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Beginning with a general discussion of bordism, Professors Madsen and Milgram present the homotopy theory of the surgery classifying spaces and the classifying spaces for the various required bundle theories. The next part covers more recent work on the maps between these spaces and the properties of the PL and Top characteristic classes, and includes integrality theorems for topological and PL manifolds. Later chapters treat the integral cohomology of BPL and Btop. The authors conclude with a discussion of the PL and topological cobordism rings and a construction of the torsion-free generators.

Algebraic topology --- 515.16 --- Classifying spaces --- Cobordism theory --- Manifolds (Mathematics) --- Surgery (Topology) --- Differential topology --- Homotopy equivalences --- Topology --- Geometry, Differential --- Spaces, Classifying --- Fiber bundles (Mathematics) --- Fiber spaces (Mathematics) --- Topology of manifolds --- Classifying spaces. --- Cobordism theory. --- Manifolds (Mathematics). --- Surgery (Topology). --- 515.16 Topology of manifolds --- Bijection. --- Calculation. --- Characteristic class. --- Classification theorem. --- Classifying space. --- Closed manifold. --- Cobordism. --- Coefficient. --- Cohomology. --- Commutative diagram. --- Commutative property. --- Complex projective space. --- Connected sum. --- Corollary. --- Cup product. --- Diagram (category theory). --- Differentiable manifold. --- Disjoint union. --- Disk (mathematics). --- Effective method. --- Eilenberg–Moore spectral sequence. --- Elaboration. --- Equivalence class. --- Exact sequence. --- Exterior algebra. --- Fiber bundle. --- Fibration. --- Function composition. --- H-space. --- Homeomorphism. --- Homomorphism. --- Homotopy fiber. --- Homotopy group. --- Homotopy. --- Hopf algebra. --- Iterative method. --- Loop space. --- Manifold. --- Massey product. --- N-sphere. --- Normal bundle. --- Obstruction theory. --- Pairing. --- Permutation. --- Piecewise linear manifold. --- Piecewise linear. --- Polynomial. --- Prime number. --- Projective space. --- Sequence. --- Simply connected space. --- Special case. --- Spin structure. --- Steenrod algebra. --- Subset. --- Summation. --- Tensor product. --- Theorem. --- Topological group. --- Topological manifold. --- Topology. --- Total order. --- Variétés topologiques --- Topologie differentielle

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Mathematical No/ex, 27Originally published in 1981.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.

Riemannian manifolds. --- Minimal surfaces. --- Surfaces, Minimal --- Maxima and minima --- Manifolds, Riemannian --- Riemannian space --- Space, Riemannian --- Geometry, Differential --- Manifolds (Mathematics) --- Differential geometry. Global analysis --- Addition. --- Analytic function. --- Branch point. --- Calculation. --- Cartesian coordinate system. --- Closed geodesic. --- Codimension. --- Coefficient. --- Compactness theorem. --- Compass-and-straightedge construction. --- Continuous function. --- Corollary. --- Counterexample. --- Covering space. --- Curvature. --- Curve. --- Decomposition theorem. --- Derivative. --- Differentiable manifold. --- Differential geometry. --- Disjoint union. --- Equation. --- Essential singularity. --- Estimation. --- Euclidean space. --- Existence theorem. --- Existential quantification. --- First variation. --- Flat topology. --- Fundamental group. --- Geometric measure theory. --- Great circle. --- Homology (mathematics). --- Homotopy group. --- Homotopy. --- Hyperbolic function. --- Hypersurface. --- Integer. --- Line–line intersection. --- Manifold. --- Measure (mathematics). --- Minimal surface. --- Monograph. --- Natural number. --- Open set. --- Parameter. --- Partition of unity. --- Pointwise. --- Quantity. --- Regularity theorem. --- Riemann surface. --- Riemannian manifold. --- Scalar curvature. --- Scientific notation. --- Second fundamental form. --- Sectional curvature. --- Sequence. --- Sign (mathematics). --- Simply connected space. --- Smoothness. --- Sobolev inequality. --- Solid torus. --- Subgroup. --- Submanifold. --- Summation. --- Theorem. --- Topology. --- Two-dimensional space. --- Unit sphere. --- Upper and lower bounds. --- Varifold. --- Weak topology.

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Five papers by distinguished American and European mathematicians describe some current trends in mathematics in the perspective of the recent past and in terms of expectations for the future. Among the subjects discussed are algebraic groups, quadratic forms, topological aspects of global analysis, variants of the index theorem, and partial differential equations.

