Fundamental group via Van Kampen

Question

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.

Answer 1

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\}) $.

Answer 2

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