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Thread: Retraction, Fundamental group

  1. #1
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    Retraction, Fundamental group

    Hello Math experts, i need some help
    let $\displaystyle X$ be a topological space.
    Let $\displaystyle A \subset X$ and let $\displaystyle r: X \rightarrow A $ be a retraction.
    Given $\displaystyle a_0 \in A$ , show that
    $\displaystyle r*: \Pi_1(X,a_0) \rightarrow \Pi_1(A,a_0)$ is surjective

    thanks
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  2. #2
    Member HappyJoe's Avatar
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    In general, if $\displaystyle f\colon Y\rightarrow Z$ and if $\displaystyle g\colon Z\rightarrow Y$ are functions, and if $\displaystyle f\circ g = \text{id}_Z$, then g is injective, and f is surjective.

    In your scenario, let $\displaystyle i\colon A\rightarrow X$ be the inclusion. Then $\displaystyle r\circ i = \text{id}_A$, and by going to the induced maps on fundamental groups, we get

    $\displaystyle r^*\circ i^* = \text{id}_{\pi_1(A,a_0)},$

    so the map induced by r is surjective.
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