Dehn–Lickorish theorem

Remarks

Dehn proved that finitely many twists generate the group, and Lickorish later gave the explicit set of $3g - 1$ curves. Humphries reduced the number to $2g + 1$, which is the smallest possible for $g \ge 2$. Idea of the proof: argue by induction on the genus with the curve complex or a connected graph of non-separating curves. Two non-separating curves can be joined by a chain of curves, each meeting the next in one point. Twists along such chains move any non-separating curve to a fixed one. This reduces the problem to the stabiliser of one curve, which is related to the mapping class group of a surface of smaller complexity.

The theorem at work: for $g = 1$ the mapping class group of the torus is $SL(2, \Z)$, acting on $H_1(T^2; \Z) = \Z^2$. The twists about the two standard curves act by $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ and $\begin{pmatrix} 1 & 0 \\ -1 & 1 \end{pmatrix}$, and these two matrices generate $SL(2, \Z)$. For $g = 0$ the mapping class group of the sphere is trivial.

Going deeper

For a student

Finite generation by twists is the starting point of most work on mapping class groups. It is the starting point for finite presentations of the group, and it lets one compute its abelianization, which is trivial for $g \ge 3$. It is the key step in the proof of the Lickorish–Wallace theorem. There a Heegaard gluing map is written as a product of twists, and each twist is realised by surgery on a curve. It is also used to show that the action on homology maps $\operatorname{Mod}(S_g)$ onto the symplectic group $Sp(2g, \Z)$.