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Using the V-K Thm to find fundamental grp of sphere union disk in R3

  1. Mar 7, 2013 #1
    Hi,

    I am trying to get my head around the Van Kampen Theorem, and how this could be applied to find the fundamental group of X = the union of the unit sphere S2 in R3 and the unit disk in x-y plane? I was thinking of splitting the sphere into 3 regions - two spherical caps each having open boundary 'disk', and a spherical cap (representing an open extension of the disk in the x-y plane through the middle of the sphere).
    I think that these regions would all then be open, and the fundamental group of each is just trivial, so the the fundamental group of the whole object X is just trivial. Is this actually the case? Or is this argument somehow flawed?

    Thanks!
     
  2. jcsd
  3. Mar 7, 2013 #2

    jgens

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    Gold Member

    Based on your description I am not exactly sure which regions you are talking about, but the argument is actually quite simple. Let U = X - N and let V = X - S where N and S are the north and south poles respectively. It is easy to see that U (resp. V) deformation retracts to a union of the southern (resp. northern) hemisphere and the disc. Each of these retracts is homeomorphic to the sphere, and therefore, it follows that U and V are simply connected. Applying van Kampen's Theorem to the cover {U,V,U∩V} now shows that X has trivial fundamental group.
     
    Last edited: Mar 7, 2013
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