I want to calculate the fundamental group of a genus-2 surface, i.e. a double torus.
Using Van-Kampen I obtain ( with the notation generators- relations) $$\Pi_1(X,p) = < \alpha, \beta, \alpha_1, \beta_1 | \alpha \beta \alpha^{-1}\beta^{-1}, \ \alpha_1 \beta_1 \alpha_1^{-1}\beta_1^{-1}, \ \beta \beta_1^{-1} >$$
Is it correct ?