Table of Contents
Preface v
Chapter 1 Categories and Functors 1
1 Classes and Sets 1
2 Pseudocategories and Categories 4
3 Morphisms, Objects, and Operations 9
4 Abelian Categories 18
5 Functors and Homology 25
6 Atlases 36
Chapter 2 Sheaves and Cohomology 43
1 Presheaves on a Topological Space 43
2 Sheaves of Sets 48
3 Sheaves with Values in a Cantorian Category 54
4 Cohomology with Coefficients in Presheaves 60
5 The Case $ = A-Mod 68
6 Cohomology with Coefficients in a Sheaf 78
Chapter 3 Fiber and Vector Bundles 93
1 Fiber Bundles 93
2 Fiber Bundles with Structure Group 101
3 Vector Bundles 113
4 Operations with Vector Bundles. Characteristic Classes 123
Chapter 4 Differential Geometry 135
1 Differentiable Manifolds 135
2 Vector and Tensor Fields 146
3 Differential Forms and Integration 155
4 Absolute Differential Calculus 165
5 Riemannian and Foliated Riemannian Manifolds 174
6 Complex and Almost Complex Manifolds 188
Chapter 5 Cohomology Classes and Differential Forms 201
1 The Theorems of de Rham and Allendoerfer-Eells 201
2 Theorems of de Rham Type for Complex and Foliated Manifolds 215
3 Characteristic Classes of Differentiable Vector Bundles 230
4 Elliptic Operators. Elliptic Complexes 247
5 Cohomology and Harmonic Forms 262
References 277
Index 281
