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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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This famous book was the first treatise on Lie groups in which a modern point of view was adopted systematically, namely, that a continuous group can be regarded as a global object. To develop this idea to its fullest extent, Chevalley incorporated a broad range of topics, such as the covering spaces of topological spaces, analytic manifolds, integration of complete systems of differential equations on a manifold, and the calculus of exterior differential forms. The book opens with a short description of the classical groups: unitary groups, orthogonal groups, symplectic groups, etc. These special groups are then used to illustrate the general properties of Lie groups, which are considered later. The general notion of a Lie group is defined and correlated with the algebraic notion of a Lie algebra; the subgroups, factor groups, and homomorphisms of Lie groups are studied by making use of the Lie algebra. The last chapter is concerned with the theory of compact groups, culminating in Peter-Weyl's theorem on the existence of representations. Given a compact group, it is shown how one can construct algebraically the corresponding Lie group with complex parameters which appears in the form of a certain algebraic variety (associated algebraic group). This construction is intimately related to the proof of the generalization given by Tannaka of Pontrjagin's duality theorem for Abelian groups. The continued importance of Lie groups in mathematics and theoretical physics make this an indispensable volume for researchers in both fields.

Continuous groups. --- Additive group. --- Adjoint representation. --- Algebra over a field. --- Algebraic extension. --- Algebraic variety. --- Algebraically closed field. --- Analytic function. --- Analytic manifold. --- Automorphism. --- Axiom of countability. --- Ball (mathematics). --- Cardinal number. --- Characteristic polynomial. --- Coefficient. --- Commutator subgroup. --- Complex number. --- Connected component (graph theory). --- Continuous function (set theory). --- Continuous function. --- Coordinate system. --- Coset. --- Countable set. --- Covering group. --- Covering space. --- Differential algebra. --- Differential calculus. --- Differential form. --- Differential of a function. --- Dual space. --- Eigenvalues and eigenvectors. --- Endomorphism. --- Equivalence class. --- Existential quantification. --- Exponential function. --- Exterior algebra. --- Fundamental group. --- Galois group. --- General topology. --- Geometry. --- Group (mathematics). --- Group theory. --- Hermitian matrix. --- Homeomorphism. --- Homogeneous space. --- Homomorphism. --- Homotopy group. --- Identity element. --- Identity matrix. --- Infinitesimal transformation. --- Integer. --- Invariant subspace. --- Irreducible representation. --- Kronecker product. --- Lie algebra. --- Lie group. --- Linear function. --- Linear map. --- Linear space (geometry). --- Linear subspace. --- Linearization. --- Locally connected space. --- Manifold. --- Mathematical induction. --- Matrix exponential. --- Modular arithmetic. --- Module (mathematics). --- Monodromy. --- Morphism. --- Open set. --- Orthogonal group. --- Parametric equation. --- Permutation. --- Power series. --- Projective plane. --- Real number. --- Regular matrix. --- Representation theory. --- Riemann surface. --- Simply connected space. --- Skew-symmetric matrix. --- Special case. --- Subalgebra. --- Subgroup. --- Submanifold. --- Subset. --- Summation. --- Symplectic geometry. --- Symplectic group. --- Tangent space. --- Theorem. --- Topological group. --- Topological space. --- Topology. --- Trigonometric polynomial. --- Union (set theory). --- Uniqueness theorem. --- Unitary group. --- Unitary matrix. --- Variable (mathematics). --- Vector space.