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

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

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."

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.

These notes are based on the course of lectures I gave at Harvard in the fall of 1964. They constitute a self-contained account of vector bundles and K-theory assuming only the rudiments of point-set topology and linear algebra. One of the features of the treatment is that no use is made of ordinary homology or cohomology theory. In fact, rational cohomology is defined in terms of K-theory.The theory is taken as far as the solution of the Hopf invariant problem and a start is mode on the J-homomorphism. In addition to the lecture notes proper, two papers of mine published since 1964 have been reproduced at the end. The first, dealing with operations, is a natural supplement to the material in Chapter III. It provides an alternative approach to operations which is less slick but more fundamental than the Grothendieck method of Chapter III, and it relates operations and filtration. Actually, the lectures deal with compact spaces, not cell-complexes, and so the skeleton-filtration does not figure in the notes. The second paper provides a new approach to K-theory and so fills an obvious gap in the lecture notes.