K-theory: an introduction

Summary of the Book by Sections
Chapter I. Vector Bundles
	1. Quasi-Vector Bundles
	2. Vector Bundles
	3. Clutching Theorems
	4. Operations on Vector Bundles
	5. Sections of Vector Bundles
	6. Algebraic Properties of the Category of Vector Bundles
	7. Homotopy Theory of Vector Bundles
	8. Metrics and Forms on Vector Bundles
	9. Exercises
	10. Historical Note
Chapter II. First Notions of K-Theory
	1. The Grothendieck Group of a Category. The Group K(X)
	2. The Grothendieck Group of a Functor. The Group K(X, Y)
	3. The Group K-¹ of a Banach Category. The Group K-¹(X)
	4. The Groups K-ⁿ(X) and K-ⁿ(X, Y)
	5. Multiplicative Structures
	6. Exercises
	7. Historical Note
Chapter III. Bott Periodicity
	1. Periodicity in Complex K-Theory
	2. First Applications of Bott Periodicity Theorem in Complex K-Theory
	3. Clifford Algebras
	4. The Functors Kᵖ,q(????) and Kᵖ,q(X)
	5. The Functors Kᵖ,q(X,Y) and the Isomorphism t. Periodicity in Real K-Theory
	6. Proof of the Fundamental Theorem
	7. Exercises
	8. Historical Note
Chapter IV. Computation of Some K-Groups
	1. The Thom Isomorphism in Complex K-Theory for Complex Vector Bundles
	2. Complex K-Theory of Complex Projective Spaces and Complex Projective Bundles
	3. Complex K-Theory of Flag Bundles and Grassmann Bundles. K-Theory of a Product
	4. Complements in Clifford Algebras
	5. The Thom Isomorphism in Real and Complex K-Theory for Real Vector Bundles
	6. Real and Complex K-Theory of Real Projective Spaces and Real Projective Bundles
	7. Operations in K-Theory
	8. Exercises
	9. Historical Note
Chapter V. Some Applications of K-Theory
	1. H-Space Structures on Spheres and the Hopf Invariant
	2. The Solution of the Vector Field Problem on the Sphere
	3. Characteristic Classes and the Chern Character
	4. The Riemann-Roch Theorem and Integrality Theorems
	5. Applications of K-Theory to Stable Homotopy
	6. Historical Note
Bibliography
List of Notation
Index