The K Book

Informally, $K$-theory is a tool for probing the structure of a mathematical object such as a ring or a topological space in terms of suitably parameterized vector spaces and producing important intrinsic invariants which are useful in the study of algebr

From the Preface: K-theory was introduced by A. Grothendieck in his formulation of the Riemann- Roch theorem. For each projective algebraic variety, Grothendieck constructed a group from the category of coherent algebraic sheaves, and showed that it had many nice properties. Atiyah and Hirzebruch considered a topological analog defined for any compact space X, a group K{X) constructed from the category of vector bundles on X. It is this ''topological K-theory" that this book will study. Topological K-theory has become an important tool in topology. Using K- theory, Adams and Atiyah were able to give a simple proof that the only spheres which can be provided with H-space structures are S1, S3 and S7. Moreover, it is possible to derive a substantial part of stable homotopy theory from K-theory. The purpose of this book is to provide advanced students and mathematicians in other fields with the fundamental material in this subject. In addition, several applications of the type described above are included. In general we have tried to make this book self-contained, beginning with elementary concepts wherever possible; however, we assume that the reader is familiar with the basic definitions of homotopy theory: homotopy classes of maps and homotopy groups.Thus this book might be regarded as a fairly self-contained introduction to a "generalized cohomology theory".

From the Introduction: "These notes are taken from a course on algebraic K-theory [given] at the University of Chicago in 1967. They also include some material from an earlier course on abelian categories, elaborating certain parts of Gabriel's thesis. The results on K-theory are mostly of a very general nature."

For fans of Hatchet and Island of the Blue Dolphins comes Theodore Taylor’s classic bestseller and Lewis Carroll Shelf Award winner, The Cay. Phillip is excited when the Germans invade the small island of Curaçao. War has always been a game to him, and he’s eager to glimpse it firsthand–until the freighter he and his mother are traveling to the United States on is torpedoed. When Phillip comes to, he is on a small raft in the middle of the sea. Besides Stew Cat, his only companion is an old West Indian, Timothy. Phillip remembers his mother’s warning about black people: “They are different, and they live differently.” But by the time the castaways arrive on a small island, Phillip’s head injury has made him blind and dependent on Timothy. “Mr. Taylor has provided an exciting story…The idea that all humanity would benefit from this special form of color blindness permeates the whole book…The result is a story with a high ethical purpose but no sermon.”—New York Times Book Review “A taut tightly compressed story of endurance and revelation…At once barbed and tender, tense and fragile—as Timothy would say, ‘outrageous good.’”—Kirkus Reviews * “Fully realized setting…artful, unobtrusive use of dialect…the representation of a hauntingly deep love, the poignancy of which is rarely achieved in children’s literature.”—School Library Journal, Starred “Starkly dramatic, believable and compelling.”—Saturday Review “A tense and moving experience in reading.”—Publishers Weekly “Eloquently underscores the intrinsic brotherhood of man.”—Booklist "This is one of the best survival stories since Robinson Crusoe."—The Washington Star · A New York Times Best Book of the Year · A School Library Journal Best Book of the Year · A Horn Book Honor Book · An American Library Association Notable Book · A Publishers Weekly Children’s Book to Remember · A Child Study Association’s Pick of Children’s Books of the Year · Jane Addams Book Award · Lewis Carroll Shelf Award · Commonwealth Club of California: Literature Award · Southern California Council on Literature for Children and Young People Award · Woodward School Annual Book Award · Friends of the Library Award, University of California at Irvine

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 K-Theory is crucial in many areas of modern mathematics, especially algebraic topology, number theory, algebraic geometry, and operator theory. This text is designed to help graduate students in other areas learn the basics of K-Theory and get a feel for its many applications. Topics include algebraic topology, homological algebra, algebraic number theory, and an introduction to cyclic homology and its interrelationship with K-Theory.

The quality of instruction is the most important factor in helping students meet the Common Core Standards. That's why Owocki's "Common Core Lesson Book" empowers teachers with a comprehensive framework for implementation that enhances existing curriculum and extends it to meet Common Core goals.

Algebraic K-theory encodes important invariants for several mathematical disciplines, spanning from geometric topology and functional analysis to number theory and algebraic geometry. As is commonly encountered, this powerful mathematical object is very hard to calculate. Apart from Quillen's calculations of finite fields and Suslin's calculation of algebraically closed fields, few complete calculations were available before the discovery of homological invariants offered by motivic cohomology and topological cyclic homology. This book covers the connection between algebraic K-theory and Bökstedt, Hsiang and Madsen's topological cyclic homology and proves that the difference between the theories are ‘locally constant’. The usefulness of this theorem stems from being more accessible for calculations than K-theory, and hence a single calculation of K-theory can be used with homological calculations to obtain a host of ‘nearby’ calculations in K-theory. For instance, Quillen's calculation of the K-theory of finite fields gives rise to Hesselholt and Madsen's calculations for local fields, and Voevodsky's calculations for the integers give insight into the diffeomorphisms of manifolds. In addition to the proof of the full integral version of the local correspondence between K-theory and topological cyclic homology, the book provides an introduction to the necessary background in algebraic K-theory and highly structured homotopy theory; collecting all necessary tools into one common framework. It relies on simplicial techniques, and contains an appendix summarizing the methods widely used in the field. The book is intended for graduate students and scientists interested in algebraic K-theory, and presupposes a basic knowledge of algebraic topology.

The theory of characteristic classes provides a meeting ground for the various disciplines of differential topology, differential and algebraic geometry, cohomology, and fiber bundle theory. As such, it is a fundamental and an essential tool in the study of differentiable manifolds. In this volume, the authors provide a thorough introduction to characteristic classes, with detailed studies of Stiefel-Whitney classes, Chern classes, Pontrjagin classes, and the Euler class. Three appendices cover the basics of cohomology theory and the differential forms approach to characteristic classes, and provide an account of Bernoulli numbers. Based on lecture notes of John Milnor, which first appeared at Princeton University in 1957 and have been widely studied by graduate students of topology ever since, this published version has been completely revised and corrected.