Euler characteristic of connected sums of projective planes with k points removed? For homework, I have to compute the Euler characteristic of connected sums of $g$ projectives planes once we removed $k$ points.
I know that the Euler characteristic of a chain complex is equal to the Euler characteristic of its homology. With this, i have compute the Euler characteristic of the connected sum of $g$ projectives planes with the homology groups of the proyective plane but now I don't know exactly how I can solve the problem once I removed $k$ points to the coneccted sum. Could someone please give me some hints?
 A: Here's a fun little trick: For any manifold $M$ and any finite subset $P \subset M$,
$$\chi(M-P) = \chi(M) - \sharp P \cdot (-1)^{\text{dim}(M)}
$$
where $\sharp P$ is the cardinality of $P$ and $\text{dim}(M)$ is the dimension of $M$.
Proof: Choose a cell decomposition of $M$ such that for each $p \in P$ there exists a top dimensional cell $C_p$ of $M$ whose interior $\text{int}(C_p)$ contains $p$, and such that $p \ne q \in P \implies C_p \ne C_q$. Consider the following subcomplex of $M$:
$$N = M - \bigcup_{p \in P} \text{int}(C_p)
$$
It should be clear that $N$ has the exact same cells as $M$ in all dimensions except that $N$ has $|P|$ fewer cells in the top dimension $\text{dim}(M)$, hence
$$\chi(N) = \chi(M) - \sharp P \cdot (-1)^{\text{dim}(M)}
$$
It should also be clear that $N$ is a deformation retract of $M-P$ and hence
$$\chi(N) = \chi(M-P)
$$
QED
To apply this, perhaps you know already that the Euler characteristic of the connected sum of $g$ projective planes is $2-g$, and so removing $k$ points the resulting manifold has Euler characteristic
$$2-g-k \cdot (-1)^2 = 2-g-k
$$
