Chapter 1 Introduction 1
Chapter 2 The Alternating Algebra 7
Chapter 3 de Rham Cohomology 15
Chapter 4 Chain Complexes and their Cohomology 25
Chapter 5 The Mayer-Vietoris Sequence 33
Chapter 6 Homotopy 39
Chapter 7 Applications of de Rham Cohomology 47
Chapter 8 Smooth Manifolds 57
Chapter 9 Differential Forms on Smoth Manifolds 65
Chapter 10 Integration on Manifolds 83
Chapter 11 Degree,Linking Numbers and Index of Vector Fields 97
Chapter 12 The Poincaré-Hopf Theorem 113
Chapter 13 PoincaréDuality 127
Chapter 14 The Complex Projective Space CPn 139
Chapter 15 Fiber Bundles and Vector Bundles 147
Chapter 16 Operations on Vector Bundles and their Sections 157
Chapter 17 Connections and Curvature 167
Chapter 18 Characteristic Classes of Complex Vector Bundles 181
Chapter 19 The Euler Class 193
Chapter 20 Cohomology of Projective and Grassmannian Bundles 199
Chapter 21 Thom Isomorphism and the General Gauss-Bonnet Formula 211
Appendix A Smooth Partition of Unity 221
Appendix B Invariant Polynomials 227
Appendix C Proof of Lemmas 12.12 and 12.13 233
Appendix D Exercises 243
References 281
Index 283