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Math Help - Fundamental group of the disk is trivial

  1. #1
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    Fundamental group of the disk is trivial

    How do we show that the fundamental group of the disk D^2={(x,y) in RxR: x^2 +y^2< or eq. to 1} is trivial?

    I know how to show that the fundamental group of the circle is isomorphic to the group of the integers under addition, but for some reason, I don't see a way to show that the fundamental group of the disk is trivial.
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  2. #2
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    Quote Originally Posted by math8 View Post
    How do we show that the fundamental group of the disk D^2={(x,y) in RxR: x^2 +y^2< or eq. to 1} is trivial?

    I know how to show that the fundamental group of the circle is isomorphic to the group of the integers under addition, but for some reason, I don't see a way to show that the fundamental group of the disk is trivial.
    Lemma 1. A convex subset A in R^n is contractible to each point x_0 in A.

    D^2 is a convex subset of R^2 and is contractible to each point x_0 in D^2 by lemma 1.

    Let X be D^2 and x_0 \in D^2; let F:X \times I \rightarrow X be a contraction such that

     F(x, 0) = x, F(x, 1)= x_0, F(x_0, t)=x_0, x \in X, t \in I.

    For [a] in pi_1(X, x_0), define a homotopy H:I \times I \rightarrow X by

     H(t,s) = F(a(t), s), (t,s) \in I \times I .

    A contraction F ensures that H is a homotopy between H( . , 0) = a and H(. , 1) = c, which is a constant loop at x_0. Thus, [a] = [c]. We conclude that pi_1(X, x_0) is a trivial group.
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