What is the Dehn twist factorization of the hyperelliptic involution on an oriented surface of genus g (with one boundary component)?



hyperelliptic involution on a surface
This is almost covered by Proposition 4.12 of Farb and Margalit's book "The primer on mapping class groups". Another useful reference is the paper "Presentations for the punctured mapping class groups in terms of Artin groups" by Labruere and Paris. See the second displayed formula in Proposition 2.12 of that paper. 


Here is an outline of the solution. Let $S$ be a surface of genus $g$ with one boundary component. Let $a\_i$ be a "chain" of $2g$ curves in $S$. That is, $a\_i$ and $a\_{i+1}$ meet in exactly one point and otherwise the $a\_i$ are disjoint from each other. This is not done cyclically - that is, $a\_1$ and $a\_{2g}$ are disjoint. Note that the boundary of a regular neighborhood of the union of the $a\_i$ is isotopic to $\partial S$. Let's also use the symbol $a\_i$ to denote a Dehn twist about the curve $a\_i$. Consider the following product of Dehn twists:


$$H = ( a\_1 a\_2 \ldots a\_{2g - 1} a\_{2g} )^{2g + 1}$$


Claim: $H$ is the desired mapping class. Proof: $H$ fixes the curve $a\_i$ (up to isotopy) but reverses the orientation of $a\_i$. Since the chain $\cup a\_i$ fills the surface, we are done. QED


Proposition 4.12 of the primer and Proposition 2.12 of [LP] says that $H^2$ is isotopic to a full Dehn twist about $\partial S$. Since you allow isotopies that move boundary points, for you $H^2$ is isotopic to the identity. 


All of these claims can be proved by drawing a few pictures, applying induction, and using the "Alexander method". Another approach (that is less general) is to consider $S$ as a double branched cover of the $2g + 1$ times marked disk, and then to think about the generator of the center of the braid group. 


One final remark - there are many ways to choose the chain of curves $a\_i$ and this might seem like a problem... however, any two ways of choosing the chain differ by a mapping class: thus the apparent choice is no choice at all. I believe that Farb and Margalit call this the "change of coordinates principle".


