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One of the great achievements of contemporary mathematics is the new understanding of four dimensions. Michael Freedman and Frank Quinn have been the principals in the geometric and topological development of this subject, proving the Poincar and Annulus conjectures respectively. Recognition for this work includes the award of the Fields Medal of the International Congress of Mathematicians to Freedman in 1986. In Topology of 4-Manifolds these authors have collaborated to give a complete and accessible account of the current state of knowledge in this field. The basic material has been considerably simplified from the original publications, and should be accessible to most graduate students. The advanced material goes well beyond the literature; nearly one-third of the book is new. This work is indispensable for any topologist whose work includes four dimensions. It is a valuable reference for geometers and physicists who need an awareness of the topological side of the field.Originally published in 1990.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Differential topology --- Four-manifolds (Topology) --- Trois-variétés (Topologie) --- Vier-menigvuldigheden (Topologie) --- 4-dimensional manifolds (Topology) --- 4-manifolds (Topology) --- Four dimensional manifolds (Topology) --- Manifolds, Four dimensional --- Low-dimensional topology --- Topological manifolds --- 4-manifold. --- Ambient isotopy. --- Annulus theorem. --- Automorphism. --- Baire category theorem. --- Bilinear form. --- Boundary (topology). --- CW complex. --- Category of manifolds. --- Central series. --- Characterization (mathematics). --- Cohomology. --- Commutative diagram. --- Commutative property. --- Commutator subgroup. --- Compactification (mathematics). --- Conformal geometry. --- Connected sum. --- Connectivity (graph theory). --- Cyclic group. --- Diagram (category theory). --- Diameter. --- Diffeomorphism. --- Differentiable manifold. --- Differential geometry. --- Dimension. --- Disk (mathematics). --- Duality (mathematics). --- Eigenvalues and eigenvectors. --- Embedding problem. --- Embedding. --- Equivariant map. --- Fiber bundle. --- Four-dimensional space. --- Fundamental group. --- General position. --- Geometry. --- H-cobordism. --- Handlebody. --- Hauptvermutung. --- Homeomorphism. --- Homology (mathematics). --- Homology sphere. --- Homomorphism. --- Homotopy group. --- Homotopy sphere. --- Homotopy. --- Hurewicz theorem. --- Hyperbolic geometry. --- Hyperbolic group. --- Hyperbolic manifold. --- Identity matrix. --- Intermediate value theorem. --- Intersection (set theory). --- Intersection curve. --- Intersection form (4-manifold). --- Intersection number (graph theory). --- Intersection number. --- J-homomorphism. --- Knot theory. --- Lefschetz duality. --- Line–line intersection. --- Manifold. --- Mapping cylinder. --- Mathematical induction. --- Metric space. --- Metrization theorem. --- Module (mathematics). --- Normal bundle. --- Parametrization. --- Parity (mathematics). --- Product topology. --- Pullback (differential geometry). --- Regular homotopy. --- Ring homomorphism. --- Rotation number. --- Seifert–van Kampen theorem. --- Sesquilinear form. --- Set (mathematics). --- Simply connected space. --- Smooth structure. --- Special case. --- Spin structure. --- Submanifold. --- Subset. --- Support (mathematics). --- Tangent bundle. --- Tangent space. --- Tensor product. --- Theorem. --- Topological category. --- Topological manifold. --- Transversal (geometry). --- Transversality (mathematics). --- Transversality theorem. --- Uniqueness theorem. --- Unit disk. --- Vector bundle. --- Whitehead torsion. --- Whitney disk.

