Basic Algebraic Topology

Chapter 12 Characteristic Classes

One of the central topics of interest in topology which is common to algebraic topology, differential geometry and algebraic geometry, etc., is the characteristic classes. In this introductory chapter, we make an attempt to present some rudimentary facts about characteristic classes. For further study, the reader may consult books such as [Milnor–Stasheff, 1974], [Husemoller, 1994], etc.

In Section 12.1, we discuss the orientation of vector bundles and the Euler class and establish the relation between the Euler class and the Euler characteristic, from which the name ‘characteristic class’ is derived. In Section 12.2, we begin the study of Steifel–Whitney classes and Chern classes, simultaneously, by actually ...