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MAPPING CLASS GROUP FACTORIZATIONS AND SYMPLECTIC 4-MANIFOLDS: SOME OPEN PROBLEMS
 

Summary: MAPPING CLASS GROUP FACTORIZATIONS AND
SYMPLECTIC 4-MANIFOLDS: SOME OPEN PROBLEMS
DENIS AUROUX
Abstract. Lefschetz fibrations 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
Definition 1. A Lefschetz fibration on an oriented compact smooth 4-mani-
fold M is a smooth map f : M S2 which is a submersion everywhere
except at finitely many non-degenerate critical points p1, . . . , pr, near which f
identifies in local orientation-preserving 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

  

Source: Auroux, Denis - Department of Mathematics, Massachusetts Institute of Technology (MIT)

 

Collections: Mathematics