This question is rather about a convention.
Is it possible (and conventional) to asign to, say the disconnected sum of two connected surfaces $X=\Sigma_h \sqcup \Sigma_g$ a genus?
Since one has $\chi(X)=\chi(\Sigma_g)+\chi(\Sigma_h)=4-2(g+h)$, is it conventional to set $\chi(X)=:2-2(\mbox{"genus of }X\!\!")$? (in which case one would get for, say $g=h=0$, negative genus $g_X=-1$ for $X=\mathbb S^2\sqcup \mathbb S^2$).