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

In summary, the person is seeking help with a paper and needs to know the curves used for Dehn twists on a genus-g orientable surface. They know that for g=1,2 the curves are known, but are having trouble figuring out the curves for g=3 and higher. They mention using a symplectic basis and needing g-1 additional curves that are "independent" from the basis curves. They also mention that twists about homologous curves cancel each other out. They eventually figure out the solution but do not have a link to share at the moment.
  • #1
Bacle
662
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?

Thanks.
 
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  • #2
Never mind, got it; unfortunately, I don't have a link for it, but I will paste it if
I do get one.
 

1. What is a generating set for the mapping-class group of a genus-g surface?

A generating set for the mapping-class group of a genus-g surface is a set of elements that, when combined through multiplication and inverses, can generate all the elements of the mapping-class group. In other words, every element in the mapping-class group can be expressed as a product of these generating elements.

2. How many elements are typically in a generating set for the mapping-class group of a genus-g surface?

The number of elements in a generating set can vary, but it is typically proportional to the genus of the surface. For example, a genus-1 surface would have a generating set of 1 element, while a genus-2 surface would have a generating set of 2 elements.

3. What is the significance of finding a generating set for the mapping-class group of a genus-g surface?

Finding a generating set for the mapping-class group allows us to understand the structure and properties of the group. It also provides a way to classify and distinguish different surfaces based on their mapping-class groups.

4. How can a generating set for the mapping-class group be used in practical applications?

The generating set can be used in practical applications such as in cryptography, where it can be used to create secure protocols for communication. It can also be used in topological data analysis to study the shapes and structures of data sets.

5. Is there a standard or well-known generating set for the mapping-class group of a genus-g surface?

Yes, there are several well-known generating sets for the mapping-class group of a genus-g surface, including the Dehn twists, Torelli group generators, and the automorphisms of the fundamental group. However, the choice of generating set may vary depending on the specific application or context.

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