Stiefel-Whitney class

Definition
The Stiefel-Whitney class is a characteristic class (or collection of characteristic classes in different dimensions) for the topological group $$GL(n,\R)$$ 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 $$E \to \R P^\infty$$, $$w_1(E)$$ is a generator of $$H^1(\R P^\infty; \mathbb{Z}_2)$$

$$w$$ 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.