Let us consider the complex line bundle $\xi$ over $\mathbb{CP}^2$ which is completely defined by its restriction on a complex projective line; this restriction is denoted by $\xi^{\prime}$ and the Euler characteristic of $\xi^{\prime}$ we denote by $e(\xi)$.
Now consider the bundle $\xi \oplus 1_{\mathbb{R}}$ over $\mathbb{CP}^2$; we are interested in the corresponding $SO(3)$-bundle that also is denoted by $\xi \oplus 1_{\mathbb{R}}$. The value of the Stiefel–Whitney class $w(\xi \oplus 1_{\mathbb{R}})$ on a projective line equals $e(\xi)\, \text{mod}\, 2$.
Assume that $e(\xi)$ is even; there is the class in $H^4(\mathbb{CP}^2)$ such that this class is the complete obstruction to the existence of a section of $\xi \oplus 1_{\mathbb{R}}$. The value of this class on $[\mathbb{CP}^2]$ we denote by $k(\xi)$.
What is the connection between the numbers $k(\xi)$ and $e(\xi)$?