I am interested in finding all the connected surfaces (up to homeomorphism) that can be described as $3$-fold covering spaces of the torus with a disc deleted (in both cases that the disc is closed or open).
1)How do we treat the "closed disc removed case" ie if we take a punctured torus with no boundary points? I am not aware of any classification of non-compact surfaces to help as with the next case...
2)If we delete an open disc, then the resulting surface $X$ is compact with boundary $S^1$. Therefore its $3$-fold cover will be also compact and with compact boundary. So the boundary has finitely many connected components. Because the covering is $3$-fold there are at most $3$ components and each is a connected compact $1$-manifold ie an $S^1$.
Also it has to be orientable as the base is.
So the $3$-fold cover is basically a closed orientable surface with $1 \leq m \leq 3$ punctures. I think (correct me if I am wrong here) that the classification of closed surfaces then implies that it will be $S^2$ or connected sum of some tori/projective planes with $m$ punctures. I am basically thinking of "filling in" these punctures by attaching $2$-cells along the boundary circles then using the classification theorem and removing the cells to go back to the original picture.
Also since $\chi (X)=-1$ the $3$-fold cover has Euler characteristic $-3$. So it could be:
A)torus with $3$ punctures.
B)connected sum of $2$ tori with $1$ puncture.
But I am not sure if all of my thoughts above are correct and if both the spaces above can actually be realized as a threefold covering space. Any help? Thanks!
EDIT: For 2) I also tried to think with the correspondence with groups but here $X$ deformation retracts to $S^1 \vee S^1$ which has $\pi_1 \cong \mathbb Z * \mathbb Z$ and I am unsure how to determine all index 3-subgroups here...