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A The fundamental group of preimage of covering map

  1. Feb 29, 2016 #1
    i: B to Y is an inclusion, p: X to Y is a covering map. Define $D=p^{-1}(B)$, we assume here B and Y are locally path-connected and semi-locally simply connected. The question 1: if B,Y, X are path-connected in what case D is path-connected (dependent on the fundamental groups)? 2 What's the fundamental group of D at some point?
     
  2. jcsd
  3. Mar 1, 2016 #2

    lavinia

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    One can always find a path connected subset whose inverse image is not path connected if the covering is non-trivial.

    Take any open set whose inverse image is a collection of disjoint open sets. Such a set always exists around any point in Y. Choose a path in this set. Its inverse image will not be path connected.
     
    Last edited: Mar 1, 2016
  4. Mar 2, 2016 #3
    My intuition is telling me that, under your assumptions, if the inclusion

    i: B → Y​

    is an isomorphism on the level of fundamental groups:

    i#: π1(B,b) → π1(Y,b),​

    then the inverse image of B under the covering map

    p: X → Y,​

    namely p-1(B), will be path-connected. I don't think the proof is difficult.
     
  5. Mar 3, 2016 #4
    The above (#3) may be true, but it is definitely not the whole story. Stay tuned.
     
  6. Mar 7, 2016 #5
    Thanks
     
  7. Mar 7, 2016 #6
    Thanks
     
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