University of Georgia
Mathematics Department Colloquium
Professor Dieter Kotschick
University of Munich
April 17, 2003
Commutator lengths and signatures of surface bundles
The geometry of four-manifolds which are surface bundles over surfaces, or
more general Lefschetz fibrations, is closely related to the algebraic
structure of the diffeomorphism group of the fiber. We shall discuss this
relationship, with special emphasis on the commutator length on some perfect
groups of diffeomorphisms.
If the genus of the fiber is at least 3, then it is known that its mapping
class group is perfect, as is the identity component of the diffeomorphism
group, and the subgroup of Hamiltonian diffeomorphisms with respect to any
area form. In a perfect group every element can be written as a product of
commutators, and it is interesting to ask how many commutators a given
element requires, e.g. whether there is a universal bound that works for all
elements. It turns our that symplectic structures and foliations on the
total spaces of fibered four-manifolds shed light on such questions and
relate to the bounded cohomology of these groups.