Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds
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A01=John W. Morgan
Affine space
Algebra bundle
Algebraic surface
Author_John W. Morgan
Automorphism
Category=PBM
Category=PBP
Clifford algebra
Cohomology
Complex dimension
Complex manifold
Complex projective space
Complex vector bundle
Complexification (Lie group)
Covariant derivative
Curvature
Differentiable manifold
Differential topology
Dimension (vector space)
Dirac operator
Division algebra
Donaldson theory
Duality (mathematics)
Elliptic operator
eq_isMigrated=1
eq_isMigrated=2
eq_nobargain
Equation
Fiber bundle
Gauge theory
Gaussian curvature
Geometry
Group homomorphism
Hodge index theorem
Homology (mathematics)
Homotopy
Identity (mathematics)
Implicit function theorem
Intersection form (4-manifold)
Inverse function theorem
Isomorphism class
Kahler manifold
Levi-Civita connection
Lie algebra
Line bundle
Linear map
Linear space (geometry)
Mathematical induction
Moduli space
Multiplication theorem
Neighbourhood (mathematics)
Parity (mathematics)
Principal bundle
Projection (linear algebra)
Pullback (category theory)
Quaternion algebra
Quotient space (topology)
Riemann surface
Riemannian manifold
Sard's theorem
Sign (mathematics)
Sobolev space
Spin representation
Spin structure
Surjective function
Symplectic geometry
Symplectic manifold
Tangent bundle
Tangent space
Tensor product
Theorem
Three-dimensional space (mathematics)
Trace (linear algebra)
Transversality (mathematics)
Two-form
Zariski tangent space
Product details
- ISBN 9780691025971
- Weight: 198g
- Dimensions: 197 x 254mm
- Publication Date: 31 Dec 1995
- Publisher: Princeton University Press
- Publication City/Country: US
- Product Form: Paperback
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The recent introduction of the Seiberg-Witten invariants of smooth four-manifolds has revolutionized the study of those manifolds. The invariants are gauge-theoretic in nature and are close cousins of the much-studied SU(2)-invariants defined over fifteen years ago by Donaldson. On a practical level, the new invariants have proved to be more powerful and have led to a vast generalization of earlier results. This book is an introduction to the Seiberg-Witten invariants. The work begins with a review of the classical material on Spin c structures and their associated Dirac operators. Next comes a discussion of the Seiberg-Witten equations, which is set in the context of nonlinear elliptic operators on an appropriate infinite dimensional space of configurations. It is demonstrated that the space of solutions to these equations, called the Seiberg-Witten moduli space, is finite dimensional, and its dimension is then computed. In contrast to the SU(2)-case, the Seiberg-Witten moduli spaces are shown to be compact. The Seiberg-Witten invariant is then essentially the homology class in the space of configurations represented by the Seiberg-Witten moduli space.
The last chapter gives a flavor for the applications of these new invariants by computing the invariants for most Kahler surfaces and then deriving some basic toological consequences for these surfaces.
John W. Morgan is Professor of Mathematics at Columbia University.
