A theorem of Edmonds (see Theorem 3.1. of ["Deformation of Maps to Branched Coverings in Dimension Two"](https://www.jstor.org/stable/1971246?seq=1)) says that


**Theorem 1**: *A degree-one map between closed orientable surfaces is homotopic to a pinch map (quotient map obtained from identifying a connected compact bordered sub-surface with one boundary component to a point).*


By Theorem 1, we have Theorem 2 below.


**Theorem 2**: *A degree-one map between closed orientable surfaces, when it induces an injective map between the fundamental groups, is homotopic to a homeomorphism.*


Edmonds' proof of Theorem 1 is based on the induction of the genus. Also, there is a proof of Theorem 2 **without** using induction; for example, see the first proof of Theorem 8.9. of "A Primer on Mapping Class Groups."



> 
> **Question:** Is there any proof of Theorem 1 **without** using induction on the genus?
> 
> 
> 


I am looking for a reference that has a proof of Theorem 1 in the flavor of the proof of Theorem 8.9. like as given in "A Primer on Mapping Class Groups. *My Question can be a little vague or weird. Sorry for this.*



Classification of degree one map between two closed orientable surfaces without using induction on the genus
The first proof of 8.9 in the Primer uses the genus (to prove that all pants decompositions of $S$ have the same number of curves). Proving that the genus of a surface is well-defined is, at some point, an induction.


It is possible (in my mind at least) that the second proof of Theorem 8.9, based on harmonic maps, gives a proof which more carefully hides the use of induction. (I mean, at some point there will be a proof that diffeomorphisms preserve dimension, which will rely on the pigeonhole principle, which is more-or-less an induction.)




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There is an area of foundations called [reverse mathematics](https://en.wikipedia.org/wiki/Reverse_mathematics). I have not heard of an analysis of low-dimensional topology (or more specifically the mapping class group) from this viewpoint. Perhaps a first step would be to formalise some of the most important tools (the Alexander trick, the Dehn-Nielsen-Baer theorem) in [Lean](https://en.wikipedia.org/wiki/Lean_(proof_assistant)) (for example).


I imagine that this would be a huge undertaking. But I am very old, and have none of the courage of youth.


