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Math Help - Disc extension (Fundamental group question)

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    Disc extension (Fundamental group question)

    Let f:S^1\rightarrow X. Show that [f]=1 \in \pi_1(X) iff f extends to the unit disc D^2

    Am I right in thinking that the fundamental group of this is trivial, so the identity map is homotopic to any "loop" f:S^1\rightarrow X? Any hints are greatly appreciated.
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  2. #2
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    Quote Originally Posted by skamoni View Post
    Let f:S^1\rightarrow X. Show that [f]=1 \in \pi_1(X) iff f extends to the unit disc D^2

    Am I right in thinking that the fundamental group of this is trivial, so the identity map is homotopic to any "loop" f:S^1\rightarrow X? Any hints are greatly appreciated.
    Denote an element of S^1 by e^{i\theta}. If [f]=1 \in \pi_1(X) then there is a homotopy h:[0,1]\times S^1\to X such that h(0,e^{i\theta}) = x_0 (a fixed point in X), and h(1,e^{i\theta}) = f(e^{i\theta}). Then the map re^{i\theta}\mapsto h(r,e^{i\theta}) is an extension of f to the unit disc.

    That construction is essentially reversible. Given an extension of f to the disc, you can use it to construct a homotopy from f to a constant map.
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