Outer Automorphisms of Mapping Class Groups of Nonorientable Surfaces

TL;DR

This paper investigates the outer automorphism groups of mapping class groups of nonorientable surfaces, revealing their structure varies with genus and connecting to automorphisms of punctured spheres.

Contribution

It characterizes the outer automorphism groups of Mod(Ng) for nonorientable surfaces, showing they are trivial, Z, or embed into a sphere with four holes depending on genus.

Findings

Outer automorphism group is trivial or Z for odd genus

Injects into mapping class group of sphere with four holes for even genus

Provides a classification based on surface genus

Abstract

Let Ng be the connected closed nonorientable surface of genus g >= 5 and Mod(Ng) denote the mapping class group of Ng. We prove that the outer automorphism group of Mod(Ng) is either trivial or Z if g is odd, and injects into the mapping class group of sphere with four holes if g is even.