This is related to this question. In the Paper On the Mordell-Weil lattices (p. 28, Lemma 10.1) it is proved that the rank $\rho$ of the Néron-Severi lattice of a rational elliptic surface is 10. In the proof the author states that $c_2 = b_2 + 2 - 2b_1$, where $c_2$ is the topological Euler number and the $b_i$ are the Betti numbers. As far as I know the actual definition of $c_2$ would be $c_2 = b_0 - b_1 + b_2$. But this surely contradicts the result of the paper, as we would then have $\rho = b_2 = 11$, as $c_2$ = 12 and $b_0$ = 1. Where is my mistake?
I think the confusion here has to do with the difference between real and complex dimension. Your formula for the Euler characteristic is true for real surfaces, but the paper is about complex surfaces, which as real manifolds are 4-dimensional. So the Euler characteristic is $b_0-b_1+b_2-b_3+b_4$; now use $b_0=1$ and Poincaré duality.