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"We develop differential algebraic K-theory for rings of integers in number fields and we construct a cycle map from geometrized bundles of modules over such a ring to the differential algebraic K-theory. We also treat some of the foundational aspects of differential cohomology, including differential function spectra and the differential Becker-Gottlieb transfer. We then state a transfer index conjecture about the equality of the Becker-Gottlieb transfer and the analytic transfer defined by Lott. In support of this conjecture, we derive some non-trivial consequences which are provable by independent means"--

K-theory. --- Differential topology. --- $K$-theory -- Topological $K$-theory -- Twisted $K$-theory; differential $K$-theory. --- Manifolds and cell complexes -- Differential topology -- Algebraic topology on manifolds.

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This book contains accounts of talks held at a symposium in honorof John C. Moore in October 1983 at Princeton University, The workincludes papers in classical homotopy theory, homological algebra,rational homotopy theory, algebraic K-theory of spaces, and othersubjects.

Algebraic topology --- K-theory --- 512.73 --- 515.14 --- 512.73 Cohomology theory of algebraic varieties and schemes --- Cohomology theory of algebraic varieties and schemes --- 515.14 Algebraic topology --- Moore, John C. --- Abelian group. --- Adams spectral sequence. --- Adjoint functors. --- Adjunction (field theory). --- Algebraic K-theory. --- Algebraic closure. --- Algebraic geometry. --- Algebraic group. --- Algebraic number field. --- Algebraic space. --- Algebraic topology. --- Algebraically closed field. --- Associative algebra. --- Boundary (topology). --- CW complex. --- Classification theorem. --- Closure (mathematics). --- Coalgebra. --- Cofibration. --- Cohomology. --- Commutative diagram. --- Commutative property. --- Coproduct. --- Deformation theory. --- Degenerate bilinear form. --- Diagram (category theory). --- Differentiable manifold. --- Dimension (vector space). --- Division algebra. --- Eilenberg–Moore spectral sequence. --- Epimorphism. --- Exterior (topology). --- Formal power series. --- Free Lie algebra. --- Free algebra. --- Freudenthal suspension theorem. --- Function (mathematics). --- Function space. --- Functor. --- G-module. --- Galois extension. --- Global dimension. --- Group cohomology. --- Group homomorphism. --- H-space. --- Hilbert's Theorem 90. --- Homology (mathematics). --- Homomorphism. --- Homotopy category. --- Homotopy group. --- Homotopy. --- Hopf algebra. --- Hopf invariant. --- Hurewicz theorem. --- Inclusion map. --- Inequality (mathematics). --- Integral domain. --- Isometry. --- Isomorphism class. --- K-theory. --- Lie algebra. --- Lie group. --- Limit (category theory). --- Loop space. --- Mathematician. --- Mathematics. --- Noetherian ring. --- Order topology. --- P-adic number. --- Polynomial ring. --- Polynomial. --- Prime number. --- Principal bundle. --- Principal ideal domain. --- Projective module. --- Projective plane. --- Pullback (category theory). --- Pushout (category theory). --- Ring of integers. --- Series (mathematics). --- Sheaf (mathematics). --- Simplicial category. --- Simplicial complex. --- Simplicial set. --- Special case. --- Spectral sequence. --- Square (algebra). --- Stable homotopy theory. --- Steenrod algebra. --- Superalgebra. --- Theorem. --- Topological K-theory. --- Topological space. --- Topology. --- Triviality (mathematics). --- Uniqueness theorem. --- Universal enveloping algebra. --- Vector bundle. --- Weak equivalence (homotopy theory). --- William Browder (mathematician). --- Géométrie algébrique --- K-théorie

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This book presents a coherent account of the current status of etale homotopy theory, a topological theory introduced into abstract algebraic geometry by M. Artin and B. Mazur. Eric M. Friedlander presents many of his own applications of this theory to algebraic topology, finite Chevalley groups, and algebraic geometry. Of particular interest are the discussions concerning the Adams Conjecture, K-theories of finite fields, and Poincare duality. Because these applications have required repeated modifications of the original formulation of etale homotopy theory, the author provides a new treatment of the foundations which is more general and more precise than previous versions.One purpose of this book is to offer the basic techniques and results of etale homotopy theory to topologists and algebraic geometers who may then apply the theory in their own work. With a view to such future applications, the author has introduced a number of new constructions (function complexes, relative homology and cohomology, generalized cohomology) which have immediately proved applicable to algebraic K-theory.

Algebraic topology --- 512.73 --- 515.14 --- Homology theory --- Homotopy theory --- Schemes (Algebraic geometry) --- Geometry, Algebraic --- Deformations, Continuous --- Topology --- Cohomology theory --- Contrahomology theory --- Cohomology theory of algebraic varieties and schemes --- 515.14 Algebraic topology --- 512.73 Cohomology theory of algebraic varieties and schemes --- Homotopy theory. --- Homology theory. --- Abelian group. --- Adams operation. --- Adjoint functors. --- Alexander Grothendieck. --- Algebraic K-theory. --- Algebraic closure. --- Algebraic geometry. --- Algebraic group. --- Algebraic number theory. --- Algebraic structure. --- Algebraic topology (object). --- Algebraic topology. --- Algebraic variety. --- Algebraically closed field. --- Automorphism. --- Base change. --- Cap product. --- Cartesian product. --- Closed immersion. --- Codimension. --- Coefficient. --- Cohomology. --- Comparison theorem. --- Complex number. --- Complex vector bundle. --- Connected component (graph theory). --- Connected space. --- Coprime integers. --- Corollary. --- Covering space. --- Derived functor. --- Dimension (vector space). --- Disjoint union. --- Embedding. --- Existence theorem. --- Ext functor. --- Exterior algebra. --- Fiber bundle. --- Fibration. --- Finite field. --- Finite group. --- Free group. --- Functor. --- Fundamental group. --- Galois cohomology. --- Galois extension. --- Geometry. --- Grothendieck topology. --- Homogeneous space. --- Homological algebra. --- Homology (mathematics). --- Homomorphism. --- Homotopy category. --- Homotopy group. --- Homotopy. --- Integral domain. --- Intersection (set theory). --- Inverse limit. --- Inverse system. --- K-theory. --- Leray spectral sequence. --- Lie group. --- Local ring. --- Mapping cylinder. --- Natural number. --- Natural transformation. --- Neighbourhood (mathematics). --- Newton polynomial. --- Noetherian ring. --- Open set. --- Opposite category. --- Pointed set. --- Presheaf (category theory). --- Reductive group. --- Regular local ring. --- Relative homology. --- Residue field. --- Riemann surface. --- Root of unity. --- Serre spectral sequence. --- Shape theory (mathematics). --- Sheaf (mathematics). --- Sheaf cohomology. --- Sheaf of spectra. --- Simplex. --- Simplicial set. --- Special case. --- Spectral sequence. --- Surjective function. --- Theorem. --- Topological K-theory. --- Topological space. --- Topology. --- Tubular neighborhood. --- Vector bundle. --- Weak equivalence (homotopy theory). --- Weil conjectures. --- Weyl group. --- Witt vector. --- Zariski topology. --- Homologie --- Topologie algebrique --- Geometrie algebrique --- Homotopie