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Suppose we have a graph embedded on a surface $Q$ and one face $F$ of the graph is not homeomorphic to an open disk. Does there exist a closed (smooth nonselfinteresecting) curve $g$ contained in $F$ such that $g$ does not divide $Q$ into two regions?

Can I use this to prove Youngs Theorem that in any minimal genus embedding all faces are homeomorphic to open disks?

Arctic Char
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Hao S
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  • So a face is a connected component of the component of the graph $G$? – Arctic Char May 17 '24 at 08:30
  • @ArcticChar yes a face is a connected component $C$ of $Q-G$ I'll also refer to the nodes of $G$ bounding $C$ as the face. – Hao S May 17 '24 at 17:01
  • There is still very little context , and I am surprised that this has been reopened. – Peter May 26 '24 at 06:14
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    @Peter first most questions on here have very little context. Second I explained exactly why I'm interested in this question. – Hao S May 26 '24 at 06:26
  • Well, the site requires context. And if users give positive feedback to questions without enough context , this is wrong behaviour , no matter how often it happens. – Peter May 26 '24 at 06:29
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    @Peter What do you mean by context here? – Hao S May 26 '24 at 06:43

2 Answers2

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I'll assume the graph, denoted $G$, is connected, and that the surface $Q$ is closed, connected, and oriented; I believe those are the prerequisites for the definition of genus.

It is not true that such a curve exists. For example, in the torus $T=S^1 \times S^1$ consider the circle graph $G = S^1 \times \{\text{1 point}\} \subset T$. The complement $T-G$ is an open annulus, homeomorphic to $S^1 \times \{\text{an open interval}\}$, and every simple closed curve in an open annulus separates the annulus into two components.

I would suggest a different way to prove that theorem about minimal genus embeddings. If some component of $Q-G$ is not homeomorphic to an open disc then there exists a simple closed curve $C \subset Q-G$ such that $C$ is homotopic to a closed curve in $G$ (possibly not simple), but $C$ is not homotopic in $Q-G$ to a point. By doing surgery along $C$ --- cutting $Q$ along $C$ and gluing in two discs --- you get a smaller genus embedding of $G$.

Lee Mosher
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    I said that the curve does not separate the surface $Q$ into two connected components not that it doesn't separate $Q \backslash G$ into two separate components. So in this case it means the curve doesn't separate the annulus into two separate components. – Hao S May 04 '24 at 18:34
  • The latter part of what you said is more or less what I want to do. How do I show such a curve $C$ exists? – Hao S May 04 '24 at 18:54
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    You can show $G$ exists using the classification of surfaces. – Lee Mosher May 04 '24 at 18:59
  • I'm not sure what you mean (I think you mean $C$ not G btw) are you applying the classification of surfaces to $Q-G$? I'm only familiar with the classification of surfaces for closed surfaces what does it say here? – Hao S May 04 '24 at 19:06
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    Yes, I meant $C$. – Lee Mosher May 04 '24 at 19:47
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    And yes, I meant applying it to $Q-G$. Every connected surface with finitely generated fundamental group is homeomorphic to some closed surface minus some finite subset. – Lee Mosher May 04 '24 at 19:49
  • Can you provide a reference? Also I'm familiar with the classification of closed surfaces without boundary but whats the more general case with boundary? – Hao S May 04 '24 at 20:04
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    Every compact surface with boundary is homeomorphic to a closed surface minus the interiors of some finitely many pairwise disjoint embedded closed discs. – Lee Mosher May 04 '24 at 20:08
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    Here is an MSE link on the subject. – Lee Mosher May 04 '24 at 20:10
  • I can't seem to find a reference for the classification of closed orientable surfaces with boundary in your MSE link. – Hao S May 06 '24 at 05:03
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    Regarding the different method, one has to be a bit careful with the details. For instance, suppose I embed a triangle $G$ into a torus $Q$ such that it is contractible. Then, deleting a simple closed curve $C$ in $Q - G$ will result in two connected pieces. – The Amplitwist May 18 '24 at 00:53
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Does there exist a closed (smooth nonselfinteresecting) curve $g$ contained in $F$ such that $g$ does not divide $Q$ into two regions?

Yes.

Can I use this to prove Youngs Theorem that in any minimal genus embedding all faces are homeomorphic to open disks?

Yes.

See the following paper where exactly this is discussed:

As Lee Mosher suggests in his answer, one would like to perform surgery along an appropriate noncontractible curve in a face $F$. However, a naive approach to this will not always work, as I mentioned in a comment. In the above paper, the authors instead use Theorem 7.2 due to Morton Brown (Locally flat imbeddings of topological manifolds, Ann. Math. (2) 75, 331–341 (1962), Zbl 0201.56202) and the "scissors theorem" (Theorem 2.3) that describes how one can cut a surface along a graph and paste it back together, in order to prove Youngs's theorem (Theorem 7.3).

The Amplitwist
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