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Universal cover of figure-8

  1. Jul 7, 2011 #1
    Ok so apparently the universal cover of the figure-8 can be represented by the cayley graph of the free group on two generators, discussed in Hatcher and here http://en.wikipedia.org/wiki/Rose_%28topology%29" [Broken]

    i can see why this is a universal cover of the figure-8. but I'm having trouble understanding why it cannot be something more simple.

    for example, create a graph with one central vertex, and then four vertices surrounding it, and then connect each vertex to only the central vertex. (so you get a plus sign with vertices on the tips and one in the middle). isn't there a correct labeling on the edges of this graph to be a cover of the figure-8?

    I'm not sure if this graph would be homeomorphic to the Cayley graph...ugh fractals.
    Last edited by a moderator: May 5, 2017
  2. jcsd
  3. Jul 7, 2011 #2


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    What is the map from your plus sign to the figure 8?

    As a sanity check, have you tried lifting some test paths on the figure 8 to your plus sign? (e.g. pick a half dozen or so paths that start at the middle of the figure 8 and proceed by randomly choosing one of the four directions and winding around until it returns to the middle and doing that a few times)
  4. Jul 7, 2011 #3
    thanks..trying to lift some paths helped me see why what i was doing didn't make sense. the map i had in mind couldn't be a covering map bc there would be no evenly covered nbhd of the vertex of the figure-8. oops :-)
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