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.

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.