So Im trying to calculate the $\pi_1$ of a perforated Torus, and Im trying to do it the most detailed as posible. So I got to show that the perforated torus deformation retracts to $\partial I^2/\sim$ where $(x,0)\sim(x,1)$ and $(0,y)\sim(1,y)$. So now I want to show that $\partial I^2/\sim \ =S^1\vee S^1$ and here is where I'm having some trouble in writing this last statement.
Could some one give me a hint in how to write this as formally as possible?