I have the following problem as a part of my homework:
Let $S$ be a closed surface (compact and connected). Show that for every $k$ exists a covering map of $k$ folds $p_k:S_k \rightarrow \mathbb{T}\sharp S$, where $\mathbb{T}$ is the torus and $S_k$ is the closed surface with the same orientability as $S$ and $\chi(S_k) = k \cdot \chi (\mathbb{T}\sharp S)$.
I've done this: as a first approach, suppose $S$ is the sphere $\mathbb{S}^2$. Then $S$ is orientable and $\mathbb{T}\sharp S = \mathbb{T} \sharp \mathbb{S}^2 = \mathbb{T}$, so $k \cdot \chi(\mathbb{T} \sharp S) = k \cdot \chi(\mathbb{T}) = k \cdot 0 = 0$, and $S_k$ has the same orientability as $S$ (orientable) and $\chi(S_k) = 0$, so necessarily $S_k = \mathbb{T}$. This means that we must find, for every $k$, a covering map $p_k:\mathbb{T} \rightarrow \mathbb{T}$ with $k$ folds. If we consider $\mathbb{S}^1$ as the subset $\{z \in \mathbb{C} : |z|=1\}$ of $\mathbb{C}$, then $\mathbb{S}^1 \rightarrow \mathbb{S}^1$ sending $z$ in $z^k$ is a $k$-folded covering for $S^1$, and if we write $\mathbb{T} = \mathbb{S}^1 \times \mathbb{S}^1$, we can construct the map $p_k: \mathbb{S}^1 \times \mathbb{S}^1 \rightarrow \mathbb{S}^1 \times \mathbb{S}^1$ with $p_k(z_1,z_2) = (z_1^k,z_2)$ which gives the $k$-folded covering.
Now, if $S$ is whatever, using Euler's characteristic is easy to show that $S_k$ must be the connected sum of a torus and $k$ copies of $S$, this is, $S_k = \mathbb{T} \sharp S \sharp ...^{k)} \sharp S$.
So we must find a $k$-folded covering $p_k:\mathbb{T} \sharp S \sharp ...^{k)} \sharp S \rightarrow \mathbb{T} \sharp S$. It seems the easiest way would be the following: use the $k$-folded covering of the torus we already have in the torus, and stack the $k$ copies of $S$ onto $S$ by an identity-like map, so that given $x \in S$, $p_k^{-1}(x)$ consists in $k$ copies of that point, one in each copy of $S$. But we can't make it that way directly, beacause the group homomorphism between fundamental groups induced by a covering map is injective, so we cant define the coverings for the torus and $S$ separately: if $S$ is, for instance, a torus, a covering map from $k$ glued torus into a torus would induce an injective homomorphism between $\mathbb{Z}^k$ and $\mathbb{Z}$, which is not possible.
I would like a clue in order to build the covering map. I can't use many arguments about fundamental groups or covering spaces clasification theorems. A more general version of this problem is 11-5 in Lee, "Introduction to topological manifolds", but I haven't found anything about it.
Thank you.