Show these two axioms of Stiefel-Whitney class are equivalent.

Question

Here is a theorem about Stiefel-Whitney class in my teacher's notes:

For a dim-$k$ real vector bundles $E$ over $B, k \geq 0,$ there are characteristic classes $w_{i}(E) \in H^{i}(B, \mathbb{Z}/2\mathbb{Z}),$ called the Stiefel-Whitney classes. They satisfy and are uniquely determined by the following axioms:

1. $w_{0}(E)=1, w_{i}(E)=0$ for $i>k$.
2. $w_{1}(L)$ is the mod $2$ Euler class $e_{2}(L)$ for any real line bundle.
3. $f^{*} w_{i}(E)=w_{i}\left(f^{*} E\right)$.
4. $w\left(E_{1} \oplus E_{2}\right)=w\left(E_{1}\right) \cup w\left(E_{2}\right)$ or equivalently $w_{i}\left(E_{1} \oplus E_{2}\right)=\sum_{j=0}^{j=i} w_{i-j}\left(E_{1}\right) \cup w_{j}\left(E_{2}\right)$.

However, in many other references, like wikipedia, the second axiom is replaced by

Normalization: The Whitney class of the tautological line bundle over the real projective space $\mathbf {P}^1(\mathbf{R})$ is nontrivial, i.e. ${w(\gamma _{1}^{1})=1+a\in H^{*}(\mathbf {P} ^{1}(\mathbf{R});\mathbf {Z} /2\mathbf{Z} )=(\mathbf {Z} /2\mathbf {Z} )[a]/(a^{2})}$.

Could you please tell me how to show that these two axioms are equivalent? Thank you for your help!

Answer 1

That the first implies the second is just the assertion that the Euler class of the tautological line bundle is nontrivial. This is easy to see because the inclusion of projective space into its Thom space is the same as the inclusion of projective space into projective space one dimension higher.

That the second implies the first is a little trickier. It is easy to show the second implies the SW class of infinite projective space is the nontrivial element. This coincides with the Euler class in this case. By naturality of the Euler and SW class and by the universality of the tautological line bundle over infinite projective space, we calculate both of these by pulling back the same cohomology class. Thus, the two are always the same.