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This book offers a systematic and comprehensive presentation of the concepts of a spin manifold, spinor fields, Dirac operators, and A-genera, which, over the last two decades, have come to play a significant role in many areas of modern mathematics. Since the deeper applications of these ideas require various general forms of the Atiyah-Singer Index Theorem, the theorems and their proofs, together with all prerequisite material, are examined here in detail. The exposition is richly embroidered with examples and applications to a wide spectrum of problems in differential geometry, topology, and mathematical physics. The authors consistently use Clifford algebras and their representations in this exposition. Clifford multiplication and Dirac operator identities are even used in place of the standard tensor calculus. This unique approach unifies all the standard elliptic operators in geometry and brings fresh insights into curvature calculations. The fundamental relationships of Clifford modules to such topics as the theory of Lie groups, K-theory, KR-theory, and Bott Periodicity also receive careful consideration. A special feature of this book is the development of the theory of Cl-linear elliptic operators and the associated index theorem, which connects certain subtle spin-corbordism invariants to classical questions in geometry and has led to some of the most profound relations known between the curvature and topology of manifolds.

Algebres de Clifford --- Clifford [Algebra's van ] --- Clifford algebras --- Fysica [Mathematische ] --- Fysica [Wiskundige ] --- Mathematische fysica --- Physics -- Mathematics --- Physics [Mathematical ] --- Physique -- Mathématiques --- Physique -- Méthodes mathématiques --- Wiskundige fysica --- Clifford, Algèbres de --- Spin, Nuclear --- Geometric algebras --- Clifford algebras. --- Spin geometry. --- Clifford, Algèbres de --- Spin geometry --- 514.76 --- Algebras, Linear --- 514.76 Geometry of differentiable manifolds and of their submanifolds --- Geometry of differentiable manifolds and of their submanifolds --- Global differential geometry --- Geometry --- Mathematical physics --- Topology --- Nuclear spin --- -Mathematics --- Géométrie --- Physique mathématique --- Spin nucléaire --- Topologie --- Mathematics --- Mathématiques --- Algebraic theory. --- Atiyah–Singer index theorem. --- Automorphism. --- Betti number. --- Binary icosahedral group. --- Binary octahedral group. --- Bundle metric. --- C*-algebra. --- Calabi conjecture. --- Calabi–Yau manifold. --- Cartesian product. --- Classification theorem. --- Clifford algebra. --- Cobordism. --- Cohomology ring. --- Cohomology. --- Cokernel. --- Complete metric space. --- Complex manifold. --- Complex vector bundle. --- Complexification (Lie group). --- Covering space. --- Diffeomorphism. --- Differential topology. --- Dimension (vector space). --- Dimension. --- Dirac operator. --- Disk (mathematics). --- Dolbeault cohomology. --- Einstein field equations. --- Elliptic operator. --- Equivariant K-theory. --- Exterior algebra. --- Fiber bundle. --- Fixed-point theorem. --- Fourier inversion theorem. --- Fundamental group. --- Gauge theory. --- Geometry. --- Hilbert scheme. --- Holonomy. --- Homotopy sphere. --- Homotopy. --- Hyperbolic manifold. --- Induced homomorphism. --- Intersection form (4-manifold). --- Isomorphism class. --- J-invariant. --- K-theory. --- Kähler manifold. --- Laplace operator. --- Lie algebra. --- Lorentz covariance. --- Lorentz group. --- Manifold. --- Mathematical induction. --- Metric connection. --- Minkowski space. --- Module (mathematics). --- N-sphere. --- Operator (physics). --- Orthonormal basis. --- Principal bundle. --- Projective space. --- Pseudo-Riemannian manifold. --- Pseudo-differential operator. --- Quadratic form. --- Quaternion. --- Quaternionic projective space. --- Ricci curvature. --- Riemann curvature tensor. --- Riemannian geometry. --- Riemannian manifold. --- Ring homomorphism. --- Scalar curvature. --- Scalar multiplication. --- Sign (mathematics). --- Space form. --- Sphere theorem. --- Spin representation. --- Spin structure. --- Spinor bundle. --- Spinor field. --- Spinor. --- Subgroup. --- Support (mathematics). --- Symplectic geometry. --- Tangent bundle. --- Tangent space. --- Tensor calculus. --- Tensor product. --- Theorem. --- Topology. --- Unit disk. --- Unit sphere. --- Variable (mathematics). --- Vector bundle. --- Vector field. --- Vector space. --- Volume form. --- Nuclear spin - - Mathematics --- -Clifford algebras.