Smooth and Symplectic 4-Manifolds

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Project Title: Smooth and Symplectic 4-ManifoldsPI: Ronald FintushelAward: 0072212Abstract: The theory of smooth 4-manifolds gains its importance both from its central location between low and high-dimensional topology and from its close interaction with high energy physics. The major problem in this field is the classification of smooth simply connected4-manifolds. The interaction between topology and physics hasstimulated the construction of invariants - at first Donaldson's invariant, and then the invariant of Seiberg and Witten - which are useful in distinguishing the diffeomorphism types of 4-manifolds. These have led to major advances, and they have allowed workers to study new constructions of 4-manifolds. These have confused the issue of classification, but also have invigorated the theory and reinforced its richness. The theory is now left without even a conjectural classification. It seems that still further examples are needed to identify a suitable classification scheme, and the proposer intends to work on such constructions. One class of 4-manifolds which have exceptionally close ties to theoretical physics are those with a symplectic structure, and the last few years have also seen progress in the theory of symplectic 4-manifolds; especially new constructions, and most notably, Donaldson's work on Lefschetz fibrations. The proposer intends to continue work on the explicit constructions of Lefschetz fibrations and related questions on symplectic submanifolds. The ultimate goal of the proposer is to develop new constructions of smooth 4 manifolds in the hope that a general picture will begin to emerge. The focus of this project will be to construct new types of examples of smooth and symplectic 4-manifolds and to study the diversity of embedded symplectic submanifolds (up to smooth isotopy) in a given homology class. In particular, is every symplectic surface in the complex projective plane smoothly isotopic to a holomorphic curve? If one allows enough blowups, this is not true.Another issue is the geography problem for simply connectedirreducible 4-manifolds. Each such manifold can be assigned a lattice point in the plane corresponding to its characteristic numbers. The problem is to study which points are realized. There has been notableprogress, but much work still remains, and the principal investigator plans to seek new methods for constructing irreducible simply connected 4-manifolds of positive signature. These techniques are related to a kind of theory of minimal genus surfaces with constraints. Also he, along with R. Stern, conjectures a replacement for the Noether inequality for symplectic 4-manifolds, and they have a promising technique for its proof, which they plan to pursue.