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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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According to Carl Friedrich Gauss (1777–1855), mathematics is the queen of the sciences—and number theory is the queen of mathematics. Numbers (integers, algebraic integers, transcendental numbers, p-adic numbers) and symmetries are investigated in the nine refereed papers of this MDPI issue. This book shows how symmetry pervades number theory. In particular, it highlights connections between symmetry and number theory, quantum computing and elementary particles (thanks to 3-manifolds), and other branches of mathematics (such as probability spaces) and revisits standard subjects (such as the Sieve procedure, primality tests, and Pascal’s triangle). The book should be of interest to all mathematicians, and physicists.

quantum computation --- IC-POVMs --- knot theory --- three-manifolds --- branch coverings --- Dehn surgeries --- zeta function --- Pólya-Hilbert conjecture --- Riemann interferometer --- prime numbers --- Prime Number Theorem (P.N.T.) --- modified Sieve procedure --- binary periodical sequences --- prime number function --- prime characteristic function --- limited intervals --- logarithmic integral estimations --- twin prime numbers --- free probability --- p-adic number fields ℚp --- Banach ∗-probability spaces --- C*-algebras --- semicircular elements --- the semicircular law --- asymptotic semicircular laws --- Kaprekar constants --- Kaprekar transformation --- fixed points for recursive functions --- Baker’s theorem --- Gel’fond–Schneider theorem --- algebraic number --- transcendental number --- standard model of elementary particles --- 4-manifold topology --- particles as 3-Braids --- branched coverings --- knots and links --- charge as Hirzebruch defect --- umbral moonshine --- number of generations --- the pe-Pascal’s triangle --- Lucas’ result on the Pascal’s triangle --- congruences of binomial expansions --- primality test --- Miller–Rabin primality test --- strong pseudoprimes --- primality witnesses

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This book gives a new foundation for the theory of links in 3-space modeled on the modern developmentby Jaco, Shalen, Johannson, Thurston et al. of the theory of 3-manifolds. The basic construction is a method of obtaining any link by "splicing" links of the simplest kinds, namely those whose exteriors are Seifert fibered or hyperbolic. This approach to link theory is particularly attractive since most invariants of links are additive under splicing.Specially distinguished from this viewpoint is the class of links, none of whose splice components is hyperbolic. It includes all links constructed by cabling and connected sums, in particular all links of singularities of complex plane curves. One of the main contributions of this monograph is the calculation of invariants of these classes of links, such as the Alexander polynomials, monodromy, and Seifert forms.

Algebraic geometry --- Differential geometry. Global analysis --- Link theory. --- Curves, Plane. --- SINGULARITIES (Mathematics) --- Curves, Plane --- Invariants --- Link theory --- Singularities (Mathematics) --- Geometry, Algebraic --- Low-dimensional topology --- Piecewise linear topology --- Higher plane curves --- Plane curves --- Invariants. --- 3-sphere. --- Alexander Grothendieck. --- Alexander polynomial. --- Algebraic curve. --- Algebraic equation. --- Algebraic geometry. --- Algebraic surface. --- Algorithm. --- Ambient space. --- Analytic function. --- Approximation. --- Big O notation. --- Call graph. --- Cartesian coordinate system. --- Characteristic polynomial. --- Closed-form expression. --- Cohomology. --- Computation. --- Conjecture. --- Connected sum. --- Contradiction. --- Coprime integers. --- Corollary. --- Curve. --- Cyclic group. --- Determinant. --- Diagram (category theory). --- Diffeomorphism. --- Dimension. --- Disjoint union. --- Eigenvalues and eigenvectors. --- Equation. --- Equivalence class. --- Euler number. --- Existential quantification. --- Exterior (topology). --- Fiber bundle. --- Fibration. --- Foliation. --- Fundamental group. --- Geometry. --- Graph (discrete mathematics). --- Ground field. --- Homeomorphism. --- Homology sphere. --- Identity matrix. --- Integer matrix. --- Intersection form (4-manifold). --- Isolated point. --- Isolated singularity. --- Jordan normal form. --- Knot theory. --- Mathematical induction. --- Monodromy matrix. --- Monodromy. --- N-sphere. --- Natural transformation. --- Newton polygon. --- Newton's method. --- Normal (geometry). --- Notation. --- Pairwise. --- Parametrization. --- Plane curve. --- Polynomial. --- Power series. --- Projective plane. --- Puiseux series. --- Quantity. --- Rational function. --- Resolution of singularities. --- Riemann sphere. --- Riemann surface. --- Root of unity. --- Scientific notation. --- Seifert surface. --- Set (mathematics). --- Sign (mathematics). --- Solid torus. --- Special case. --- Stereographic projection. --- Submanifold. --- Summation. --- Theorem. --- Three-dimensional space (mathematics). --- Topology. --- Torus knot. --- Torus. --- Tubular neighborhood. --- Unit circle. --- Unit vector. --- Unknot. --- Variable (mathematics).

