Description of regular covering maps between surfaces. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T21:47:40Z http://mathoverflow.net/feeds/question/109052 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/109052/description-of-regular-covering-maps-between-surfaces Description of regular covering maps between surfaces. George 2012-10-07T11:58:40Z 2012-11-18T20:22:00Z <p>This is an improved and hopefully a more precise version of the question <a href="http://mathoverflow.net/questions/104718/covering-spaces-of-surfaces" rel="nofollow">http://mathoverflow.net/questions/104718/covering-spaces-of-surfaces</a>.</p> <p><strong>Question:</strong> 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? </p> <p>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 $\Sigma_g$. </p> <p>Any other explicit description would likely also be useful.</p> <p>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)?</p> http://mathoverflow.net/questions/109052/description-of-regular-covering-maps-between-surfaces/109093#109093 Answer by Igor Rivin for Description of regular covering maps between surfaces. Igor Rivin 2012-10-07T19:21:28Z 2012-10-07T19:21:28Z <p>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.</p>