Let $M$ be a compact connected 3-manifold and let $S$ be a closed connected surface in $\partial M$. Let $G$ be the image of the map $\pi_1(S) \to \pi_1(M)$ induced by inclusion. I was reading the first chapter of Jaco's "Lectures on 3-Manifolds" and as a corollary to the Loop theorem, he states that $G \cong F \ast G_1 \ast \cdots \ast G_k$ where the $G_i$ are all isomorphic to fundamental groups of closed surfaces.
Let $K$ be the kernel of the map $\pi_1(S) \to \pi_1(M)$ induced by the inclusion. Using the loop theorem to prove that $K$ is normally generated by a finite pairwise disjoint collection of simple closed curves in $\pi_1(S)$. I think that I roughly understand the argument for this - except for the pairwase nonintersection.
If $K = \{1\}$, then we are all set, so assume $K \neq \{ 1 \}$, then by the loop theorem we get an embedding $f_1 : (B^2,\partial B^2) \to (M, S)$ with $[f_1(\partial B^2)] \in K$ and $[f_1(\partial B^2)] \neq 1$. Let $N_1$ be the normal subgroup generated by $[f_1(\partial B^2)]$. If $K \setminus N_1 = \emptyset$ then the claim follows so assume that it is not. Then by applying the loop theorem again, we get an embedding $f_2 : (B^2, \partial B^2) \to (M, S)$ with $[f_2(\partial B^2)] \in K \setminus N_1$. Why is it that we can make $f_1(\partial B^2) \cap f_2(\partial B^2) = \emptyset$?
I also do not understand how, given that we have shown $K$ is normally generated by a set of a pairwise nonintersecting, simple closed curves, how do I see that $G \cong F \ast G_1 \ast \cdots \ast G_k$ as claimed?
I would imagine that we claim that $\pi_1(S)/K \cong \pi_1(S')$ where $S'$ is the space obtained by attaching disks to all of the simple closed curves that normally generate $K$ (from some sort of Seifert-van Kampen argument) and then claim that any space obtained by attaching disks to a such a surface along disjoint simple closed curves is homotopy equivalent to a wedge of circles and surfaces.
