Stiefel-Whitney class

This article defines a characteristic class

Definition

The Stiefel-Whitney class is a characteristic class (or collection of characteristic classes in different dimensions) for the topological group $GL(n,\mathbb {R} )$ with coefficients mod 2. It can be defined axiomatically as follows.

To each real vector bundle $p:E\to B$, an element $w\in H^{*}(B;\mathbb {Z} _{2})$ such that if $w_{i}$ denotes the component of $w$ in $H^{i}(B;\mathbb {Z} _{2})$, we have:

1. $w_{i}(f^{*}(E))=f^{*}(w_{i}(E))$ where $f^{*}$ denotes the pullback (this is the condition for being a natural transformation, and is part of the definition of a characteristic class)
2. $w(E_{1}\oplus E_{2})=w(E_{1})\smile w(E_{2})$ where $\smile$ denotes the cap product (this is a Whitney sum formula)
3. $w_{i}(E)=0$ if $i$ is greater than the dimension of $E$

• For the canonical real line bundle $E\to \mathbb {R} P^{\infty }$, $w_{1}(E)$ is a generator of $H^{1}(\mathbb {R} P^{\infty };\mathbb {Z} _{2})$

$w$ is termed the total Stiefel Whitney-class and $w_{i}$ is termed the $i^{th}$ Stiefel-Whitney class.

Related notions

• First Stiefel-Whitney class
• Chern class
• Euler class

Facts

The ring of characteristic classes on real vector bundles with $\mathbb {Z} _{2}$ coefficients, is a polynomial ring in the Stiefel-Whitney classes.