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Thread: Disc extension (Fundamental group question)

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    Disc extension (Fundamental group question)

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

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

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