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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 surface), 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?

Edit: 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.

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    By the twists cancelling each other out, I mean that the effect of one after the other (in opposite directions to each other)on a non-trivial curve would leave the curve unchanged.2011-07-18

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You are talking about a bounding pair map, which is one of the standard generators for the Torelli group. These do indeed induce the trivial map on homology. To see this, consider a curve $\gamma$ representing a homology class. If this doesn't hit either of the twisting curves, then obviously this gets sent to itself by the BP map. On the other hand, if the curve does intersect, it must do so algebraically trivially. Basically, once it enters the region bounded by the pair, it must leave it, so that the intersections come in canceling pairs. However, these could be distributed between the two boundary components, but I claim that's irrelevant for the proof. Note that each of the two twist curves for the BP map represent the same homology class $\alpha$. Then $\gamma\mapsto \gamma+(\gamma\cdot \alpha) \alpha$, where $\gamma\cdot\alpha$ is the algebraic intersection number between the two curves. Since this is zero, $\gamma\mapsto\gamma$.

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    Well, I'd like to understand some basics like: why, if C1~C2, (and neither of C1,C2 is trivial) then any non-trivial C3 must have algebraic intersection number 0 with $C1\cup C2$, and why, if C1'~/C2' , then a non-trivial curve may have algebraic intersection number +1. I am thinking these results are independent of the choice of basis, so that we could choose, e.g., a symplectic basis for $S_1$ (the torus), and then I can see this is true, but I would like to see an actual argument explaining this.2011-07-18