Let $E \to X$ be a vector bundle. We can associate to $E$ several invariants: among them are the *Stiefel-Whitney* classes $w\_i(E) \in H^i(X;\mathbb{Z}\_2)$. These classes may be defined using the axioms:  

0. $w\_0(E)=1$ and $w\_i(E) \in H^i(X;\mathbb{Z}\_2)$.  

1. $w(f^\*E)=f^\*w(E)$ for continuous maps $f$ (here $w=1+w\_1+w\_2+\cdots$ is the total class)  

2. $w(E \oplus F)=w(E) \cup w(F)$  

3. $w\_1(\gamma\_1) \neq 0$ for the tautological bundle $\gamma\_1$ over $\mathbb{R}P^{\infty}=BO(1)$ 


In particular, we can rephrase these axioms to recognize the *first* Stiefel Whitney class:  

1'. $w\_1(f^\*E)=f^\*w\_1(E)$ for continuous maps $f$  

2'. $w\_1(E \oplus F)=w\_1(E) + w\_1(F)$  

3'. $w\_1(\gamma\_1) \neq 0$ for a tautological bundle $\gamma\_1$ over $\mathbb{R}\mathbb{P}^{\infty}=BO(1)$.


However, in Lawson-Michelsohn's book *"Spin Geometry"* it is stated that in order for a cohomology class $v\_1$ to equal $w\_1$ we only need to check:  

1''. $v\_1(f^\*E)=f^\*v\_1(E)$ for continuous maps $f$  

2''. $v\_1(\gamma\_n) \neq 0$ for the tautological bundle $\gamma\_n$ over $BO(n)$ for every natural number $n$. 



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> How do we prove that these two sets of axioms are equivalent (and therefore characterize $w\_1$)?
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Concerning higher Stiefel-Whitney classes, does a set of axioms like 1''-2'' define $w\_k$ for each $k$? 



Two set of axioms for Stiefel-Whitney classes
The naturality condition 1'' reduces the question of whether $v\_1=w\_1$ to the special case of the tautological bundles $\gamma\_n$. In this case both $v\_1$ and $w\_1$ lie in $H^1(BO(n);{\mathbb Z}\_2)$. This group is just ${\mathbb Z}\_2$, as shown in the book of Milnor and Stasheff for example, or just from the fact that $\pi\_1BO(n)=\pi\_0O(n)={\mathbb Z}\_2$. So if $v\_1(\gamma\_n)$ is nonzero it must equal the nonzero class $w\_1(\gamma\_n)$.


This argument does not work for $w\_2$ since $H^2(BO(n);{\mathbb Z}\_2)= {\mathbb Z}\_2 \times {\mathbb Z}\_2 $ with basis $w\_2$ and $w\_1^2$, assuming $n>1$. Thus to characterize $w\_2$ one needs to know more than just that $w\_2(\gamma\_n)$ is nonzero since $w\_1^2$ also has this property, as does $w\_1^2+w\_2$. On the other hand, if one considers only oriented vector bundles then $H^2(BSO(n);{\mathbb Z}\_2)={\mathbb Z}\_2$ when $n>2$ so in this situation $w\_2$ is characterized by naturality and being nonzero for the tautological bundle over $BSO(n)$.


