Vector bundles provide the background for the proper formulation of many classical and modern problems of differential topology, whose solutions via characteristic classes and K-theory are among the mathematical triumphs of the past few decades. These volumes are an introduction to vector bundles, characteristic classes, and K-theory and to some of their applications:
Volume 1: Foundations and Stiefel- Whitney Classes Volume
Volume 2: Euler, Chern, and Pontrjagin Classes Volume
Volume 3: K-Theory and Integrality Theorems