 
Summary: MAPPING CLASS GROUP FACTORIZATIONS AND
SYMPLECTIC 4MANIFOLDS: SOME OPEN PROBLEMS
DENIS AUROUX
Abstract. Lefschetz fibrations and their monodromy establish a bridge
between the world of symplectic 4manifolds and that of factorizations in
mapping class groups. We outline various open problems about mapping
class group factorizations which translate to topological questions and
conjectures about symplectic 4manifolds.
1. Lefschetz fibrations and symplectic 4manifolds
Definition 1. A Lefschetz fibration on an oriented compact smooth 4mani
fold M is a smooth map f : M S2 which is a submersion everywhere
except at finitely many nondegenerate critical points p1, . . . , pr, near which f
identifies in local orientationpreserving complex coordinates with the model
map (z1, z2) z2
1 + z2
2.
The fibers of a Lefschetz fibration f are compact oriented surfaces, smooth
except for finitely many of them. The fiber through pi presents a transverse
double point, or node, at pi. Without loss of generality, we can assume after
perturbing f slightly that the critical values qi = f(pi) are all distinct. Fix a
