# Genus-g surface with Boundary and Dehn twists.

1. Jul 17, 2011

### 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?