I am trying to compute the Stiefel-Whitney Class of the tautological line bundle, $\gamma$, over $\mathbb{CP}^1$.
Since $H^1(\mathbb{CP}^1,\mathbb Z/2)$ and $H^3(\mathbb{CP}^1,\mathbb Z/2)$ are trivial, the only possible non trivial Stiefel-Whitney Class is $\omega_2(\gamma)$.
Since $\mathbb{CP}^1\cong S^2$ and there is the quotient map from $S^1\rightarrow \mathbb{RP}^1$ it seems we might be able to relate $\gamma$ to the tautological line bundle over $\mathbb{RP}^1$. Which by the normalization axiom the tautological line bundle over $\mathbb{RP}^1$ has a nontrivial first Stiefel-Whitney Class.
Is this along the right lines and if so could you give some hints on how to continue?