Let $\Sigma_n$ be a genus $n$ surface, let $\mathcal{H}_n$ be a genus $n$ handle body, and let $F_n$ be a free group of rank $n$. Fix an identification of $\pi_1(\mathcal{H}_n)$ with $F_n$. I know several proofs of the following result:
Theorem: Let $\phi\colon \pi_1(\Sigma_n) \rightarrow F_n$ be a surjection. Then there exists an orientation-preserving homeomorphism $\psi\colon \Sigma_n \rightarrow \partial \mathcal{H}_n$ such that $\phi$ factors as
$$\pi_1(\Sigma_n) \stackrel{\psi_{\ast}}{\longrightarrow} \pi_1(\partial \mathcal{H}_n) \longrightarrow \pi_1(\mathcal{H}_n) = F_n.$$
However, I do not know any references for it, nor who to attribute it to. Does anyone know any references, preferably the original one?
Best Answer
I think the first formal proof is due to Zieschang, Stallings probably knew it:
https://mathscinet.ams.org/mathscinet/search/publdoc.html?pg1=INDI&s1=187195&sort=Newest&vfpref=html&r=101&mx-pid=161901
There is a discussion at the end of the paper that refers to a correspondence with Lyndon. There it is mentioned that the claim is implicit in Satz 2.