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The description for this book, Topics in Transcendental Algebraic Geometry. (AM-106), Volume 106, will be forthcoming.

Geometry, Algebraic. --- Hodge theory. --- Torelli theorem. --- Géométrie algébrique --- Théorie de Hodge --- Geometry, Algebraic --- Hodge theory --- Torelli theorem --- 512.7 --- Torelli's theorem --- Curves, Algebraic --- Jacobians --- Complex manifolds --- Differentiable manifolds --- Homology theory --- Algebraic geometry --- Geometry --- Algebraic geometry. Commutative rings and algebras --- 512.7 Algebraic geometry. Commutative rings and algebras --- Géométrie algébrique --- Théorie de Hodge --- Abelian integral. --- Algebraic curve. --- Algebraic cycle. --- Algebraic equation. --- Algebraic geometry. --- Algebraic integer. --- Algebraic structure. --- Algebraic surface. --- Arithmetic genus. --- Arithmetic group. --- Asymptotic analysis. --- Automorphism. --- Base change. --- Bilinear form. --- Bilinear map. --- Cohomology. --- Combinatorics. --- Commutative diagram. --- Compactification (mathematics). --- Complete intersection. --- Complex manifold. --- Complex number. --- Computation. --- Deformation theory. --- Degeneracy (mathematics). --- Differentiable manifold. --- Dimension (vector space). --- Divisor (algebraic geometry). --- Divisor. --- Elliptic curve. --- Elliptic surface. --- Equation. --- Exact sequence. --- Fiber bundle. --- Function (mathematics). --- Fundamental class. --- Geometric genus. --- Geometry. --- Hermitian symmetric space. --- Hodge structure. --- Homology (mathematics). --- Homomorphism. --- Homotopy. --- Hypersurface. --- Intersection form (4-manifold). --- Intersection number. --- Irreducibility (mathematics). --- Isomorphism class. --- Jacobian variety. --- K3 surface. --- Kodaira dimension. --- Kronecker's theorem. --- Kummer surface. --- Kähler manifold. --- Lie algebra bundle. --- Lie algebra. --- Linear algebra. --- Linear algebraic group. --- Line–line intersection. --- Mathematical induction. --- Mathematical proof. --- Mathematics. --- Modular arithmetic. --- Module (mathematics). --- Moduli space. --- Monodromy matrix. --- Monodromy theorem. --- Monodromy. --- Nilpotent orbit. --- Normal function. --- Open set. --- Period mapping. --- Permutation group. --- Phillip Griffiths. --- Point at infinity. --- Pole (complex analysis). --- Polynomial. --- Projective space. --- Pullback (category theory). --- Quadric. --- Regular singular point. --- Resolution of singularities. --- Riemann–Roch theorem for surfaces. --- Scientific notation. --- Set (mathematics). --- Special case. --- Spectral sequence. --- Subgroup. --- Submanifold. --- Surface of general type. --- Surjective function. --- Tangent bundle. --- Theorem. --- Topology. --- Transcendental number. --- Vector space. --- Zariski topology. --- Zariski's main theorem.

