Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

Since its introduction by Friedhelm Waldhausen in the 1970's, the algebraic K-theory of spaces has been recognized as the main tool for studying parametrized phenomena in the theory of manifolds. However, a full proof of the equivalence relating the two areas has not appeared until now. This book presents such a proof, essentially completing Waldhausen's program from more than thirty years ago. The main result is a stable parametrized h-cobordism theorem, derived from a homotopy equivalence between a space of PL h-cobordisms on a space X and the classifying space of a category of simple maps of spaces having X as deformation retract. The smooth and topological results then follow by smoothing and triangulation theory. The proof has two main parts. The essence of the first part is a "desingularization," improving arbitrary finite simplicial sets to polyhedra. The second part compares polyhedra with PL manifolds by a thickening procedure. Many of the techniques and results developed should be useful in other connections.

Piecewise linear topology. --- Mappings (Mathematics) --- Maps (Mathematics) --- Functions --- Functions, Continuous --- Topology --- Transformations (Mathematics) --- PL topology --- Manifolds (Mathematics) --- CAT Whitehead space. --- CAT manifold. --- Euclidean neighborhood retract. --- Euclidean polyhedra. --- PL Serre fibration. --- PL Whitehead space. --- PL homeomorphism. --- PL manifold. --- Quillen's Theorem A. --- Quillen's Theorem B. --- absolute neighborhood retract. --- algebraic K-theory of spaces. --- algebraic K-theory. --- approximate lifting property. --- cofibration. --- degeneracy operator. --- desingularization. --- finite simplicial set. --- h-cobordism theorem. --- homotopy equivalence. --- homotopy fiber sequence. --- horn. --- manifold space. --- manifold. --- n-manifolds. --- normal subdivision. --- polyhedral realization. --- q-simplex. --- reduced mapping cylinder. --- simple map. --- simplicial set. --- space. --- stabilization map. --- stable parametrized h-cobordism theorem. --- tangent microbundle. --- thickening. --- triangulation. --- weak homotopy equivalence. --- Čech homotopy type.

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

This book presents the complete proof of the Bloch-Kato conjecture and several related conjectures of Beilinson and Lichtenbaum in algebraic geometry. Brought together here for the first time, these conjectures describe the structure of étale cohomology and its relation to motivic cohomology and Chow groups.Although the proof relies on the work of several people, it is credited primarily to Vladimir Voevodsky. The authors draw on a multitude of published and unpublished sources to explain the large-scale structure of Voevodsky's proof and introduce the key figures behind its development. They go on to describe the highly innovative geometric constructions of Markus Rost, including the construction of norm varieties, which play a crucial role in the proof. The book then addresses symmetric powers of motives and motivic cohomology operations.Comprehensive and self-contained, The Norm Residue Theorem in Motivic Cohomology unites various components of the proof that until now were scattered across many sources of varying accessibility, often with differing hypotheses, definitions, and language.

Homology theory. --- Cohomology theory --- Contrahomology theory --- Algebraic topology --- 5P. --- Abelian group. --- Addition. --- Additive category. --- Adjoint functors. --- Alexander Grothendieck. --- Algebraic closure. --- Algebraic cobordism. --- Algebraic cycle. --- Algebraic extension. --- Algebraic geometry. --- Algebraic topology. --- Andrei Suslin. --- Axiom. --- Characteristic class. --- Classifying space. --- Closed set. --- Codimension. --- Cofibration. --- Cohomology operation. --- Cohomology. --- Conjecture. --- Corollary. --- Diagram (category theory). --- Direct limit. --- Exact sequence. --- Factorization. --- Fibration. --- Functor. --- Galois cohomology. --- Galois extension. --- Group object. --- Homology (mathematics). --- Homotopy category. --- Homotopy. --- Hypersurface. --- Inverse function. --- Mathematical induction. --- Mathematics. --- Milnor K-theory. --- Model category. --- Module (mathematics). --- Monoid. --- Monomorphism. --- Morphism. --- Motivic cohomology. --- Natural number. --- Normal bundle. --- Open set. --- Presheaf (category theory). --- Pushout (category theory). --- Quantity. --- Quillen adjunction. --- Rational point. --- Regular representation. --- Remainder. --- Retract. --- Separable extension. --- Sheaf (mathematics). --- Smooth scheme. --- Special case. --- Subgroup. --- Summation. --- Tangent space. --- Theorem. --- Trivial representation. --- Vladimir Voevodsky. --- Weak equivalence (homotopy theory).

