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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 Appendices provide a full discussion of the classification of homotopy tori, including Casson's unpublished work and a consideration of periodicity in topological surgery.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

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One of the most important mathematical achievements of the past several decades has been A. Grothendieck's work on algebraic geometry. In the early 1960s, he and M. Artin introduced étale cohomology in order to extend the methods of sheaf-theoretic cohomology from complex varieties to more general schemes. This work found many applications, not only in algebraic geometry, but also in several different branches of number theory and in the representation theory of finite and p-adic groups. Yet until now, the work has been available only in the original massive and difficult papers. In order to provide an accessible introduction to étale cohomology, J. S. Milne offers this more elementary account covering the essential features of the theory. The author begins with a review of the basic properties of flat and étale morphisms and of the algebraic fundamental group. The next two chapters concern the basic theory of étale sheaves and elementary étale cohomology, and are followed by an application of the cohomology to the study of the Brauer group. After a detailed analysis of the cohomology of curves and surfaces, Professor Milne proves the fundamental theorems in étale cohomology -- those of base change, purity, Poincaré duality, and the Lefschetz trace formula. He then applies these theorems to show the rationality of some very general L-series.Originally published in 1980.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.

Ordered algebraic structures --- 512.73 --- 512.66 --- Geometry, Algebraic --- Homology theory --- Sheaf theory --- Cohomology, Sheaf --- Sheaf cohomology --- Sheaves, Theory of --- Sheaves (Algebraic topology) --- Algebraic topology --- Cohomology theory --- Contrahomology theory --- Algebraic geometry --- Geometry --- Cohomology theory of algebraic varieties and schemes --- Homological algebra --- Geometry, Algebraic. --- Homology theory. --- Sheaf theory. --- 512.66 Homological algebra --- 512.73 Cohomology theory of algebraic varieties and schemes --- Abelian category. --- Abelian group. --- Adjoint functors. --- Affine variety. --- Alexander Grothendieck. --- Algebraic closure. --- Algebraic cycle. --- Algebraic equation. --- Algebraic space. --- Algebraically closed field. --- Artinian. --- Automorphism. --- Base change. --- Brauer group. --- CW complex. --- Cardinal number. --- Category of sets. --- Central simple algebra. --- Chow's lemma. --- Closed immersion. --- Codimension. --- Cohomology ring. --- Cohomology. --- Cokernel. --- Commutative diagram. --- Complex number. --- Dedekind domain. --- Derived category. --- Diagram (category theory). --- Direct limit. --- Discrete valuation ring. --- Divisor. --- Epimorphism. --- Equivalence class. --- Existential quantification. --- Fibration. --- Field of fractions. --- Fine topology (potential theory). --- Finite field. --- Finite morphism. --- Flat morphism. --- Functor. --- Fundamental class. --- Fundamental group. --- G-module. --- Galois cohomology. --- Galois extension. --- Galois group. --- Generic point. --- Group scheme. --- Gysin sequence. --- Henselian ring. --- Identity element. --- Inclusion map. --- Integral domain. --- Intersection (set theory). --- Inverse limit. --- Invertible sheaf. --- Isomorphism class. --- Lefschetz pencil. --- Local ring. --- Maximal ideal. --- Module (mathematics). --- Morphism of schemes. --- Morphism. --- Noetherian. --- Open set. --- Power series. --- Presheaf (category theory). --- Prime ideal. --- Prime number. --- Principal homogeneous space. --- Profinite group. --- Projection (mathematics). --- Projective variety. --- Quasi-compact morphism. --- Residue field. --- Riemann surface. --- Sheaf (mathematics). --- Sheaf of modules. --- Special case. --- Spectral sequence. --- Stein factorization. --- Subalgebra. --- Subcategory. --- Subgroup. --- Subring. --- Subset. --- Surjective function. --- Tangent space. --- Theorem. --- Topological space. --- Topology. --- Torsion sheaf. --- Torsor (algebraic geometry). --- Vector bundle. --- Weil conjecture. --- Yoneda lemma. --- Zariski topology. --- Zariski's main theorem. --- Geometrie algebrique --- Cohomologie