The Ultimate Stiefel-Whitney class - American History Information Guide and Reference

Stiefel-Whitney classes arise in mathematics as a type of characteristic class associated to real vector bundles $E\rightarrow X$ . They are denoted $w i (E)$ , taking values in $H^i(X,\mathbb Z_2)$ , the cohomology groups with mod $2$ coefficients. Naturally enough, we say that $w i (E)$ is the $i$ th Stiefel-Whitney class of $E$ . As an example, over the circle, $S 1$ , there is a line bundle that is topologically non-trivial: that is, the line bundle associated to the Möbius band, usually thought of as having fibres $[0,1]$ . The cohomology group

$H^1(S^1,\mathbb Z/2\mathbb Z)$

has just one element other than $0$ , this element being the first Steifel-Whitney class, $w 1$ , of that line bundle.

Contents

Axioms

Throughout, $H^i(\;\cdot\;;G)$ denotes singular cohomology with coefficient group $G$ .

For every real vector bundle $E\rightarrow X$ , there exist $w i (E)$ in $H^i(X;\mathbb Z/2\mathbb Z)$ which are natural, i.e., characteristic classes.
$w 0 (E) = 1$ in $H^0(X;\mathbb Z/2\mathbb Z)$ .
$w i (E) = 0$ whenever $i > rank(E)$ .
$w 1 (γ 1) = x$ in $H^1(\mathbb RP^1;\mathbb Z/2\mathbb Z)=\mathbb Z/2\mathbb Z$ (normalization condition). Here, $γ n$ is the canonical line bundle.
$w_k(E\oplus F)=\sum_{i+j=k}w_i(E)\cup w_j(F)$ .
If $E k$ has $s_1,\ldots,s_{\ell}$ sections which are everywhere linearly independent then $w_{k-\ell+1}=\cdots=w_k=0$ .