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I want to know what techniques are known to present a diffeomorphism on a surface with boundary (the diffeomorphism is not necessarily the identity restricted to the boundary) as product of Dehn twists (of course, up to isotopy). I know there are a few books I can look at but can someone briefly describe the methods, or at least refer to exact pages of a book? Perhaps this is not easy to do in general, can you choose a diffeomorphism and show me how you present it as Dehn twists on a low genus surface?

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Did you read the Lickorish proof that mcg is generated by Dehn twists? It is very constructive. – Misha Sep 17 '13 at 3:06
As Misha said, the usual proof that the mapping class group is generated by Dehn twists (as presented in many sources, including Farb-Margalit's primer) is constructive in a straightforward way. This question is too vague to admit of an answer beyond that. How is your diffeomorphism presented to you? – Andy Putman Sep 17 '13 at 3:15
If your surface diffeomorphism is given as a product of Dehn twists, the empty procedure works. The Lickorish proof is given in terms of the action of the mapping class group on the collection of closed curves in the surface (and the homology of the surface). – Ryan Budney Sep 17 '13 at 3:21
for example, how do you calculate it for these family diffeomorphisms: take an axis (resp. a plane) such that the surface is symmetric about that, rotate the points by 180 degrees (resp. reflect) about the axis (resp. plane). – nikita Sep 17 '13 at 3:22
It seems this question is a duplicate. See the following --… – Sam Nead Sep 17 '13 at 4:27
up vote 6 down vote accepted

$\newcommand{\RR}{\mathbb{R}}$Dehn twists are orientation preserving, so you can never write a reflection as a product of Dehn twists. On the other hand writing an orientation preserving, periodic element $\phi$ as a product of Dehn twists is a non-trivial problem -- that is to say, there are mechanical ways to get the desired product, but often the number of Dehn twists required is surprisingly large! For a slick (read, comprehensible) solution we need more details about how the surface and periodic element are given.

In general, the mechanical solution is as follows.

  1. Draw a filling collection of curves on the surface, so that the collection is permuted by $\phi$. It will also be helpful to make the number of intersection points as small as possible.

  2. Now move the curves back to their original positions using Dehn twists. This is described in Lickorish's paper. To get you started - show that if a pair of curves $a$ and $b$ meet once then you can send $a$ to $b$ using first a twist on $b$ and then a twist on $a$.

Sometimes a slick solution can be found using the braid relation and its generalization, the "chain relation". In your comment, it sounds like you are drawing the surface in $\RR^3$ and the periodic element $\phi$ is induced by a rotation of $\RR^3$, so I will just point out the following nice fact, given as an exercise in Thurston's book: not every periodic element is induced by a symmetric embedding of the surface into $\RR^3$. Consider the element of order six in the mapping class group of the torus...

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