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Genus-g surface with Boundary and Dehn twists.

  1. Jul 17, 2011 #1
    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.

  2. jcsd
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