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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?
Thanks.
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?
Thanks.