A proof of the braid relation that is satisfied by the Dehn twist

Question

I'm studying the mapping class group of a closed compact surface and I'm trying to prove the braid relation among the Dehn twists but the only reference I have do that with a draw. How can I prove it formally?

Answer 1

Let $a$ and $b$ be two isotopy classes of curves. Denote by $i(a, b)$ the minimal number of intersections between a representative curve in class $a$ and representative curve in class $b$. Denote by $T_a$ Dehn twist along $a$.

The following facts are proved in the book Farb Margalit "A primer on mapping class groups".

• If $i(a, b) = 0$ then $T_a T_b = T_b T_a$ (obvious)
• If $i(a,b)=1$ then $T_a T_b T_a = T_b T_a T_b$ (proposition 3.11)
• If $i(a, b) \geq 2$ then there are no relations between $T_a$ and $T_b$ (proposition 3.14)

Proposition 3.14 is a hard one. But 3.11 is quite easy. I'll sketch the proof

$$T_a T_b T_a = T_b T_a T_b \Leftrightarrow (T_a T_b) T_a (T_a T_b)^{-1} = T_b \Leftrightarrow T_{T_a T_b (a)} = T_b \Leftrightarrow T_a T_b (a) = b$$

The last assertion is indeed should be proved by a drawing. It is quite easy (by the way, it is figure 3.9 in the aforementioned book).