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Complements of Curves in Closed Surfaces: Homeomorphic?

  1. Oct 8, 2014 #1

    WWGD

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    Hi, let ## \alpha, \gamma ## be non-isotopic curves in a compact, oriented surface S. There is a result to the effect that ## S-\alpha## is homeo. to ## S- \gamma ## . This is not true as stated; we can , e.g., remove a disk (trivial class) in a copy of S and then remove a meridian ( a generator of ## H^1 (T^2) ##). Does anyone know the corrected result? This is not true either if we only remove essential curves. It seems like if ##\alpha, \gamma## actually _were_ isotopic , then this would be true. Or maybe if they were homologous?
    Thanks.
     
  2. jcsd
  3. Oct 11, 2014 #2
    I seem to remember coming across the result you are looking for in Farb and Margalit's Primer on Mapping Class Groups, in chapter 1, if I am not mistaken. I think maybe they call it the change of coordinates principle.
     
  4. Oct 11, 2014 #3
    http://www.math.ethz.ch/~bgabi/Farb%20Magalit%20January%202011%20version.pdf [Broken]
     
    Last edited by a moderator: May 7, 2017
  5. Oct 14, 2014 #4

    WWGD

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    Thanks; in case anyone else is interested, here it is, towards the bottom of p.39:

    There is an orientation-preserving homeo. of a surface S taking one s.c.c * ## \gamma## to ## \beta## iff the
    corresponding cut surfaces ##S -\gamma ## , ##S- \beta ## are homeomorphic.

    *s.c.c: Simple-Closed Curve.
     
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