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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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The theory of infinite loop spaces has been the center of much recent activity in algebraic topology. Frank Adams surveys this extensive work for researchers and students. Among the major topics covered are generalized cohomology theories and spectra; infinite-loop space machines in the sense of Boadman-Vogt, May, and Segal; localization and group completion; the transfer; the Adams conjecture and several proofs of it; and the recent theories of Adams and Priddy and of Madsen, Snaith, and Tornehave.

Algebraic topology --- Loop spaces --- Espaces de lacets --- Infinite loop spaces. --- Abelian group. --- Adams spectral sequence. --- Adjoint functors. --- Algebraic K-theory. --- Algebraic topology. --- Automorphism. --- Axiom. --- Bott periodicity theorem. --- CW complex. --- Calculation. --- Cartesian product. --- Cobordism. --- Coefficient. --- Cofibration. --- Cohomology operation. --- Cohomology ring. --- Cohomology. --- Commutative diagram. --- Continuous function. --- Counterexample. --- De Rham cohomology. --- Diagram (category theory). --- Differentiable manifold. --- Dimension. --- Discrete space. --- Disjoint union. --- Double coset. --- Eilenberg. --- Eilenberg–Steenrod axioms. --- Endomorphism. --- Epimorphism. --- Equivalence class. --- Euler class. --- Existential quantification. --- Explicit formulae (L-function). --- Exterior algebra. --- F-space. --- Fiber bundle. --- Fibration. --- Finite group. --- Function composition. --- Function space. --- Functor. --- Fundamental class. --- Fundamental group. --- Geometry. --- H-space. --- Homology (mathematics). --- Homomorphism. --- Homotopy category. --- Homotopy group. --- Homotopy. --- Hurewicz theorem. --- Inverse limit. --- J-homomorphism. --- K-theory. --- Limit (mathematics). --- Loop space. --- Mathematical induction. --- Maximal torus. --- Module (mathematics). --- Monoid. --- Monoidal category. --- Moore space. --- Morphism. --- Multiplication. --- Natural transformation. --- P-adic number. --- P-complete. --- Parameter space. --- Permutation. --- Prime number. --- Principal bundle. --- Principal ideal domain. --- Pullback (category theory). --- Quotient space (topology). --- Reduced homology. --- Riemannian manifold. --- Ring spectrum. --- Serre spectral sequence. --- Simplicial set. --- Simplicial space. --- Special case. --- Spectral sequence. --- Stable homotopy theory. --- Steenrod algebra. --- Subalgebra. --- Subring. --- Subset. --- Surjective function. --- Theorem. --- Theory. --- Topological K-theory. --- Topological ring. --- Topological space. --- Topology. --- Universal bundle. --- Universal coefficient theorem. --- Vector bundle. --- Weak equivalence (homotopy theory). --- Topologie algébrique

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Written by Arthur Ogus on the basis of notes from Pierre Berthelot's seminar on crystalline cohomology at Princeton University in the spring of 1974, this book constitutes an informal introduction to a significant branch of algebraic geometry. Specifically, it provides the basic tools used in the study of crystalline cohomology of algebraic varieties in positive characteristic.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.

