# Description of regular covering maps between surfaces.

This is an improved and hopefully a more precise version of the question Covering spaces of surfaces.

Question: Given a regular covering map $\pi:\Sigma_g\to\Sigma_h$, where $\Sigma_n$ denotes a surface of genus $n$, is it possible to describe the covering map?

One example of such a description is the following. In the decomposition of $\Sigma_h$ into the connected sum of tori $T_1$#...#$T_h$, one torus, say $T_1$, is covered $k$-times by another torus $T_1'$, which appears in a similar decomposition of $\Sigma_g$; every other torus $T_i$ in the decomposition of $\Sigma_h$ is covered by $k$ different tori $T_{i_1}'$,...,$T_{i_k}'$ (each covering $T_i$ identically) in the decomposition of $\Sigma_g$.

Any other explicit description would likely also be useful.

A weaker version of the question would be the following: given a regular covering $\rho:\Sigma_l\to \Sigma_h$, is there another regular covering $\rho':\Sigma_g\to \Sigma_l$, such that the composition $\pi=\rho\rho'$ has a description as above (for example)?

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1. What do you mean by "describe"? If you mean "realize surface in 3-space so that the covering group action extends", then the answer is "of course not". I do not see a real question here. 2. Your weaker question has obvious negative answer: just take any $\rho$ with nonabelian finite covering group. –  Misha Oct 7 '12 at 12:17
I am interested in any kind of description, really, the simpler the better. In particular, is the description stated in the example above valid in general? –  George Oct 7 '12 at 12:54
Cut your surface $\Sigma_h$ into a $2h$-sided polygon $P$ glued in the traditional pattern $a_1b_1a_1^{-1}b_1^{-1}\ldots$. Letting the degree of the covering by $D$ (which could be infinity), take $D$ copies of $P$. Glue them up appropriately to form $\Sigma_g$. Map each copy to $P$. The covering map has been described. –  Lee Mosher Oct 7 '12 at 12:58
Not every torus is necessarily covered by another torus. If the circles that separate the toruses in the connect-sum decomposition pull back to longer circles in the cover, rather than just disjoint unions of a bunch of copies of themselves, then the covering of the torus, which you get by gluing discs onto those circles, will be a ramified covering (ramified at the center of the disc), which necessarily increases the genus of the torus. –  Will Sawin Oct 7 '12 at 16:08
George: It is a good exercise to see that in the example of a covering (call it $q$) that you gave, the covering group is abelian, see Will's comments. Therefore, no covering $\rho$, with nonabelian deck-group, appears in $q=\rho\circ \rho'$. –  Misha Oct 7 '12 at 21:37

Igor, I think you just mean that the deck-group $G$ of a regular covering $M\to N$ admits a fundamental domain $F$ in $M$ (translates $g(F), g\in G$, are the "sheets" of the covering). This is valid for arbitrary (not necessarily finite) regular coverings of arbitrary smooth/PL/topological manifolds. I guess, in the surface case you can also say that permutation representation is transitive and is given by a collection of permutations s.t. product of their commutators is $1$. Is there more to it? –  Misha Oct 7 '12 at 21:33