Theorem 1.7 in "Curves on 2-manifolds and isotopies" by Epstein says that "If a simple closed curve $S$ on a surface $M$ is homotopic to zero, then it bounds a disk". At one point in the proof the situation gets reduced to the case when $M$ is simply connected and has no boundary. At this point the proof says "By the Appendix we may assume $S$ is a subcomplex of some triangulation of $M$". There are theorems in the Appendix that say hat any embedding of a circle is ambient isotopic to a piecewise-linear embedding and that any homeomorphism of $M$ is isotopic to a piecewise-linear one (everything above being with fixed based point). As far as I understand from these two theorems combined one can extract a homeomorphism of $M$ which will give the needed triangulation where $S$ is a subcomplex. But it also seems that for this fact the isotopies are not needed. So is there a simpler proof of this fact?
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- $\begingroup$ It's a little hard to know the exact setting in what you are reading... but, from the assumption that $M$ is simply connected one can conclude that it is either the plane or the sphere, and then one can apply the Schönflies Theorem. $\endgroup$Lee Mosher– Lee Mosher2025-10-09 19:10:00 +00:00Commented Oct 9 at 19:10
- $\begingroup$ @LeeMosher The corollary after the theorem is the fact that any simply connected noncompact surface is a plane. So it seems at this point in the proof we have to do it without knowing that $M$ is a plane or a sphere. $\endgroup$Юрій Ярош– Юрій Ярош2025-10-09 19:32:49 +00:00Commented Oct 9 at 19:32
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