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Gauge Theory and the Topology of Four-manifolds By Robert Friedman

Gauge Theory and the Topology of Four-manifolds by Robert Friedman

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Summary

Offers a perspective on how 4-manifold theory was studied before the discovery of modern-day Seiberg-Witten theory. This book predates Donaldson's applications of the subject to 4-manifold topology, where the central concern was the geometry of the moduli space.

Gauge Theory and the Topology of Four-manifolds Summary

Gauge Theory and the Topology of Four-manifolds by Robert Friedman

The lectures in this volume provide a perspective on how 4-manifold theory was studied before the discovery of modern-day Seiberg-Witten theory. One reason the progress using the Seiberg-Witten invariants was so spectacular was that those studying $SU(2)$-gauge theory had more than ten years' experience with the subject. The tools had been honed, the correct questions formulated, and the basic strategies well understood. The knowledge immediately bore fruit in the technically simpler environment of the Seiberg-Witten theory. Gauge theory long predates Donaldson's applications of the subject to 4-manifold topology, where the central concern was the geometry of the moduli space.One reason for the interest in this study is the connection between the gauge theory moduli spaces of a Kahler manifold and the algebro-geometric moduli space of stable holomorphic bundles over the manifold. The extra geometric richness of the $SU(2)$ - moduli spaces may one day be important for purposes beyond the algebraic invariants that have been studied to date. It is for this reason that the results presented in this volume will be essential.

Table of Contents

Geometric invariant theory and the moduli of bundles: Geometric invariant theory by D. Gieseker The numerical criterion by D. Gieseker The moduli of stable bundles by D. Gieseker References by D. Gieseker Anti-self-dual connections and stable vector bundles: Hermitian bundles, Hermitian connections and their curvatures by J. Li Hermitian-Einstein connections and stable vector bundles by J. Li The existence of Hermitian-Einstein metrics by J. Li References by J. Li An introduction to gauge theory: The context of Gauge theory by J. W. Morgan Principal bundles and connections by J. W. Morgan Curvature and characteristic classes by J. W. Morgan The space of connections by J. W. Morgan The ASD equations and the moduli space by J. W. Morgan Compactness and gluing theorems by J. W. Morgan The Donaldson polynomial invariants by J. W. Morgan The connected sum theorem by J. W. Morgan References by J. W. Morgan Computing Donaldson invariants: Overview by R. J. Stern -2 spheres and the blowup formula by R. J. Stern Simple-type criteria and elliptic surfaces by R. J. Stern Elementary rational blowdowns by R. J. Stern Taut configurations and Horikowa surfaces by R. J. Stern References by R. J. Stern Donaldson-Floer theory: Introduction by C. Taubes and J. A. Bryan Quantization by C. Taubes and J. A. Bryan Simplicial decomposition of $\Cal{M}^0_X$ by C. Taubes and J. A. Bryan Half-infinite dimensional spaces by C. Taubes and J. A. Bryan References by C. Taubes and J. A. Bryan.

Additional information

NGR9780821805916
9780821805916
0821805916
Gauge Theory and the Topology of Four-manifolds by Robert Friedman
New
Hardback
American Mathematical Society
1997-12-15
221
N/A
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