Commutators in the Extended
Mapping Class Group

In his paper, "Quasi-Homomorphisms and Stable Lengths in Mapping Class Groups", D. Kotschick, using results about the Seiberg-Witten invariants of symplectic four-manifolds, proved a srong result which implies, in particular, that no nontrivial power of a right-handed Dehn twist about an essential curve on a closed oriented surface S of genus g at least 2 is a single commutator in the mapping class group of S.

It is a rather straightforward observation that this particular consequence of Kotschick's strong result is false for the extended mapping class group of S, in which the square of any right-handed Dehn twist is a commutator. Indeed, as we shall explain in this talk, for genus at least 3, a right handed Dehn twist about a nonseparating circle is a commutator of two pseudo-Anosov maps S ----> S. We shall give explicit examples in genus 3, using well-known constructions of pseudo-Anosov maps.

As far as we know, there is no known purely surface theoretic proof of Kotschick's strong result, although the result is a purely group theoretic statement about some topologically special elements in the mapping class group of S. The failure of his result for the extended mapping class group indicates that this result depends upon "positivity" phenomena in the mapping class group, phenomena which is reflected through the connection between commutator relations in the mapping class group of S and Lefschetz fibrations over closed surfaces with generic fiber S.