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

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 and Chern Classes  26
5. Algebraic Vector Bundles  38

II. The Grothendieck Group K_{0} (101 pp)

1. The Group Completion of a monoid  1
2. K_{0} of a ring  5
3. K(X), KO(X) and KU(X) of a topological space 17
4. Lambda and Adams Operations  24
5. K_{0} of a Symmetric Monoidal Category  37
6. K_{0} of an Abelian Category  45
7. K_{0} of an Exact Category  59
8. K_{0} of Schemes and Varieties  73
9. K_{0} of a Waldhausen category  83
Appendix. Localizing by categories of fractions  97

III. K_{1} and K_{2} of a ring (70pp)

1. The Whitehead Group K_{1} of a ring  1
2. Relative K_{1}  13
3. The Fundamental Theorems for K_{1} and K_{0}  17
4. Negative K-theory  27
5. K_{2} of a ring  33
6. K_{2} of fields  45
7. Milnor K-theory of fields  58

IV. Definitions of higher K-theory (78 pp)

1. The BGL^{+} definition for Rings  2
2. Geometric realization of a small category  15
3. Symmetric Monoidal Categories  26
4. Quillen’s Q-construction  38
5. The “+ = Q” Theorem  45
6. Waldhausen’s wS. construction  49
7. The Gillet-Grayson construction  58
8. Non-connective spectra for algebraic K-theory  61
9. Karoubi-Villamayor K-theory  64
9. Homotopy K-theory  71
10 K-theory with finite coefficients  75

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  6
3. The Resolution Theorem  12

4. Devissage

5. Localization Sequences for G_{∗} 6.

7. Localization Sequences for K-theory

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