Geometric Realization of Möbius Triangulations

Abstract

A Mobius triangulation is a triangulation on the Mobius band. A geometric realization of a map $M$ on a surface $\Sigma$ is an embedding of $\Sigma$ into a Euclidean 3-space $\mathbb{R}^3$ such that each face of $M$ is a flat polygon. In this paper, we shall prove that every 5-connected triangulation on the Mobius band has a geometric realization. In order to prove it, we prove that if $G$ is a 5-connected triangulation on the projective plane, then for any face $f$ of $G$, the Mobius triangulation $G-f$ obtained from $G$ by removing the interior of $f$ has a geometric realization.