= Dehn-Nielsen-Baer theorem for closed orientable surfaces
{c}
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 $GL_2(\mathbb Z)$; 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>.
Back to article page
