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Let $F$ be a compact, oriented surface. A Dehn move, $D_\beta$, on a simple closed curve (scc) $\alpha$ is a Dehn twist applied to $\alpha$ along a scc $\beta$ which intersects $\alpha$ once.

Is there a finite collection of scc's in $F$ such that every non-separating scc $\alpha$ in $F$ can be reduced to one of them by Dehn moves $D_\beta$ as above?

Can such reduction be made by $D_\beta^2$-moves?

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    $\begingroup$ It's been a while since I've thought about this topic, but isn't this question more or less implicitly answered in Lickorish's paper on the generators of the mapping class group? $\endgroup$ – Ryan Budney Mar 15 at 18:46
  • $\begingroup$ I think I misread your question, although still not sure I understand it. You ask if there is a finite collection F st every α∈F can be reduced to one of them? What does that mean? α is already in F $\endgroup$ – Paul Plummer Mar 15 at 23:21
  • $\begingroup$ @Paul: First of all, F is my surface :-). I edited the question slightly, which perhaps will help. $\endgroup$ – Adam Mar 16 at 0:10
  • $\begingroup$ Okay, that is why I misinterpreted it(skipped the first part and thought it was the $F$inite collection). $F$ is a strange name for a surface :) $\endgroup$ – Paul Plummer Mar 16 at 0:20
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    $\begingroup$ @PaulPlummer: $F$ is a perfectly natural name for a surface if one's native language is German. $\endgroup$ – Misha Mar 16 at 14:32

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