Mathematics --- Mathématiques --- Congresses --- Congrès --- 51 --- -Math --- Science --- Congresses. --- -Mathematics --- 51 Mathematics --- -51 Mathematics --- Math --- Mathématiques --- Congrès --- A priori estimate. --- Addition. --- Additive group. --- Affine space. --- Algebraic geometry. --- Algebraic group. --- Atiyah–Singer index theorem. --- Bernoulli number. --- Boundary value problem. --- Bounded operator. --- C*-algebra. --- Canonical transformation. --- Cauchy problem. --- Characteristic class. --- Clifford algebra. --- Coefficient. --- Cohomology. --- Commutative property. --- Commutative ring. --- Complex manifold. --- Complex number. --- Complex vector bundle. --- Dedekind sum. --- Degenerate bilinear form. --- Diagram (category theory). --- Diffeomorphism. --- Differentiable manifold. --- Differential operator. --- Dimension (vector space). --- Ellipse. --- Elliptic operator. --- Equation. --- Euler characteristic. --- Euler number. --- Existence theorem. --- Exotic sphere. --- Finite difference. --- Finite group. --- Fourier integral operator. --- Fourier transform. --- Fourier. --- Fredholm operator. --- Hardy space. --- Hilbert space. --- Holomorphic vector bundle. --- Homogeneous coordinates. --- Homomorphism. --- Homotopy. --- Hyperbolic partial differential equation. --- Identity component. --- Integer. --- Integral transform. --- Isomorphism class. --- John Milnor. --- K-theory. --- Lebesgue measure. --- Line bundle. --- Local ring. --- Mathematics. --- Maximal ideal. --- Modular form. --- Module (mathematics). --- Monoid. --- Normal bundle. --- Number theory. --- Open set. --- Parametrix. --- Parity (mathematics). --- Partial differential equation. --- Piecewise linear manifold. --- Poisson bracket. --- Polynomial ring. --- Polynomial. --- Prime number. --- Principal part. --- Projective space. --- Pseudo-differential operator. --- Quadratic form. --- Rational variety. --- Real number. --- Reciprocity law. --- Resolution of singularities. --- Riemann–Roch theorem. --- Shift operator. --- Simply connected space. --- Special case. --- Square-integrable function. --- Subalgebra. --- Submanifold. --- Support (mathematics). --- Surjective function. --- Symmetric bilinear form. --- Symplectic vector space. --- Tangent space. --- Theorem. --- Topology. --- Variable (mathematics). --- Vector bundle. --- Vector space. --- Winding number. --- Mathematics - Congresses

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The intention of the authors is to examine the relationship between piecewise linear structure and differential structure: a relationship, they assert, that can be understood as a homotopy obstruction theory, and, hence, can be studied by using the traditional techniques of algebraic topology.Thus the book attacks the problem of existence and classification (up to isotopy) of differential structures compatible with a given combinatorial structure on a manifold. The problem is completely "solved" in the sense that it is reduced to standard problems of algebraic topology.The first part of the book is purely geometrical; it proves that every smoothing of the product of a manifold M and an interval is derived from an essentially unique smoothing of M. In the second part this result is used to translate the classification of smoothings into the problem of putting a linear structure on the tangent microbundle of M. This in turn is converted to the homotopy problem of classifying maps from M into a certain space PL/O. The set of equivalence classes of smoothings on M is given a natural abelian group structure.

Algebraic topology --- Piecewise linear topology --- Manifolds (Mathematics) --- Topologie linéaire par morceaux --- Variétés (Mathématiques) --- 515.16 --- PL topology --- Topology --- Geometry, Differential --- Topology of manifolds --- Piecewise linear topology. --- Manifolds (Mathematics). --- 515.16 Topology of manifolds --- Topologie linéaire par morceaux --- Variétés (Mathématiques) --- Affine transformation. --- Approximation. --- Associative property. --- Bijection. --- Bundle map. --- Classification theorem. --- Codimension. --- Coefficient. --- Cohomology. --- Commutative property. --- Computation. --- Convex cone. --- Convolution. --- Corollary. --- Counterexample. --- Diffeomorphism. --- Differentiable function. --- Differentiable manifold. --- Differential structure. --- Dimension. --- Direct proof. --- Division by zero. --- Embedding. --- Empty set. --- Equivalence class. --- Equivalence relation. --- Euclidean space. --- Existential quantification. --- Exponential map (Lie theory). --- Fiber bundle. --- Fibration. --- Functor. --- Grassmannian. --- H-space. --- Homeomorphism. --- Homotopy. --- Integral curve. --- Inverse problem. --- Isomorphism class. --- K0. --- Linearization. --- Manifold. --- Mathematical induction. --- Milnor conjecture. --- Natural transformation. --- Neighbourhood (mathematics). --- Normal bundle. --- Obstruction theory. --- Open set. --- Partition of unity. --- Piecewise linear. --- Polyhedron. --- Reflexive relation. --- Regular map (graph theory). --- Sheaf (mathematics). --- Smoothing. --- Smoothness. --- Special case. --- Submanifold. --- Tangent bundle. --- Tangent vector. --- Theorem. --- Topological manifold. --- Topological space. --- Topology. --- Transition function. --- Transitive relation. --- Vector bundle. --- Vector field. --- Variétés topologiques

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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.