The book contains a complete proof of the remarkable result relating the Seiberg-Witten and Gromov invariants of symplectic four manifolds. The first chapter proves that pseudo-holomorphic curves can be sonstruction from solutions to the Seiberg-Witten equations. The second chapter describes how the Gromov invariant for compact symplectic 4-manifolds assigns an integer to each dimension 2-cohomology class (or roughly speaking, counts with suitable weights, compact, pseudo-holomorphic submanifolds whose fundamental class is Poincare dual to the cohomology class in question. The third chapter continues the proof of the equivalence of Gromov invariants and the Seiberg-Witten invariants. In this chapter, a construction which associates solutions of the one-parameter family of Weiber-Witten equations to certain symplectic submanifolds in X (Pseudo-holomorphic ones.) The final chapter shows that counting for the two invariants yeilds the same answer.
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This work offers the proof of the remarkable relationship between Seiberg-Witten and Gromov invariants on symplectic 4-manifolds. It is a companion to Topics on Symplectic 4-Manifolds published in 1998 and brings together articles published in two American journals.About the Author:
Clifford Henry Taubes is a senior mathematics professor at Harvard University. He has published extensively on differential geometry and related mathematical subjects.
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