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Let $F$ be a surface of genus $g$ that is decorated with $g-$many $\alpha$ curves (in red) and $g-$many $\beta$ curves (in blue). If we like, we can take each curve to be non-separating. Now suppose that $\alpha'$ is an entirely different collection of $g-$many non-separating red curves. Is there necessarily a homeomorphism $\phi:F\to F$ which takes $\alpha$ to $\alpha'$? Is it unique?

If so, is there an algorithm to compute what the new blue curves will be? Something like cutting up the original surface by the red curves and keeping track of intersections.

Thanks

J. Moeller
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  • I suspect you want more assumptions, such as $F\setminus (\alpha_1\cup ...\cup \alpha_g)$ and $F\setminus (\alpha'_1\cup ...\cup \alpha'_g)$ are connected. Otherwise, there are obvious counter-examples. If you assume this, then such $\phi$ does exist. Also, your curves $\beta$ do not seem to play any role. – Moishe Kohan May 22 '19 at 19:10
  • @MoisheKohan the question is, what happens to the given blue curves. But this question only makes sense if there is a unique homeomorphism taking $\alpha$ to $\alpha'$. I meant to say this, but I didn't: I want the red curves to be pairwise disjoint. Same with the blue.

    In any case, I now see that this is not true. Sending $g$ curves to any other $g$ curves of the same type can be achieved by infinitely many distinct homeomorphisms.

     – J. Moeller May 22 '19 at 19:57

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