# Homeomorphism

• June 11th 2011, 10:42 AM
AMA
Homeomorphism
Hello,

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

Thank you.
• June 11th 2011, 11:29 AM
hatsoff
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.
• June 11th 2011, 12:00 PM
AMA
Thank you very much!