Algebraic geometry --- Geometry, Algebraic. --- Homology theory. --- Functions, Zeta. --- Zeta functions --- Cohomology theory --- Contrahomology theory --- Algebraic topology --- Geometry --- Abelian category. --- Additive map. --- Adjoint functors. --- Adjunction (field theory). --- Adjunction formula. --- Alexander Grothendieck. --- Algebra homomorphism. --- Artinian. --- Automorphism. --- Axiom. --- Banach space. --- Base change map. --- Base change. --- Betti number. --- Calculation. --- Cartesian product. --- Category of abelian groups. --- Characteristic polynomial. --- Characterization (mathematics). --- Closed immersion. --- Codimension. --- Coefficient. --- Cohomology. --- Cokernel. --- Commutative diagram. --- Commutative property. --- Commutative ring. --- Compact space. --- Corollary. --- Crystalline cohomology. --- De Rham cohomology. --- Degeneracy (mathematics). --- Derived category. --- Diagram (category theory). --- Differential operator. --- Discrete valuation ring. --- Divisibility rule. --- Dual basis. --- Eigenvalues and eigenvectors. --- Endomorphism. --- Epimorphism. --- Equation. --- Equivalence of categories. --- Exact sequence. --- Existential quantification. --- Explicit formula. --- Explicit formulae (L-function). --- Exponential type. --- Exterior algebra. --- Exterior derivative. --- Formal power series. --- Formal scheme. --- Frobenius endomorphism. --- Functor. --- Fundamental theorem. --- Hasse invariant. --- Hodge theory. --- Homotopy. --- Ideal (ring theory). --- Initial and terminal objects. --- Inverse image functor. --- Inverse limit. --- Inverse system. --- K-theory. --- Leray spectral sequence. --- Linear map. --- Linearization. --- Locally constant function. --- Mapping cone (homological algebra). --- Mathematical induction. --- Maximal ideal. --- Module (mathematics). --- Monomial. --- Monotonic function. --- Morphism. --- Natural transformation. --- Newton polygon. --- Noetherian ring. --- Noetherian. --- P-adic number. --- Polynomial. --- Power series. --- Presheaf (category theory). --- Projective module. --- Scientific notation. --- Series (mathematics). --- Sheaf (mathematics). --- Sheaf of modules. --- Special case. --- Spectral sequence. --- Subring. --- Subset. --- Symmetric algebra. --- Theorem. --- Topological space. --- Topology. --- Topos. --- Transitive relation. --- Universal property. --- Zariski topology. --- Geometrie algebrique --- Topologie algebrique --- Varietes algebriques --- Cohomologie

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In Hypo-Analytic Structures Franois Treves provides a systematic approach to the study of the differential structures on manifolds defined by systems of complex vector fields. Serving as his main examples are the elliptic complexes, among which the De Rham and Dolbeault are the best known, and the tangential Cauchy-Riemann operators. Basic geometric entities attached to those structures are isolated, such as maximally real submanifolds and orbits of the system. Treves discusses the existence, uniqueness, and approximation of local solutions to homogeneous and inhomogeneous equations and delimits their supports. The contents of this book consist of many results accumulated in the last decade by the author and his collaborators, but also include classical results, such as the Newlander-Nirenberg theorem. The reader will find an elementary description of the FBI transform, as well as examples of its use. Treves extends the main approximation and uniqueness results to first-order nonlinear equations by means of the Hamiltonian lift.Originally published in 1993.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 equations, Partial. --- Manifolds (Mathematics) --- Vector fields. --- Direction fields (Mathematics) --- Fields, Direction (Mathematics) --- Fields, Slope (Mathematics) --- Fields, Vector --- Slope fields (Mathematics) --- Vector analysis --- Geometry, Differential --- Topology --- Partial differential equations --- Algebra homomorphism. --- Analytic function. --- Automorphism. --- Basis (linear algebra). --- Bijection. --- Bounded operator. --- C0. --- CR manifold. --- Cauchy problem. --- Cauchy sequence. --- Cauchy–Riemann equations. --- Characterization (mathematics). --- Coefficient. --- Cohomology. --- Commutative property. --- Commutator. --- Complex dimension. --- Complex manifold. --- Complex number. --- Complex space. --- Complex-analytic variety. --- Continuous function (set theory). --- Corollary. --- Coset. --- De Rham cohomology. --- Diagram (category theory). --- Diffeomorphism. --- Differential form. --- Differential operator. --- Dimension (vector space). --- Dirac delta function. --- Dirac measure. --- Eigenvalues and eigenvectors. --- Embedding. --- Equation. --- Exact differential. --- Existential quantification. --- Exterior algebra. --- F-space. --- Formal power series. --- Frobenius theorem (differential topology). --- Frobenius theorem (real division algebras). --- H-vector. --- Hadamard three-circle theorem. --- Hahn–Banach theorem. --- Holomorphic function. --- Hypersurface. --- Hölder condition. --- Identity matrix. --- Infimum and supremum. --- Integer. --- Integral equation. --- Integral transform. --- Intersection (set theory). --- Jacobian matrix and determinant. --- Linear differential equation. --- Linear equation. --- Linear map. --- Lipschitz continuity. --- Manifold. --- Mean value theorem. --- Method of characteristics. --- Monomial. --- Multi-index notation. --- Neighbourhood (mathematics). --- Norm (mathematics). --- One-form. --- Open mapping theorem (complex analysis). --- Open mapping theorem. --- Open set. --- Ordinary differential equation. --- Partial differential equation. --- Poisson bracket. --- Polynomial. --- Power series. --- Projection (linear algebra). --- Pullback (category theory). --- Pullback (differential geometry). --- Pullback. --- Riemann mapping theorem. --- Riemann surface. --- Ring homomorphism. --- Sesquilinear form. --- Sobolev space. --- Special case. --- Stokes' theorem. --- Stone–Weierstrass theorem. --- Submanifold. --- Subset. --- Support (mathematics). --- Surjective function. --- Symplectic geometry. --- Symplectic vector space. --- Taylor series. --- Theorem. --- Unit disk. --- Upper half-plane. --- Vector bundle. --- Vector field. --- Volume form.

