I just read some things about van Kampen's theorem that threat this one from a different perspective than we discussed in class and this brought up a few questions:
It was said that the images of the canonical embedding $\pi_1(X_1) \rightarrow \pi_1(X_1 \cup X_2)$ and $\pi_1(X_2) \rightarrow \pi_1(X_1 \cup X_2)$ completely define the fundamental group $\pi_1(X_1 \cup X_2)$? Why is this so? Why do these two subgroups completely define $\pi_1(X_1 \cup X_2)$?
I noticed when we were threating fundamental polygons that you can easily read off wikipedia this $ABAB$ stuff and so on that you also find in this wikipedia reference. Now, I was wondering whether this is somehow related to the free product of groups that shows up in the context of van Kampen's theorem?
Just to be clear about language, the maps $\pi_1 (X_i) \to \pi_1(X_1 \cup X_2)$ ($i=1,2$) are not necessarily injections, but rather homomorphisms induced by the embeddings $ X_i \hookrightarrow X_1 \cup X_2$. As mentioned above, the images of these homomorphisms generate $\pi_1(X_1 \cup X_2)$. The kernels of these homomorphisms describe the ways in which loops in $X_1$ and loops in $X_2$ "interact".
Regarding fundamental polygons, the "words" such as $ABAB$ can be interpreted through free products of groups in some sense. A word $ABAB$ represents a relation in the group $G=\langle A,B : ABAB=1 \rangle$, i.e. the free group on letters $A$ and $B$ quotiented out by the subgroup generated by $ABAB$. This group can also be viewed as a free product of two groups, $G_A = \langle A \rangle$ and $G_B= \langle B\rangle$, amalgamated over the subgroup $\langle ABAB\rangle$. However, with fundamental polygons, these relations arise from identification of edges. When applying Van Kampen's theorem, the relations arise from inclusions of loops in $X_1$ and $X_2$ into the intersection $X_1 \cap X_2$.