Mapping class group factorizations and symplectic 4-manifolds: some open problems, Problems on mapping class groups and related topics

Abstract. Lefschetz ¯brations and their monodromy establish a bridge between the world of symplectic 4-manifolds 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 4-manifolds. 1. Lefschetz fibrations and symplectic 4-manifolds De¯nition 1. A Lefschetz ¯bration on an oriented compact smooth 4-mani-fold M is a smooth map f: M! S2 which is a submersion everywhere except at ¯nitely many non-degenerate critical points p1; : : : ; pr, near which f identi¯es in local orientation-preserving complex coordinates with the model map (z1; z2) 7! z 2 1 +