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?



Surjections from genus $n$ surface group to free group of rank $n$
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.


