Genus-g surface with Boundary and Dehn twists.

[Bacle]

Hi, All:

Let S g,2 be the orientable genus-g surface with two boundary components, and let C be a

simple-closed curve in S g,2 .

If C is homologically non-trivial (i.e., C does not bound a subsurface of Sg,2), and C

intersects one of the boundary components

, must C also intersect the other boundary component, i.e., can a non-trivial

curve on S g,2 intersect only one of the boundary components?

The question I am trying to answer is whether Dehn twists about the boundary

curves are in the Torelli group , i.e., if these twists (twists in opposite

directions in each boundary component ) induce the identity map on homology.

If the answer is yes, the curve must go through both, then I think the Dehn

twists (both about the same curve C) in one component will cancel out

the effect of the other twist, so that the composition of these twists will have

no effect on homology.

Any Ideas?

Thanks in Advance.

Thanks.

 

[jvicens]

This is an interesting question. Let's break it down step by step.

First, let's define some terms for those who may not be familiar with them. A surface of genus g is a surface with g holes in it. For example, a sphere has genus 0, a torus has genus 1, a double torus has genus 2, and so on. The orientable genus-g surface with two boundary components is a surface with g holes and two edges that are not identified with each other.

A simple-closed curve is a curve that does not intersect itself and has no endpoints. In this case, we are talking about a curve on the surface S g,2 that starts and ends on the boundary components.

Now, to answer the main question: can a non-trivial curve on S g,2 intersect only one of the boundary components? The answer is no. This is because if C intersects one boundary component, it must also intersect the other by the definition of a simple-closed curve. Imagine drawing a curve on a piece of paper with two holes. If the curve starts and ends on one hole, it must also go through the other hole.

Now, let's talk about Dehn twists. A Dehn twist is a geometric operation that twists a curve around a fixed point on a surface. In this case, we are talking about Dehn twists about the boundary curves. The Torelli group is a group of geometric transformations that preserve the topology of a surface. So, the question is whether Dehn twists about the boundary curves are in this group.

The answer is yes, Dehn twists about the boundary curves are in the Torelli group. This is because these twists do not change the topology of the surface. As mentioned earlier, a simple-closed curve on S g,2 must intersect both boundary components. So, even if we twist the curve around one boundary component, it will still intersect the other one.

In terms of homology, the Dehn twists will cancel out each other's effects. This is because homology is a topological invariant, meaning it does not change under continuous deformations. So, the composition of these twists will have no effect on homology.

I hope this helps answer your question. If you have any further questions or need clarification, please don't hesitate to ask.