Description of regular covering maps between surfaces.

2012-10-07T11:58:40Z

This is an improved and hopefully a more precise version of the question http://mathoverflow.net/questions/104718/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)?

Answer by Igor Rivin for Description of regular covering maps between surfaces.

2012-10-07T19:21:28Z

I agree with the commenters that the question is a bit vague, but the most concrete description I know of is the permutation representation of the fundamental group (permuting the "sheets" of the covering. This tends to give a rather concrete "cut and paste" description of the cover (and in addition, I believe that this description, going back to Hurrwitz, might have been historically the first way to look at covering maps). For more along these lines, check out "On extension of coverings" by M. Droste and I. Rivin, and references therein.

Description of regular covering maps between surfaces. - MathOverflow

Description of regular covering maps between surfaces.

Answer by Igor Rivin for Description of regular covering maps between surfaces.