Euler characteristic of connected sums of projective planes and tori? For homework, I was told to prove the following equalities.
$$\begin{gather} \chi _{i=1}^n(\# T^2 _i)=2-2n \\ \chi _{i=1}^n(\# \mathbb RP^2 )=2-2n \end{gather}$$
First of all, the notation is strange, but I take them as:
$$\begin{gather} \chi (\#_{i=1}^n T^2 )=2-2n \\ \chi(\#_{i=1}^n \mathbb RP^2 )=2-2n \end{gather}$$
I know $\chi(M_1 \#M_2)=\chi(M_1)+\chi(M_2)-2$ and I'm trying to use this to solve my problems, but I'm getting different answers: Since $\chi(\mathbb RP^2)=1$ and $\chi(T^2)=0$ I find $$\begin{gather} \chi (\#_{i=1}^n T^2 )=2-2n \\ \chi(\#_{i=1}^n \mathbb RP^2 )=2-n \end{gather}$$What is my mistake?

The given inequality for $\Bbb R P^2$ is incorrect: it should be $2 - n$ (as you compute). Indeed, $\chi(\Bbb R P^2) = 1$, as we can quickly compute using the polygonal presentation in this answer.
