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Math Help - Fundamental group of the circle S^1

  1. #1
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    Fundamental group of the circle S^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 .
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  2. #2
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    Quote Originally Posted by math8 View Post
    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 .
    Lemma. For loops f, g in S^1 with base point (1,0), [f]=[g] iff deg(f)=deg(g).

    If [f]=[g], then Phi([f])=deg(f)=deg(g)=Phi([g]) by lemma. Thus, Phi is well-defined.
    If Phi([f])=Phi([g]), then deg(f) = deg(g). By lemma, if deg(f)=deg(g), then [f]=[g]. Thus, Phi is one-to-one.
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