I came to know that the statement below could be proved using *Stallings' binding tie* argument, though I have no reference article proving the statement by the binding tie argument. Can anyone help me telling something related to this? Another question, except this, does there exist any other application of binding tie argument, maybe in other dimensions?


*Given a $\pi\_1$-surjective map $f\colon S\to S'$ between two closed orientable surfaces and a smoothly embedded circle $C$ in $S$, we can homotope $f$ to make it transverse to $C$ so that $f^{-1}(C)$ is either empty or exactly one circle.*



Stallings' binding tie
Several of the papers citing Stallings paper *A topological proof of Grushko's theorem on free products* are using the binding tie argument. Here are the ones I found using a Google search. More can probably be found using MathSciNet.


1. Jaco [1968] *Constructing 3-manifolds from group homomorphisms*
2. Heil [1972] *On Kneser’s conjecture for bounded 3-manifolds*
3. Feustel [1972] *A splitting theorem for closed orientable 3-manifolds*
4. Bowditch [1999] *Connectedness properties of limit sets*
5. Bellettini, Paolini, Wang [2021] *A complete invariant for closed surfaces in the three-sphere*


Several searches failed to find the precise statement you mention. However, Lemma 3.2 in Jaco's paper comes close (he maps a surface to a graph, pulls back midpoints, and so on).


