Fundamental group of a surface (${\pi }_1(\Sigma )$)

The fundamental group of a surface. For a closed orientable genus-$g$ surface the fundamental group of a closed orientable surface has $2g$ generators and one relation, the product of their paired commutators.

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.

For a closed orientable aspherical topological surface, every outer automorphism of its fundamental group is induced by a surface homeomorphism. The mapping class group allowing orientation reversal is therefore the full outer automorphism group. For the torus the statement is realized by linear maps in $G{L}_2(\mathbb{Z})$; for higher genus it is the Dehn-Nielsen-Baer theorem. A semidirect product by the infinite cyclic group is consequently the fundamental group of the corresponding mapping torus.