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Let $S_1$ and $S_2$ be two surfaces with compact boundary and the same number of boundary components. Let $M_1$ and $M_2$ be the surfaces obtained by glueing closed disks to the boundary of $S_1$ and $S_2$, respectively. Is there a direct proof that if $M_1$ and $M_2$ are homeomorphic then $S_1$ and $S_2$ are also homeomorphic? I know that this can be done in the compact case using normal forms, as in Massey's book. And if the surfaces are not compact?

Fernando Oliveira
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    One can always isotope the homemorphism such that it is identity on a disk. – Anubhav Mukherjee Feb 21 '21 at 23:39
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    You need equal number of boundary components for this to hold. – Moishe Kohan Feb 22 '21 at 00:53
  • Thanks, of course the same number of boundary components. I forgot to write it. – Fernando Oliveira Feb 22 '21 at 04:12
  • As for the isotopy idea, would you please explain more carefully how it helps to solve the problem? Thanks – Fernando Oliveira Feb 22 '21 at 04:15
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    What you need ultimately is the "unique disc lemma": if $M$ is a surface without boundary, any two embeddings $i,j: D^2 \to M$ are nearly isotopic: there is a family of homomorphisms $F_t: M \to M$ so that $F_0$ is the identity and $i F_1 = j$ or $i F_1 = jr$, where r is reflection of the disc. In particular there is an ambient isotopy taking the image of one to the image of the other. You can then extend this to embeddings of k disjoint discs. Now you recover S from M by deleting the interiors of these discs, and the isotopies above justify that the result is unique up to homeomorphism. – mme Feb 22 '21 at 11:40
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    The unique disc lemma is basically a form of the Schoenflies theorem (or more accurately a form of the annulus theorem in dimension 2). – mme Feb 22 '21 at 11:42
  • @MikeMiller Thank you. I am not a topologist, but I thought that a result like this should already exist. Would you please give me a reference for the unique disk lemma? Thanks again – Fernando Oliveira Feb 23 '21 at 14:43
  • I am sure there are many nice references but due to my own ignorance I don't know any (other than my own unpublished course notes). I can write a proof if you want; if a reference is more important I will have to punt on that. – mme Feb 23 '21 at 14:47
  • @MikeMiller I am trying to write a paper on the dynamics of homeomorphisms of surfaces and I would like to explain why Kerekjarto / Ian Richards' characterisation of homeomorphic surfaces without boundary in terms of genus, orientability and ends is also true for surfaces with compact boundary. I don't want bother you or make you waist time. A reference that I could understand with the existence of a homeomorphism of a surface that maps a collection of n disjoint to another would be perfect. If you could sketch a proof with little effort I could go through the details. Thanks – Fernando Oliveira Feb 23 '21 at 23:15
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    For a single disc, the question is answered in all dimensions here (see the "For the rest..." paragraph), as an application of the Annulus Theorem. That proof should easily generalize to a finite disjoint union of discs. – Lee Mosher Feb 26 '21 at 14:44
  • @LeeMosher I read the argument for one disk, and I think I understand the idea. As for details, I am not familiar with topology in higher dimensions. When you say "nicely embedded disk" I hope that in dimension two topological obstructions do not exist and all embedded disks are nicely embedded thanks to the Jordan - Schoenflies Theorem. Is it like this? – Fernando Oliveira Feb 28 '21 at 16:11
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    That's correct. And you don't need any higher dimensional topology to understand the application of the annulus lemma. – Lee Mosher Mar 01 '21 at 00:29
  • @Lee Mosher . I wrote a detailed proof (using your ideas) that given two collections of pairwise disjoint disks, there is a homeomorphism of the surface that maps one collection to the other. Just to fill safe. I used two facts, that I believe are correct: 1) If $C$ is a simple closed curve contained in the surface, then $C$ has a neighborhood $N$ homeomorphic to an open annulus or a Mobius band. 2) If $C$ and $D$ are two simple closed curves contained in the plane and $C$ is contained in the region bounded by $D$, then the region bounded by $C$ and $D$ is homeomorphic to an annulus. – Fernando Oliveira Mar 22 '21 at 19:46
  • @Lee Mosher. I don't really know how to prove this two facts, but believe that a proof could be written using triangulations. – Fernando Oliveira Mar 22 '21 at 19:46

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