The Dehn-Nielsen-Baer Theorem for Bounded Surfaces

TL;DR

This paper extends the Dehn-Nielsen-Baer theorem to bounded surfaces, establishing an isomorphism between the mapping class group and certain automorphisms of the fundamental groupoid.

Contribution

It proves the Dehn-Nielsen-Baer theorem specifically for bounded surfaces, linking the mapping class group to boundary-fixing automorphisms of the fundamental groupoid.

Findings

Established the isomorphism for bounded surfaces.

Connected the mapping class group with automorphisms fixing boundary loops.

Extended classical theorem to new class of surfaces.

Abstract

Let $\Sigma$ be a bounded surface. We prove the Dehn-Nielsen-Baer theorem for bounded surfaces to show that the mapping class group of $\Sigma$ is isomorphic to the automorphisms of the fundamental groupoid of $\Sigma$ that fix loops around the boundary.