نظریه K (چاپ بر اساس تقاضا) (کتاب‌های کلاسیک پیشرفته)

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. Cover 1 Title Page 2 Copyright Page 3 Publisher’s Foreword 8 Vita 12 Special Preface 14 Table of Contents 18 Introduction 20 CHAPTER I: Vector Bundles 22 1.1 Basic definitions 22 1.2 Operations on vector bundles 27 1.3 Sub-bundles and quotient bundles 31 1.4 Vector bundles on compact spaces 36 1.5 Additional structures 53 1.6 G-bundles over G-spaces 56 CHAPTER II: K-Theory 63 2.1 Definitions 63 2.2 Elementary properties 67 2.3 The Bott periodicity theorem 78 2.4 KG(X) 100 2.5 Computations of K*(X) for some X 100 2.6 Multiplication in K*(X,Y) 106 2.7 The Thom isomorphism 124 CHAPTER III: Operations 140 3.1 Exterior powers 140 3.2 The Adams operations 158 3.3 The group J(X) 169 APPENDIX: The space of Fredholm operators 176 REFERENCES 189 REPRINTS 190 Power operations in K-Theory 190 K-Theory and reality 220 We shall develop the theory of complex vector bundles only, though much of the elementary theory is the same for real and symplectic bundles. M.f. Atiyah ; Notes By D.w. Anderson. Reprint. Originally Published: New York : W.a. Benjamin, 1967. Includes Bibliographical References.