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. $w(E_1\oplus E_2)=w(E_1)⌣w(E_2)$ where $⌣$ 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 $E\to \mathbb{R}{P}^{\mathrm{\infty }}$, $w_1(E)$ is a generator of $H^1(\mathbb{R}{P}^{\mathrm{\infty }};{\mathbb{Z}}_2)$

$w$ 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.