the result of Herman (which solves the case of the torus)
this provides a new and elementary proof of the fact that the group of
diffeomorphisms is a perfect and hence simple group.
This is joint work with Josef Teichmann from the Technical
University of Vienna.