I'm studying Farb and Margalit's A primer on mapping class groups and trying to understand Wajnryb's finite presentation of Mod(S). I understand that There exists a finite presentation, but I can't understand how they got explicit Wajnryb's finite presentation. More precisely, I want to know how they knew that those 5 kinds of relation generate all relations. 



About the proof of Wajnryb's finite presentation of Mod(S)
It is still an open problem to find a short and simple way to derive a finite presentation for the mapping class group. The book by Farb and Margalit (in the recent preliminary version 4.00) gives a clear sketch of the known derivations, which are rather long and complicated. For the details one must consult the original papers cited in the book. There are two main steps. The first is to construct a certain 2-dimensional cell complex that the mapping class group acts on and prove that this complex is simply-connected. This was originally done in a 1980 paper of Thurston and myself. Basic properties of this complex make it clear that writing down an actual presentation is then just a matter of carrying out some long and difficult calculations, with perhaps some ingenuity to reduce the work. The second step is then to do the work and write down a presentation. Harer carried out this work shortly thereafter, and Wajnryb greatly simplified Harer's presentation in a 1983 paper. In a later 1999 paper Wajnryb gave a more elementary, self-contained exposition of the whole story, but this is a 60-page paper. The exact references are in the Farb-Margalit book. I certainly hope that a more efficient approach is found someday.