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Algebraic K-theory describes a branch of algebra that centers about two functors. K0 and K1, which assign to each associative ring ∧ an abelian group K0∧ or K1∧ respectively. Professor Milnor sets out, in the present work, to define and study an analogous functor K2, also from associative rings to abelian groups. Just as functors K0 and K1 are important to geometric topologists, K2 is now considered to have similar topological applications. The exposition includes, besides K-theory, a considerable amount of related arithmetic.

Algebraic geometry --- Ordered algebraic structures --- Associative rings --- Abelian groups --- Functor theory --- Anneaux associatifs --- Groupes abéliens --- Foncteurs, Théorie des --- 512.73 --- 515.14 --- Functorial representation --- Algebra, Homological --- Categories (Mathematics) --- Functional analysis --- Transformations (Mathematics) --- Commutative groups --- Group theory --- Rings (Algebra) --- Cohomology theory of algebraic varieties and schemes --- Algebraic topology --- Abelian groups. --- Associative rings. --- Functor theory. --- 515.14 Algebraic topology --- 512.73 Cohomology theory of algebraic varieties and schemes --- Groupes abéliens --- Foncteurs, Théorie des --- Abelian group. --- Absolute value. --- Addition. --- Algebraic K-theory. --- Algebraic equation. --- Algebraic integer. --- Banach algebra. --- Basis (linear algebra). --- Big O notation. --- Circle group. --- Coefficient. --- Commutative property. --- Commutative ring. --- Commutator. --- Complex number. --- Computation. --- Congruence subgroup. --- Coprime integers. --- Cyclic group. --- Dedekind domain. --- Direct limit. --- Direct proof. --- Direct sum. --- Discrete valuation. --- Division algebra. --- Division ring. --- Elementary matrix. --- Elliptic function. --- Exact sequence. --- Existential quantification. --- Exterior algebra. --- Factorization. --- Finite group. --- Free abelian group. --- Function (mathematics). --- Fundamental group. --- Galois extension. --- Galois group. --- General linear group. --- Group extension. --- Hausdorff space. --- Homological algebra. --- Homomorphism. --- Homotopy. --- Ideal (ring theory). --- Ideal class group. --- Identity element. --- Identity matrix. --- Integral domain. --- Invertible matrix. --- Isomorphism class. --- K-theory. --- Kummer theory. --- Lattice (group). --- Left inverse. --- Local field. --- Local ring. --- Mathematics. --- Matsumoto's theorem. --- Maximal ideal. --- Meromorphic function. --- Monomial. --- Natural number. --- Noetherian. --- Normal subgroup. --- Number theory. --- Open set. --- Picard group. --- Polynomial. --- Prime element. --- Prime ideal. --- Projective module. --- Quadratic form. --- Quaternion. --- Quotient ring. --- Rational number. --- Real number. --- Right inverse. --- Ring of integers. --- Root of unity. --- Schur multiplier. --- Scientific notation. --- Simple algebra. --- Special case. --- Special linear group. --- Subgroup. --- Summation. --- Surjective function. --- Tensor product. --- Theorem. --- Topological K-theory. --- Topological group. --- Topological space. --- Topology. --- Torsion group. --- Variable (mathematics). --- Vector space. --- Wedderburn's theorem. --- Weierstrass function. --- Whitehead torsion. --- K-théorie