Smooth Four-Manifolds and Complex Surfaces

1. Front Matter

   Pages I-X

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2. Introduction

   • Robert Friedman, John W. Morgan

   Pages 1-13

3. The Kodaira Classification of Surfaces

   • Robert Friedman, John W. Morgan

   Pages 14-137

4. The Topology of Elliptic Surfaces

   • Robert Friedman, John W. Morgan

   Pages 138-225

5. Definition of the Polynomial Invariants

   • Robert Friedman, John W. Morgan

   Pages 226-278

6. Holomorphic Vector Bundles, Stability, and Gauge Theory

   • Robert Friedman, John W. Morgan

   Pages 279-339

7. Donaldson Polynomials of Algebraic Surfaces

   • Robert Friedman, John W. Morgan

   Pages 340-392

8. Big Diffeomorphism Groups and Minimal Models

   • Robert Friedman, John W. Morgan

   Pages 393-441

9. Donaldson Polynomials of Elliptic Surfaces

   • Robert Friedman, John W. Morgan

   Pages 442-499

10. Back Matter

    Pages 500-522

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In 1961 Smale established the generalized Poincare Conjecture in dimensions greater than or equal to 5 [129] and proceeded to prove the h-cobordism theorem [130]. This result inaugurated a major effort to classify all possible smooth and topological structures on manifolds of dimension at least 5. By the mid 1970's the main outlines of this theory were complete, and explicit answers (especially concerning simply connected manifolds) as well as general qualitative results had been obtained. As an example of such a qualitative result, a closed, simply connected manifold of dimension 2: 5 is determined up to finitely many diffeomorphism possibilities by its homotopy type and its Pontrjagin classes. There are similar results for self-diffeomorphisms, which, at least in the simply connected case, say that the group of self-diffeomorphisms of a closed manifold M of dimension at least 5 is commensurate with an arithmetic subgroup of the linear algebraic group of all automorphisms of its so-called rational minimal model which preserve the Pontrjagin classes [131]. Once the high dimensional theory was in good shape, attention shifted to the remaining, and seemingly exceptional, dimensions 3 and 4. The theory behind the results for manifolds of dimension at least 5 does not carryover to manifolds of these low dimensions, essentially because there is no longer enough room to maneuver. Thus new ideas are necessary to study manifolds of these "low" dimensions.

• 4-Manifold
• 4-Mannigfaltigkeiten
• Anti-Self-Dual Connection
• Complex Surface
• Donaldson Polynomial
• Donaldson'sches Polynom
• Elliptic Surface
• Elliptische Fläche
• Four-Manifold
• Gauge Theory
• Komplexe Fläche
• algebraic geometry
• diffeomorphism
• manifold
• topology

• Department of Mathematics, Columbia University, New York, USA

  Robert Friedman, John W. Morgan

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