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Math Help - Retraction, Fundamental group

  1. #1
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    Retraction, Fundamental group

    Hello Math experts, i need some help
    let X be a topological space.
    Let A \subset X and let  r: X \rightarrow  A be a retraction.
    Given a_0 \in A , show that
    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 f\colon Y\rightarrow Z and if g\colon Z\rightarrow Y are functions, and if f\circ g = \text{id}_Z, then g is injective, and f is surjective.

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

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

    so the map induced by r is surjective.
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