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Let $ M $ be a topological $ 2 $-manifold (possibly with boundary), $ C $ an arc in the interior of $ M $ (i.e., an injective continuous function from $ [- 1,1] $ into $ \operatorname{Int}(M) $), and $ K $ a compact subset of $ M $ that is disjoint from $ \operatorname{Range}(C) $.

Problem. Prove that there exists a continuous embedding $ f: [- 2,2] \times [- 1,1] \to M $ with the following properties:

  • For all $ x \in [- 1,1] $, we have $ f(x,0) = C(x) $.
  • $ \operatorname{Range}(f) \cap K = \varnothing $.

I think that the Jordan-Schoenflies Theorem is needed to solve this problem, but I do not know how to do it. Thank you!

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    $\begingroup$ See D.B.A. Epstein, Curves on 2-manifolds and isotopies, Acta Math., 1966. $\endgroup$
    – Moishe Kohan
    Commented Aug 19, 2020 at 15:03
  • $\begingroup$ @MoisheKohan: Thanks! However, Epstein does assume that the topological $ 2 $-manifold comes with a triangulation, which I can’t. $\endgroup$
    – Transcendental
    Commented Aug 20, 2020 at 9:15
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    $\begingroup$ Every surface admits a triangulation. This is a theorem due to Rado. A standard reference is "Riemann Surfaces" by Ahlfors and Sario. $\endgroup$
    – Moishe Kohan
    Commented Aug 20, 2020 at 13:55
  • $\begingroup$ @MoisheKohan: Ah! I forgot to mention that I’m supposed to solve the problem without assuming that topological $ 2 $-manifolds (with or without boundary) can be triangulated. The reason for my post is that a similar result was used by P. H. Doyle and D. A. Moran to establish the main objective of their paper A Short Proof that Compact $ 2 $-Manifolds Can Be Triangulated, but they neglected to prove that result, merely saying that it was a standard result in geometric topology. $\endgroup$
    – Transcendental
    Commented Aug 20, 2020 at 21:14
  • $\begingroup$ I see. Then (unless they/you are doing something seriously wrong) you should not need the result about topological arcs in full generality, only for arcs contained in coordinate neighborhoods, which are already known to be homeomorphic to open planar sets in which case you can quote the planar simple arc theorem. $\endgroup$
    – Moishe Kohan
    Commented Aug 21, 2020 at 15:49

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