January 29th, 2019, 05:45 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 07:10 AM. 
January 29th, 2019, 08:10 AM  #2 
Senior Member Joined: Aug 2017 From: United Kingdom Posts: 313 Thanks: 112 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, 12:20 PM  #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, 01:22 PM  #4  
Senior Member Joined: Aug 2017 From: United Kingdom Posts: 313 Thanks: 112 Math Focus: Number Theory, Algebraic Geometry  Quote:
 
January 30th, 2019, 10:28 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, 10:37 AM  #6 
Senior Member Joined: Aug 2017 From: United Kingdom Posts: 313 Thanks: 112 Math Focus: Number Theory, Algebraic Geometry  
January 30th, 2019, 11:01 AM  #7 
Newbie Joined: Jan 2019 From: italy Posts: 6 Thanks: 0 
Thanks 

Tags 
fundamental, group, topology 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Last nontrivial group in the derived series of a solvable group  fromage  Abstract Algebra  0  December 4th, 2016 03:39 AM 
Is there a topological space with fundamental group that does not contain anything?  Hayato  Topology  3  September 8th, 2015 04:52 PM 
Group Theory Proofs, least common multiples, and group operations HELP!  msv  Abstract Algebra  1  February 19th, 2015 12:19 PM 
Fundamental Group of the Projective Plane....  TTB3  Real Analysis  2  April 19th, 2009 09:52 AM 
fundamental group, free group  mingcai6172  Real Analysis  0  March 21st, 2009 03:35 PM 