Commutativity in Mapping Class Groups I am trying to understand the behavior of finite order mapping classes for surfaces of genus g>=2. 
After fiddling for a while I started to think that no finite order mapping class commutes with any Dehn twist for g>=2. Is there a simple proof or disproof? Otherwise I'd appreciate a pointer to a reference! 
 A: The thing to know is that, if $D_\alpha$ denotes the Dehn twist along a simple loop $\alpha$, then for an orientation-preserving homeomorphism $f$ of the surface $S$ we have $f D_\alpha f^{-1}= D_{f(\alpha)}$ (up to isotopy, of course). Thus, the issue reduces to finding a reducible mapping class of finite order $\ge 2$, i.e. a finite order homeomorphism which preserves an essential simple loop on the surface. One, quite famous, example is the hyperelliptic involution $\theta$ of $S$. For instance, if $S$ is closed and oriented of genus $\le 2$ then   $\theta$ commutes with every element of the mapping class group. If you want higher order examples, take (similarly what Mike Miller suggested) an order $n$ diffeomorphism $h$ of a surface $F$  which has at least two distinct fixed points $x, y\in F$. Next, remove small $h$-invariant disks $D_x, D_y$ around these fixed points and identify the boundary components of $F'=F- (D_x\cup D_y)$ by a suitable orientation-reversing diffeomorphism of the boundary curves ("suitable" simply means that it commutes with $h$ restricted to the boundary of $F'$). The result is a surface $S$ of genus equal to 1+ genus($F$). Then $h$ yields a reducible homeomorphism $f: S\to S$ of order $n$.   
