stiefel-whitney class

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.

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 $i>\mathrm{rank}(E)$ .
$w_1(\gamma^1)=x$ in $H^1(\mathbb RP^1;\mathbb$

Z/2\mathbb Z)=\mathbb Z/2\mathbb Z (normalization condition). Here,

\gamma^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 $w_{k-\ell+1}=\cdots=w_k=0$ .

Some work is required to show that such classes do indeed exist and are unique.

Properties

The first Stiefel-Whitney class is zero if and only if the bundle is orientable. The second Stiefel-Whitney class is zero if and only if the bundle admits a spin structure.

References

J. Milnor & J. Stasheff, Characteristic Classes, Princeton, 1974.

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