Stiefel-Whitney Classes

Abstract

In Sect. 4.7.6, we encountered the Euler class $e(\xi )\in H^n(B)$ of a vector bundle $\xi$ of rank $n$ over $B$, which sometimes distinguishes between non-isomorphic vector bundles. This is an example of a characteristic class , meaning that it is natural with respect to induced bundles: $e(f^*\xi )=H^*f(e(\xi ))$ for any continuous map $f:B'\rightarrow B$. Characteristic classes play a major role in the study of vector bundles.