I have the following question:
$X$ is a topological space, $X=A\cup B$, where $A,B$ are open simply connected subspaces and $A \cap B$ contains $n\geq 1$ path connected components. Show that $\pi _1(X) \simeq F_{n-1} $ which is the free group on $n-1$ free generators.
First, its easy to see that its enough to prove for $n=1,2$ and from there you could use induction and the regular Van Kampen theorem to proceed. The case $n=1$ is clear from Van Kampen. Now about the case $n=2$ I had a problem. I tried to use the Van Kampen theorem for fundamental groupoids, and I chose to fix a couple of points: they are chosen from the 2 path connected components of $A \cap B$. I ended up not knowing how to calculate the corresponding pushout diagram.
I would be happy for an explanation on how to continue, or a solution which doesn't use groupoids, if there is one.
Thank you.