TABLE OF CONTENTS (IN A STATE OF FLUX!) I. Projective modules and vector bundles (52pp)

1. Free modules, GL_{n} and stably free modules  1
2. Projective Modules  6
3. The Picard Group of a commutative ring  15
4. Topological Vector Bundles  26
5. Algebraic Vector Bundles  38

II. The Grothendieck Group K_{0} (96pp)

1. The Group Completion of a monoid  1
2. K_{0} of a ring  4
3. K(X), KO(X) and KU(X) of a topological space 16
4. Lambda and Adams Operations  22
5. Symmetric Monoidal Categories  35
6. Abelian Categories  43
7. Exact Categories  57
8. Schemes  70
9. Waldhausen categories  80
Appendix. Localizing by categories of fractions  93

III. K_{1} and K_{2} of a ring (64pp)

1. The Whitehead Group K_{1} of a ring  1
2. Relative K_{1}  10
3. The Fundamental Theorems for K_{1} and K_{0}  14
4. Negative K-theory  22
5. Milnor’s K_{2} of a ring  27
6. K_{2} of fields  38
7. Milnor K-theory of fields  51

IV. Definitions of higher K-theory

1. Geometric realization of a small category  8
2. The BGL^{+} definition for Rings  2
3. Symmetric Monoidal Categories  17
4. Quillen’s Q-construction  26
5. The “+ = Q” Theorem  30 (TK)
6. Waldhausen’s wS. construction  33
7. The Gillet-Grayson construction  38
8. Karoubi-Villamayor K-theory  40
9. Mod ℓ K-theory  41

Typeset by AMS-TEX 2 TABLE OF CONTENTS (IN A STATE OF FLUX!)

V. The Fundamental Theorems of higher K-theory

1. The Additivity theorem  1

2. Changes in Waldhausen Structure

3. The Resolution Theorem

4. Devissage

5. Localization Sequences for G_{∗} 6.

7. Localization for K_{∗}

VI. Higher K-theory of Fields VII. Applications to Algebraic Geometry VIII. Higher Chow Groups and Motivic Cohomology References