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

Since Poincaré's time, topologists have been most concerned with three species of manifold. The most primitive of these--the TOP manifolds--remained rather mysterious until 1968, when Kirby discovered his now famous torus unfurling device. A period of rapid progress with TOP manifolds ensued, including, in 1969, Siebenmann's refutation of the Hauptvermutung and the Triangulation Conjecture. Here is the first connected account of Kirby's and Siebenmann's basic research in this area.The five sections of this book are introduced by three articles by the authors that initially appeared between 1968 and 1970. Appendices provide a full discussion of the classification of homotopy tori, including Casson's unpublished work and a consideration of periodicity in topological surgery.

Differential geometry. Global analysis --- Manifolds (Mathematics) --- Piecewise linear topology --- Triangulating manifolds --- Variétés (Mathématiques) --- Topologie linéaire par morceaux --- 515.16 --- Manifolds, Triangulating --- PL topology --- Topology --- Geometry, Differential --- Topology of manifolds --- Piecewise linear topology. --- Triangulating manifolds. --- Manifolds (Mathematics). --- 515.16 Topology of manifolds --- Variétés (Mathématiques) --- Topologie linéaire par morceaux --- Triangulation. --- Triangulation --- Affine space. --- Algebraic topology (object). --- Approximation. --- Associative property. --- Automorphism. --- Big O notation. --- CW complex. --- Calculation. --- Cap product. --- Cartesian product. --- Category of sets. --- Chain complex. --- Classification theorem. --- Classifying space. --- Cobordism. --- Codimension. --- Cofibration. --- Cohomology. --- Connected space. --- Continuous function (set theory). --- Continuous function. --- Counterexample. --- Diffeomorphism. --- Differentiable manifold. --- Differential structure. --- Differential topology. --- Dimension (vector space). --- Direct proof. --- Disjoint union. --- Elementary proof. --- Embedding. --- Euclidean space. --- Existence theorem. --- Existential quantification. --- Fiber bundle. --- Fibration. --- General position. --- Geometry. --- Group homomorphism. --- H-cobordism. --- H-space. --- Handle decomposition. --- Handlebody. --- Hauptvermutung. --- Hausdorff space. --- Hilbert cube. --- Homeomorphism group. --- Homeomorphism. --- Homomorphism. --- Homotopy group. --- Homotopy. --- Inclusion map. --- Injective function. --- Invertible matrix. --- K-cell (mathematics). --- Kan extension. --- Linear subspace. --- Linear topology. --- Manifold. --- Mapping cylinder. --- Mathematical induction. --- Mathematician. --- Metric space. --- Morse theory. --- Neighbourhood (mathematics). --- Open set. --- Partition of unity. --- Piecewise linear manifold. --- Piecewise linear. --- Poincaré conjecture. --- Polyhedron. --- Principal bundle. --- Product metric. --- Pushout (category theory). --- Regular homotopy. --- Retract. --- Sheaf (mathematics). --- Simplicial complex. --- Smoothing. --- Spin structure. --- Stability theory. --- Stable manifold. --- Standard map. --- Submanifold. --- Submersion (mathematics). --- Subset. --- Surgery exact sequence. --- Surjective function. --- Theorem. --- Topological group. --- Topological manifold. --- Topological space. --- Topology. --- Transversal (geometry). --- Transversality (mathematics). --- Transversality theorem. --- Union (set theory). --- Uniqueness theorem. --- Vector bundle. --- Zorn's lemma. --- Variétés topologiques

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

Kenneth Brakke studies in general dimensions a dynamic system of surfaces of no inertial mass driven by the force of surface tension and opposed by a frictional force proportional to velocity. He formulates his study in terms of varifold surfaces and uses the methods of geometric measure theory to develop a mathematical description of the motion of a surface by its mean curvature. This mathematical description encompasses, among other subtleties, those of changing geometries and instantaneous mass losses.Originally published in 1978.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.

