Periodic mapping classes of the genus two orientable surface - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T05:14:36Z http://mathoverflow.net/feeds/question/7746 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/7746/periodic-mapping-classes-of-the-genus-two-orientable-surface Periodic mapping classes of the genus two orientable surface ivane 2009-12-04T04:14:19Z 2009-12-17T20:31:26Z <p>Please, any information on the periodic mapping classes of the genus two orientable surface, \$O_2\$, will be greatly thanked. We had been studying the topological structure of 3d surface bundles and reintrepreting them as a circle bundles over orbifolds. </p> <p>In the <a href="http://web.archive.org/web/20070316045651/http://www.smm.org.mx/SMMP/html/modules/Publicaciones/AM/Cm/35/artExp08.pdf" rel="nofollow">http://web.archive.org/web/20070316045651/http://www.smm.org.mx/SMMP/html/modules/Publicaciones/AM/Cm/35/artExp08.pdf</a> -work you would like see the cases \$O_1\$, among \$N_1\$ and \$N_2\$, solved. Any feedback on the results and conjectures, some of them obviously false, will bring a lot of happiness :) </p> http://mathoverflow.net/questions/7746/periodic-mapping-classes-of-the-genus-two-orientable-surface/7747#7747 Answer by Ryan Budney for Periodic mapping classes of the genus two orientable surface Ryan Budney 2009-12-04T04:29:30Z 2009-12-04T04:34:47Z <p>If you want to enumerate the finite-order automorphisms (up to conjugacy) I suggest the following exercise. The associated 3-manifold is Seifert fibred. So determine how the genus 2 surface is sitting in the Seifert manifold (horizontal incompressible surface). </p> <p>This will give you a formula relating the various branch points of the monodromy to the Seifert data. Moreover, you should be able to go back-and-forth between the description of the Seifert-fibred space (unnormalized Seifert data, fibred over a genus 0, 1 or 2 surface) and the monodromy of the surface. So the classification of Seifert-fibred spaces basically gives you a dictionary for walking-through the finite-order automorphisms of a mapping class group. </p> http://mathoverflow.net/questions/7746/periodic-mapping-classes-of-the-genus-two-orientable-surface/7753#7753 Answer by Allen Hatcher for Periodic mapping classes of the genus two orientable surface Allen Hatcher 2009-12-04T05:50:52Z 2009-12-04T05:50:52Z <p>In the paper listed below there is a calculation of all the finite group actions on a genus 2 surface. There are 20 of them, with the groups ranging from order 2 to order 48. Nine of the actions are of cyclic groups, of orders 2,2,3,4,5,6,6,8,10 respectively. The paper also does the genus 3 case. The techniques are mostly algebraic. It is an interesting exercise to try to find nice geometric pictures of all the actions.</p> <p>S.A.Broughton, Classifying finite group actions on surfaces of low genus, J.Pure Appl.Alg. 69 (1991), 233-270.</p>