# Dehn twist generators for mapping class group of a genus zero surface with boundary

Can you help me find a reference or explain how to find explicit Dehn twist generators for $MCG(S_{0,n})$, the mapping class group of a genus $0$ surface with $n$ boundary components, fixing the boundary components pointwise? (for the problem I am working on, $n=4,5$ would be sufficient.)

PS: My original post was about the mapping class group of a $n$-punctured sphere, but then I realized what I am looking for is Dehn twist generators for the mapping class group of a sphere with $n$ boundary components, so I edited the problem accordingly.

• It is not generated by Dehn twists. The reason is that Dehn twist act trivially on the puncures, so the induced permutation is the identity. In other words, they are pure braids. On the other hand, a half twist induces a transposition. – Daniele Zuddas Jan 12 '18 at 23:11
• All you want to know is contained here: math.stackexchange.com/questions/616981/… – Igor Rivin Jan 12 '18 at 23:14
• @DanieleZuddas So does that mean that Dehn' theorem, that the mapping class group of a surface is generated by Dehn twists, is only valid for closed surfaces? – braid rep Jan 13 '18 at 0:09
• I suspect I did not specify what I meant by a punctured disc. I am trying to find Dehn twist generators for a disc with $n$ holes, i.e. a disc with $n$ inner boundary components, or equivalenty a $2$-sphere with $n+1$ boundary components. – braid rep Jan 13 '18 at 0:18
• To be precise... by punctures we mean points you remove, and they can be permuted by a mapping class. On the other hand, for a compact genus-0 surface with n boundary components, the mapping class group acts trivially on the boundary, so half twists do not occur, and this mapping class group is not the braid group. – Daniele Zuddas Jan 13 '18 at 1:45