A theorem of Edmonds (see Theorem 3.1. of "Deformation of Maps to Branched Coverings in Dimension Two") 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.
Best Answer
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.)
There is an area of foundations called 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 (for example).
I imagine that this would be a huge undertaking. But I am very old, and have none of the courage of youth.