On certain cases of Seifert Van Kampen Given two connected open sets $U,V \subset X$ such that $U \cap V$ is path connected and $U \cup V = X$, then $\pi_1(U) \ast_{\pi_1(U \cap V)} \pi_1(V) \cong \pi_1(X)$. This is of course the Seifert Van Kampen theorem, a question in Munkres asks if the homomorphism induced by the inclusion map of $i: V \rightarrow X$ is trivial what can you say about the homomorphism induced by $j: U \rightarrow X$.
It's clear to me that $j_\ast$ must be surjective and therefore $\pi_1(X) \cong \pi_1(U)/ker(j_\ast)$. My question is can you say anything more? Does the kernel of $j_\ast$ relate at all to $F$, as given in the usual diagram. Would the kernel be at all related to the normal closure of the image of the induced homomorphism of the inclusion $k: U \cap V \rightarrow U$?

Indeed, $j_*$ must be surjective.  There are certainly many possibilities for the kernel:
(a)  $j_*$ could be the zero map, for instance if $X$ is simply connected ($U$ need not be simply connected here);
(b)  $j_*$ could be injective, for instance if $V$ is a point;
(c)  or it could be somewhere in between, for instance if $X$ is a torus, $U$ a punctured torus, $V$ a disk, and $U\cap V$ a circle.  (Here $j_*$ is the "abelianization" map from $F_2$ to $Z^2$.)
There must be many other simple examples.  I would guess that Munkres is looking for "$j_*$ is surjective."