Geometric measure theory. --- Surfaces. --- Curvature. --- Measure theory --- Calculus --- Curves --- Surfaces --- Curved surfaces --- Geometry --- Shapes --- Affine transformation. --- Approximation. --- Asymptote. --- Barrier function. --- Besicovitch covering theorem. --- Big O notation. --- Bounded set (topological vector space). --- Boundedness. --- Calculation. --- Cauchy–Schwarz inequality. --- Characteristic function (probability theory). --- Compactness theorem. --- Completing the square. --- Concave function. --- Convex set. --- Convolution. --- Crystal structure. --- Curve. --- Derivative. --- Diameter. --- Differentiable function. --- Differentiable manifold. --- Differential geometry. --- Dimension. --- Domain of a function. --- Dyadic rational. --- Equivalence relation. --- Estimation. --- Euclidean space. --- Existential quantification. --- Exterior (topology). --- First variation. --- Gaussian curvature. --- Geometry. --- Grain boundary. --- Graph of a function. --- Grassmannian. --- Harmonic function. --- Hausdorff measure. --- Heat equation. --- Heat kernel. --- Heat transfer. --- Homotopy. --- Hypersurface. --- Hölder's inequality. --- Infimum and supremum. --- Initial condition. --- Lebesgue measure. --- Lebesgue point. --- Linear space (geometry). --- Lipschitz continuity. --- Mean curvature. --- Melting point. --- Microstructure. --- Monotonic function. --- Natural number. --- Nonparametric statistics. --- Order of integration (calculus). --- Order of integration. --- Order of magnitude. --- Parabolic partial differential equation. --- Paraboloid. --- Partial differential equation. --- Permutation. --- Perpendicular. --- Pointwise. --- Probability. --- Quantity. --- Quotient space (topology). --- Radon measure. --- Regularity theorem. --- Retract. --- Rewriting. --- Riemannian manifold. --- Right angle. --- Second derivative. --- Sectional curvature. --- Semi-continuity. --- Smoothness. --- Subsequence. --- Subset. --- Support (mathematics). --- Tangent space. --- Taylor's theorem. --- Theorem. --- Theory. --- Topology. --- Total curvature. --- Translational symmetry. --- Uniform boundedness. --- Unit circle. --- Unit vector. --- Upper and lower bounds. --- Variable (mathematics). --- Varifold. --- Vector field. --- Weight function. --- Without loss of generality.

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

One of the most cited books in mathematics, John Milnor's exposition of Morse theory has been the most important book on the subject for more than forty years. Morse theory was developed in the 1920s by mathematician Marston Morse. (Morse was on the faculty of the Institute for Advanced Study, and Princeton published his Topological Methods in the Theory of Functions of a Complex Variable in the Annals of Mathematics Studies series in 1947.) One classical application of Morse theory includes the attempt to understand, with only limited information, the large-scale structure of an object. This kind of problem occurs in mathematical physics, dynamic systems, and mechanical engineering. Morse theory has received much attention in the last two decades as a result of a famous paper in which theoretical physicist Edward Witten relates Morse theory to quantum field theory. Milnor was awarded the Fields Medal (the mathematical equivalent of a Nobel Prize) in 1962 for his work in differential topology. He has since received the National Medal of Science (1967) and the Steele Prize from the American Mathematical Society twice (1982 and 2004) in recognition of his explanations of mathematical concepts across a wide range of scienti.c disciplines. The citation reads, "The phrase sublime elegance is rarely associated with mathematical exposition, but it applies to all of Milnor's writings. Reading his books, one is struck with the ease with which the subject is unfolding and it only becomes apparent after re.ection that this ease is the mark of a master.? Milnor has published five books with Princeton University Press.

Homotopy theory. --- Geometry, Differential. --- Affine connection. --- Banach algebra. --- Betti number. --- Bott periodicity theorem. --- Bounded set. --- Calculus of variations. --- Cauchy sequence. --- Characteristic class. --- Clifford algebra. --- Compact space. --- Complex number. --- Conjugate points. --- Coordinate system. --- Corollary. --- Covariant derivative. --- Covering space. --- Critical point (mathematics). --- Curvature. --- Cyclic group. --- Derivative. --- Diagram (category theory). --- Diffeomorphism. --- Differentiable function. --- Differentiable manifold. --- Differential geometry. --- Differential structure. --- Differential topology. --- Dimension (vector space). --- Dirichlet problem. --- Elementary proof. --- Euclidean space. --- Euler characteristic. --- Exact sequence. --- Exponentiation. --- First variation. --- Function (mathematics). --- Fundamental lemma (Langlands program). --- Fundamental theorem. --- General position. --- Geometry. --- Great circle. --- Hessian matrix. --- Hilbert space. --- Homomorphism. --- Homotopy group. --- Homotopy. --- Implicit function theorem. --- Inclusion map. --- Infimum and supremum. --- Jacobi field. --- Lie algebra. --- Lie group. --- Line segment. --- Linear equation. --- Linear map. --- Loop space. --- Manifold. --- Mathematical induction. --- Metric connection. --- Metric space. --- Morse theory. --- N-sphere. --- Order of approximation. --- Orthogonal group. --- Orthogonal transformation. --- Paraboloid. --- Path space. --- Piecewise. --- Projective plane. --- Real number. --- Retract. --- Ricci curvature. --- Riemannian geometry. --- Riemannian manifold. --- Sard's theorem. --- Second fundamental form. --- Sectional curvature. --- Sequence. --- Simply connected space. --- Skew-Hermitian matrix. --- Smoothness. --- Special unitary group. --- Square-integrable function. --- Subgroup. --- Submanifold. --- Subset. --- Symmetric space. --- Tangent space. --- Tangent vector. --- Theorem. --- Topological group. --- Topological space. --- Topology. --- Torus. --- Unit sphere. --- Unit vector. --- Unitary group. --- Vector bundle. --- Vector field. --- Vector space.