Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Generating Set for Mapping-Class Group of Genus-g Surface.

  1. Aug 17, 2011 #1
    Hi, All:

    I am sorry for such a simple request, but I need to turn-in a paper soon so I am posting here instead of Googling to try to have an authoritative answer:

    I know that the mapping -class-group of the genus-g orientable surface has a generating set of size 3g-1 (best possible), and that all the generators are Dehn twists. I know the curves about which we do the twists for g=1,2, but I am having trouble figuring out the curves for g=3 and higher. I know we use twists about a symplectic basis {x1,y1;x2,y2;...;xg,yg} , so that (xi,yj)=del_ij ; where (xi,yj) is the intersection number, but this only gives us 2g curves about which to do the twists. What other g-1 curves do we use to define the twists on? Clearly, any new curve should be "independent" (i.e., not homologous to) any of the basis curves, since twists about homologous curves cancel each other out.
    Any Ideas?

  2. jcsd
  3. Aug 17, 2011 #2
    Never mind, got it; unfortunately, I don't have a link for it, but I will paste it if
    I do get one.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook