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SOME OPEN QUESTIONS ABOUT SYMPLECTIC 4-MANIFOLDS, SINGULAR PLANE CURVES, AND
 

Summary: SOME OPEN QUESTIONS ABOUT SYMPLECTIC
4-MANIFOLDS, SINGULAR PLANE CURVES, AND
BRAID GROUP FACTORIZATIONS
DENIS AUROUX
Abstract. The topology of symplectic 4-manifolds is related to that of
singular plane curves via the concept of branched covers. Thus, various
classification problems concerning symplectic 4-manifolds can be refor-
mulated as questions about singular plane curves. Moreover, using braid
monodromy, these can in turn be reformulated in the language of braid
group factorizations. While the results mentioned in this paper are not
new, we hope that they will stimulate interest in these questions, which
remain essentially wide open.
1. Introduction
An important problem in 4-manifold topology is to understand which
manifolds carry symplectic structures (i.e., closed non-degenerate 2-forms),
and to develop invariants that can distinguish symplectic manifolds. Ad-
ditionally, one would like to understand to what extent the category of
symplectic manifolds is richer than that of Kšahler (or complex projective)
manifolds. For example, one would like to identify a set of surgery operations
that can be used to turn an arbitrary symplectic 4-manifold into a Kšahler

  

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

 

Collections: Mathematics