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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.

Four-manifolds (Topology) --- Seiberg-Witten invariants. --- Mathematical physics. --- Physical mathematics --- Physics --- Invariants --- 4-dimensional manifolds (Topology) --- 4-manifolds (Topology) --- Four dimensional manifolds (Topology) --- Manifolds, Four dimensional --- Low-dimensional topology --- Topological manifolds --- Mathematics --- Affine space. --- Affine transformation. --- Algebra bundle. --- Algebraic surface. --- Almost complex manifold. --- Automorphism. --- Banach space. --- Clifford algebra. --- Cohomology. --- Cokernel. --- Complex dimension. --- Complex manifold. --- Complex plane. --- Complex projective space. --- Complex vector bundle. --- Complexification (Lie group). --- Computation. --- Configuration space. --- Conjugate transpose. --- Covariant derivative. --- Curvature form. --- Curvature. --- Differentiable manifold. --- Differential topology. --- Dimension (vector space). --- Dirac equation. --- Dirac operator. --- Division algebra. --- Donaldson theory. --- Duality (mathematics). --- Eigenvalues and eigenvectors. --- Elliptic operator. --- Elliptic surface. --- Equation. --- Fiber bundle. --- Frenet–Serret formulas. --- Gauge fixing. --- Gauge theory. --- Gaussian curvature. --- Geometry. --- Group homomorphism. --- Hilbert space. --- Hodge index theorem. --- Homology (mathematics). --- Homotopy. --- Identity (mathematics). --- Implicit function theorem. --- Intersection form (4-manifold). --- Inverse function theorem. --- Isomorphism class. --- K3 surface. --- Kähler manifold. --- Levi-Civita connection. --- Lie algebra. --- Line bundle. --- Linear map. --- Linear space (geometry). --- Linearization. --- Manifold. --- Mathematical induction. --- Moduli space. --- Multiplication theorem. --- Neighbourhood (mathematics). --- One-form. --- Open set. --- Orientability. --- Orthonormal basis. --- Parameter space. --- Parametric equation. --- Parity (mathematics). --- Partial derivative. --- Principal bundle. --- Projection (linear algebra). --- Pullback (category theory). --- Quadratic form. --- Quaternion algebra. --- Quotient space (topology). --- Riemann surface. --- Riemannian manifold. --- Sard's theorem. --- Sign (mathematics). --- Sobolev space. --- Spin group. --- Spin representation. --- Spin structure. --- Spinor field. --- Subgroup. --- Submanifold. --- 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.