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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- Jun 11th 2011, 09:42 AMAMAHomeomorphism
Hello,

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

Thank you. - Jun 11th 2011, 10:29 AMhatsoff
Let $\displaystyle h:X\to Y$ be continuous. Then $\displaystyle h_*:\pi_1(X,x_0)\to\pi_2(Y,y_0)$ defined by $\displaystyle h_*([f])=[h\circ f]$ is a homomorphism, where $\displaystyle \pi_1(X,x_0)$ and $\displaystyle \pi(Y,y_0)$ are the fundamental groups of $\displaystyle X$ and $\displaystyle Y$, respectively, given base points $\displaystyle x_0$ and $\displaystyle y_0$. So if $\displaystyle h$ is a homeomorphism there is a homomorphism $\displaystyle (h^{-1})_*$ which we can show is the inverse of $\displaystyle h_*$ by noting that $\displaystyle h_*\circ(h^{-1})_*$ and $\displaystyle (h^{-1})_*\circ h_*$ are the identity maps.

- Jun 11th 2011, 11:00 AMAMA
Thank you very much!