# Stiefel Whitney Class Tautological Line Bundle

Let $\gamma$ be the tautological line bundle over $\mathbb{CP}^1$. Is it possible to compute the second Stiefel Whitney Class $w_2(\gamma)$ from the axioms, and if so, can you give a hint how?

• You do not want to use the fact that the stiefel whitney classes are the mod 2 reductions of the chern classes? – Thomas Rot Nov 23 '16 at 15:17
• @ThomasRot We haven't covered those yet (and we will), so I would rather do it without – Pepijn Nov 23 '16 at 19:24
• One idea: There is a homotopically non-trivial map $RP^2\rightarrow S^2$, by mapping everything in the one skeleton to the southpole. Maybe one can understand what happens to the tautological bundle under this map. If this pullback has a non-zero $w_2$, one knows that the original bundle has a non-zero $w_2$. – Thomas Rot Nov 24 '16 at 15:35