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. 



Images of boundary surfaces in 3-manifold groups
Here's the reference I mentioned at tea today:


[MR0732345 (85j:57011)  

Bonahon, Francis
Cobordism of automorphisms of surfaces. 
Ann. Sci. École Norm. Sup. (4) 16 (1983), no. 2, 237–270.](http://www.numdam.org/item?id=ASENS_1983_4_16_2_237_0) 


See section 2, which shows that given any irreducible manifold 
$M$, there is a compression body $V\subset M$ which has the property
that $M-V$ is incompressible. Moreover $V$ is unique up to isotopy
(following from Proposition 2.2 (d)). Hence by the loop theorem
the boundary of $M-V$ is $\pi\_1$-injective.


For any component $S$ of $\partial M$, $\pi\_1 S$ will surject the fundamental group of the component of $V$ in which it lies in the boundary. Moreover, each component has fundamental group given by a product of a free group and some surface groups, as Jaco claims. 


