Fundamental group of a once-punctured torus (${\pi }_1(T^2\setminus B^2)\cong F_2$)

Removing an embedded open disk from the torus leaves a topological surface with one boundary circle. In the square model, radial expansion away from the removed disk gives a deformation retraction onto the square boundary with opposite sides identified, a wedge of two circles. Its fundamental group is therefore the free group on two group generators.

In the standard free basis of a group $a,b$ for the fundamental group of a once-punctured torus, its oriented boundary loop represents $ab{a}^{-1}{b}^{-1}$ up to orientation and conjugation. The square model proves this: radial deformation retraction takes the inner boundary once around the four sides, whose orientations read $a,b,a^{-1},b^{-1}$. Attaching a simply connected space along that boundary kills the normal closure of the commutator of group elements, whenever the hypotheses of the Seifert-van Kampen theorem hold.