January 29th, 2019, 04:45 AM  #1 
Newbie Joined: Jan 2019 From: italy Posts: 6 Thanks: 0  Fundamental group
Hi everybody, I have to solve this exercise. I have to find two topological spaces $Y_1$, $Y_2$ such that $\pi (Y_1)=\pi (Y_2)=(0)$ but $\pi (Y_1 \cup Y_2)\ne (0)$. Can you help me? Last edited by skipjack; January 29th, 2019 at 06:10 AM. 
January 29th, 2019, 07:10 AM  #2 
Senior Member Joined: Aug 2017 From: United Kingdom Posts: 312 Thanks: 111 Math Focus: Number Theory, Algebraic Geometry 
Hint: you might like $Y_1 \cup Y_2$ to be a circle. It's hard to say much more without giving the whole game away.

January 29th, 2019, 11:20 AM  #3 
Newbie Joined: Jan 2019 From: italy Posts: 6 Thanks: 0 
Maybe I have an idea. $S'−{p_0}$ (where $S'$ is the circonference with radius 1 and $p_0$ is the point with coordinates (1,0) ) is homeomorphic (through the projection function) to $ \mathbb R $ so it has trivial fundamental group. The singleton ${p_0}$ has trivial fundamental group but S' has fundamental group isomorphic to Z. So I can take $Y_1=S'−{p_0}$ and $Y_2={p_0}$. Am I right?

January 29th, 2019, 12:22 PM  #4  
Senior Member Joined: Aug 2017 From: United Kingdom Posts: 312 Thanks: 111 Math Focus: Number Theory, Algebraic Geometry  Quote:
 
January 30th, 2019, 09:28 AM  #5 
Newbie Joined: Jan 2019 From: italy Posts: 6 Thanks: 0 
I meant the stereographic projection. Am I right? Thank you very much 
January 30th, 2019, 09:37 AM  #6 
Senior Member Joined: Aug 2017 From: United Kingdom Posts: 312 Thanks: 111 Math Focus: Number Theory, Algebraic Geometry  
January 30th, 2019, 10:01 AM  #7 
Newbie Joined: Jan 2019 From: italy Posts: 6 Thanks: 0 
Thanks 

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fundamental, group, topology 
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