$\beta_1 \gamma_1 {\beta_1}^{-1}{\gamma_1}^{-1}$ is not null-homotopic in the two-holed torus.

Question

$\pi_1(\mathbb{T}^2\#\mathbb{T}^2) \cong <\beta_1, \gamma_1, \beta_2, \gamma_2|\beta_1 \gamma_1 {\beta_1}^{-1}{\gamma_1}^{-1}\beta_2 \gamma_2 {\beta_2}^{-1}{\gamma_2}^{-1}=1>$

My question : How do I know $\beta_1 \gamma_1 {\beta_1}^{-1}{\gamma_1}^{-1}$ is not null-homotopic in the two-holed torus. Can I have some proof, hint or source site?

Answer 1

It suffices to give an example of a group $G$ with elements $a,b,c,d\in G$ such that $[a,b][c,d]=1$ but $[a,b]\not=1$. This is not hard; for instance, you can find such elements in $S_3$.