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Let $\Sigma_g$ be a Riemann surface of genus g, and $C$ is a cut curve of $\Sigma_g$, i.e. an oriented simple close curve.

What is a right-handed Dehn twist of $C$ of $\Sigma_g$?

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Please see the primer on mapping class groups. math.utah.edu/~margalit/primer –  Sam Nead Jun 24 '10 at 12:30

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up vote 2 down vote accepted

Cut the curve with a scalpel, going along the curve (it is oriented), rotate the right side 360 degrees, and glue it back in...

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That's correct, but it turns out that the answer depends not on the orientation of the curve, but on the orientation of the surface. In particular, "the right side" doesn't make sense unless the surface is also oriented. And it turns out that the isotopy class of the diffeomorphism doesn't depend on the orientation of the curve. This is discussed in the primer Sam Nead linked to. –  Ben Wieland Jun 24 '10 at 15:21
Agree, but it suffices to have an orientation on the normal bundle (on the curve to the surface). –  Bugs Bunny Jun 28 '10 at 10:58

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