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Fundamental group of the circle S^1

  1. Mar 18, 2009 #1
    The question is to prove that the fundamental group of the circle S^1 is isomorphic to the group of integers under addition.

    So I think I should show that the following map Phi is an isomorphism.

    Phi: F(S^1, (1,0)) --> Z defined by Phi([f])= f*(1) where f* is the lifting path of f ( pof*=f) and f*(1) is the degree of f and p is the map
    p:Reals--> S^1 defined by p(t)=(cos 2 pi t, sin 2 pi t).

    I am able to show that Phi is onto, but I am having trouble showing that it is 1-1 and that it is well defined.
     
  2. jcsd
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