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Surgery theory, the basis for the classification theory of manifolds, is now about forty years old. The sixtieth birthday (on December 14, 1996) of C.T.C. Wall, a leading member of the subject's founding generation, led the editors of this volume to reflect on the extraordinary accomplishments of surgery theory as well as its current enormously varied interactions with algebra, analysis, and geometry. Workers in many of these areas have often lamented the lack of a single source surveying surgery theory and its applications. Because no one person could write such a survey, the editors asked a variety of experts to report on the areas of current interest. This is the second of two volumes resulting from that collective effort. It will be useful to topologists, to other interested researchers, and to advanced students. The topics covered include current applications of surgery, Wall's finiteness obstruction, algebraic surgery, automorphisms and embeddings of manifolds, surgery theoretic methods for the study of group actions and stratified spaces, metrics of positive scalar curvature, and surgery in dimension four. In addition to the editors, the contributors are S. Ferry, M. Weiss, B. Williams, T. Goodwillie, J. Klein, S. Weinberger, B. Hughes, S. Stolz, R. Kirby, L. Taylor, and F. Quinn.

Chirurgie (Topologie) --- Heelkunde (Topologie) --- Surgery (Topology) --- Differential topology --- Homotopy equivalences --- Manifolds (Mathematics) --- Topology --- Algebraic topology (object). --- Algebraic topology. --- Ambient isotopy. --- Assembly map. --- Atiyah–Hirzebruch spectral sequence. --- Atiyah–Singer index theorem. --- Automorphism. --- Banach algebra. --- Borsuk–Ulam theorem. --- C*-algebra. --- CW complex. --- Calculation. --- Category of manifolds. --- Characterization (mathematics). --- Chern class. --- Cobordism. --- Codimension. --- Cohomology. --- Compactification (mathematics). --- Conjecture. --- Contact geometry. --- Degeneracy (mathematics). --- Diagram (category theory). --- Diffeomorphism. --- Differentiable manifold. --- Differential geometry. --- Dirac operator. --- Disk (mathematics). --- Donaldson theory. --- Duality (mathematics). --- Embedding. --- Epimorphism. --- Excision theorem. --- Exponential map (Riemannian geometry). --- Fiber bundle. --- Fibration. --- Fundamental group. --- Group action. --- Group homomorphism. --- H-cobordism. --- Handle decomposition. --- Handlebody. --- Homeomorphism group. --- Homeomorphism. --- Homology (mathematics). --- Homomorphism. --- Homotopy extension property. --- Homotopy fiber. --- Homotopy group. --- Homotopy. --- Hypersurface. --- Intersection form (4-manifold). --- Intersection homology. --- Isomorphism class. --- K3 surface. --- L-theory. --- Limit (category theory). --- Manifold. --- Mapping cone (homological algebra). --- Mapping cylinder. --- Mostow rigidity theorem. --- Orthonormal basis. --- Parallelizable manifold. --- Poincaré conjecture. --- Product metric. --- Projection (linear algebra). --- Pushout (category theory). --- Quaternionic projective space. --- Quotient space (topology). --- Resolution of singularities. --- Ricci curvature. --- Riemann surface. --- Riemannian geometry. --- Riemannian manifold. --- Ring homomorphism. --- Scalar curvature. --- Semisimple algebra. --- Sheaf (mathematics). --- Sign (mathematics). --- Special case. --- Sub"ient. --- Subgroup. --- Submanifold. --- Support (mathematics). --- Surgery exact sequence. --- Surgery obstruction. --- Surgery theory. --- Symplectic geometry. --- Symplectic vector space. --- Theorem. --- Topological conjugacy. --- Topological manifold. --- Topology. --- Transversality (mathematics). --- Transversality theorem. --- Vector bundle. --- Waldhausen category. --- Whitehead torsion. --- Whitney embedding theorem. --- Yamabe invariant.