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Math Help - Homeomorphism

  1. #1
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    Homeomorphism

    Hello,

    How do I show that two topological spaces X and Y which are homeomorphic have same fundamental group ?


    Thank you.
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  2. #2
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    Let h:X\to Y be continuous. Then h_*:\pi_1(X,x_0)\to\pi_2(Y,y_0) defined by h_*([f])=[h\circ f] is a homomorphism, where \pi_1(X,x_0) and \pi(Y,y_0) are the fundamental groups of X and Y, respectively, given base points x_0 and y_0. So if h is a homeomorphism there is a homomorphism (h^{-1})_* which we can show is the inverse of h_* by noting that h_*\circ(h^{-1})_* and (h^{-1})_*\circ h_* are the identity maps.
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  3. #3
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    Thank you very much!
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