# Fundamental group via Van Kampen

I want to compute the fundamental group of the set C defined below :

$A_{1}:=[0,1]²,A_2:=[-1,0]\times[0,1],C=\partial A_1 \cup \partial A_2$.

I have to use the Van Kampen theorem and so I know that I must exhibit two open sets that are arc-connected [and cover the space] but I do not see how.

• There are lots of open sets that are arc-connected. What exactly are you trying to do? Mar 9, 2015 at 13:58
• @anomaly i am triying to compute the fundamental group of C via Van Kampen theorem.
– Far
Mar 9, 2015 at 14:12

consider $U =(-1/2,1]$x$[0,1] \cap C$ and $V= [-1,1/2)$x$[0,1] \cap C$ then $U$ & $V$ is d.r to a circle...and $U\cap V$ is contractible...so fundametal group will be $\mathbb{Z*Z}$...to see this draw pictures

• Thank you. Your U and V are they open sets ? I have tried to draw a picture but it seems complicated.
– Far
Mar 9, 2015 at 14:15
• they are open in $C$...why it is complicated $C$ is basically looks like two square are gluing together with respect to an edge...can you see this??? Mar 9, 2015 at 14:17
• You are right! They are open in C.So they do not have to be open sets in $\mathbb{R}^2$?. Thank you.
– Far
Mar 9, 2015 at 14:25
• So $U\cap V = ]-1/2,1/2[\times [0,1]\cup C$ and $U\cup V=A_1 \cup A_2$ not equal to $\partial A_1 \cup \partial A_2$.Sorry i just started learning this. So i understand slowly!
– Far
Mar 9, 2015 at 14:50
• sorrry...that was a big TYPO Mar 9, 2015 at 17:24

Ok. I can take $U:= \partial A_1 \cup ([0,-1/2[\times \{0\})\cup ( [0,-1/2[\times \{1\})$ idem for V : $V:= \partial A_2 \cup ([0,1/2[\times \{0\})\cup ([0,1/2[\times \{1\})$.