User george - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T07:03:35Z http://mathoverflow.net/feeds/user/25711 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/104718/covering-spaces-of-surfaces Covering spaces of surfaces George 2012-08-14T20:47:42Z 2012-08-20T11:04:34Z <p>Let $\Sigma_g$ be a surface of genus $g\ge 2$, and let $\Sigma_k$ be an $m$-sheeted covering space of $\Sigma_g$. It is known that $k=m(g-1)+1$. </p> <p>An example of such a covering space is a regular covering obtained by choosing one hole" as the center of the symmetry and take $\Sigma_k$ to have $m$-fold rotational symmetry around that chosen center (as on standard pictures in your favorite topology book).</p> <p><strong>Question:</strong> Does every finite sheeted regular covering space of $\Sigma_g$ arise in this way? </p> <p>It feels like this should be known/standard, but I can't find an argument or a reference. </p> <p><strong>Edit</strong>: I agree, the question is not very precise as stated, I certainly didn't have in mind that every finite cover arises via cyclic symmetry as in the example above. The reason for the lack of precision is that I do not have a particular result in mind, but I would like to know if there is a simple geometric description of the covering <em>map</em>, as Misha points out, between two surfaces? The example with the cyclic group gives such a simple description of the covering map. Or, given any covering map $\Sigma_h\to \Sigma_g$ between two surfaces, is there some kind of a standard" covering $M\to \Sigma_g$, which factors through $\Sigma_h$?</p> http://mathoverflow.net/questions/109052/description-of-regular-covering-maps-between-surfaces Comment by George George 2012-10-07T12:54:33Z 2012-10-07T12:54:33Z 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? http://mathoverflow.net/questions/104718/covering-spaces-of-surfaces/105085#105085 Comment by George George 2012-08-23T18:00:01Z 2012-08-23T18:00:01Z Ok, so this is the closest to what I had in mind and I accept this answer, but I think the actual question I am interested in (and the whole discussion above helped me to realize this as well, so thanks) is the following: given a finite covering space $\Sigma_h \to \Sigma_g$, is there a way to describe the covering map? The example with the cyclic group gives such a description. http://mathoverflow.net/questions/104718/covering-spaces-of-surfaces Comment by George George 2012-08-14T21:26:02Z 2012-08-14T21:26:02Z I do not mean just cyclic groups, it is clear how this could work for other finite groups, and part of the question is: does it?