Ramified covering of a torus

Question

Let $M\to T$ be a double covering of a torus $T=S^1\times S^1$ ramified over $n$ disjoint circles. Then what is the Euler characteristics of $M$?

Answer 1

Hint : if $\chi(A) = 0$ and $p :X \to Y$ is a $2$-fold covering with branch locus $A$ then $\chi(X) = 2 \chi(Y)$. (To prove it, just triangulate appropriately everything. )

Remark : As mentioned by Lee Mosher in the comments, this can only happens if $n=0$. Indeed, if $f : M \to N$ is a covering between surfaces ramified say along a circle $C \subset N$, and $U$ is a little neighborhood of some $c \in C$, $f^{-1}(U) \cong \{(x,y,z) : xy = 0$ in $\Bbb R^3\}$ and in particular $M$ is not a manifold. The only possibility for $n \neq 0$ is when $N$ has boundary and the ramification locus is contained in the boundary $\partial N$.