# Stiefel-Whitney class

Stiefel-Whitney classes arise in mathematics as a type of characteristic class associated to real vector bundles [itex]E\rightarrow X[itex]. They are denoted [itex]w_i(E)[itex], taking values in [itex]H^i(X,\mathbb Z_2)[itex], the cohomology groups with mod [itex]2[itex] coefficients. Naturally enough, we say that [itex]w_i(E)[itex] is the [itex]i[itex]th Stiefel-Whitney class of [itex]E[itex]. As an example, over the circle, [itex]S^1[itex], 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 [itex][0,1][itex]. The cohomology group

[itex]H^1(S^1,\mathbb Z/2\mathbb Z)[itex]

has just one element other than [itex]0[itex], this element being the first Steifel-Whitney class, [itex]w_1[itex], of that line bundle.

 Contents

## Axioms

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

1. For every real vector bundle [itex]E\rightarrow X[itex], there exist [itex]w_i(E)[itex] in [itex]H^i(X;\mathbb Z/2\mathbb Z)[itex] which are natural, i.e., characteristic classes.
2. [itex]w_0(E)=1[itex] in [itex]H^0(X;\mathbb Z/2\mathbb Z)[itex].
3. [itex]w_ i(E)=0[itex] whenever [itex]i>\mathrm{rank}(E)[itex].
4. [itex]w_1(\gamma^1)=x[itex] in [itex]H^1(\mathbb RP^1;\mathbb

Z/2\mathbb Z)=\mathbb Z/2\mathbb Z[itex] (normalization condition). Here, [itex]\gamma^n[itex] is the canonical line bundle.

1. [itex]w_k(E\oplus F)=\sum_{i+j=k}w_i(E)\cup w_j(F)[itex].
2. If [itex]E^k[itex] has [itex]s_1,\ldots,s_{\ell}[itex] sections which are everywhere linearly independent then [itex]w_{k-\ell+1}=\cdots=w_k=0[itex].

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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