0
$\begingroup$

Let S be a compact orientable surface. Let A and B be two subsurfaces of S that have the same signature. How to check if there is a homeomorphism of S that sends A to B and if so, find one?

Here a subsurface of $S$ is a component of S \ simple closed curves.

I saw this question Mapping-Class Groups of Subsurfaces of a Hyperbolic Surface and https://arxiv.org/pdf/math/9906122.pdf but I didn't find the part related to my question yet.

Improve this question
$\endgroup$
1
  • 1
    $\begingroup$ If you're willing to quote results about the classification of surfaces, then it's much easier to "check if there is a homeomorphism of S that sends A to B" than to "construct" that homeomorphism. So the question in the body isn't quite the same as the question in the title. $\endgroup$
    – HJRW
    Commented Mar 9, 2022 at 17:01

1 Answer 1

Reset to default
1
$\begingroup$

We are assuming that $A$ and $B$ are both connected and contain their boundary points. (If we do not assume $A$ and $B$ are connected then the problem becomes much harder.) Let $(A_i)$ be the closures of the components of $S - A$. Let $(B_j)$ be closures of the components of $S - B$. Define $(g_i, b_i, s_i)$ to be the genus of $A_i$, the size $|\partial A_i|$, and the size $|A \cap A_i|$, respectively.

We now sort the list $(A_i)$ using the complexities $(g, b, s)$. We do the same for the list $(B_j)$. Finally, there is a homeomorphism of $S$ sending $A$ to $B$ if and only if the two lists of complexities are identical.

To produce an "explicit" homeomorphism is much more work. Any algorithm will be sensitive to the details of how the data (of $S$, $A$, and $B$) are presented. One technique is to carefully order the boundary components of $A$ and $B$, compute all pairwise intersection numbers, and then perform Dehn twists to reduce the intersection numbers to zero. This is morally similar to Lee Mosher's automatic structure for the mapping class group.

Improve this answer
$\endgroup$
6
  • 1
    $\begingroup$ This follows from the classification of surfaces, and from the fact that orientation-preserving mapping classes (correctly defined) can arbitrarily permute boundary components. $\endgroup$
    – Sam Nead
    Commented Mar 9, 2022 at 22:47
  • $\begingroup$ @Osiris - The "correct" definition of mapping class in this case is "homeomorphisms up to isotopy". Thus a mapping class can permute boundary components. $\endgroup$
    – Sam Nead
    Commented Mar 11, 2022 at 15:11
  • $\begingroup$ If you don't like this definition of mapping class, then you could call it something else. What matters is having the ability to permute boundary components (of $B$, say) according to the "instructions" given by the list of complexities. $\endgroup$
    – Sam Nead
    Commented Mar 11, 2022 at 15:12
  • $\begingroup$ Regarding Lee Mosher's paper - I was thinking of Lemma "combing terminates" on page 321 of his 1995 Annal's paper. You will need to read a bit more than just that to understand the context. $\endgroup$
    – Sam Nead
    Commented Mar 11, 2022 at 15:15
  • $\begingroup$ Finally, if my answer answers your question, it would be polite to accept it (by checking the tick mark near the voting buttons). $\endgroup$
    – Sam Nead
    Commented Mar 11, 2022 at 15:17

You must log in to answer this question.