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Math Help - Algebraic topology question

  1. #1
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    Algebraic topology question

    Hey, I need some help with part 2 of this question.

    i) compute  \Pi_1(\Re P^2)
    I got  Z_2 for this one.

    here is part 2.
    ii) Describe a map  S^1 \rightarrow \Re P^2 with i* not equal to 0. Assume  \Pi_1 (S^1)=Z and A \rightarrow X is a retraction means  r \cdot i = I_{da} .

    The i is suppose to be on top of the arrow, but I'm not sure how to do that with latex.

    Thanks!
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  2. #2
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    if I understood your question correctly, you need to find a non-trivial loop in RP^2. Consider a curve c connecting the north pole pn=(0,0,1) and the south pole ps=(0,0,-1) on a sphere S^2, and let p(c) be the image of this curve by the projection p: S^2 -> RP^2 which indetifies all antipodal points. Then p(c) is a closed loop since p(pn)=p(ps). If p(c) is trivial, there is a homotopy, that is, a family of closed curves c_t with the common base point p0=p(pn), with c_0=p0 the constant map and c_1 = p(c). Lift this family of curves we get a homotopy on S^2 that continously deform the c to a pole, while keeping the ending points of c fixed. This is a contradiction.
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