Geometric realization of a triangulation on the projective plane with one face removed

A geometric realization of a map on a surface is an embedding of the map into a 3-space such that each face is a flat polygon. In my talk, we prove that every triangulation G on the projective plane has a face f such that the triangulation G−f of the Möbius band obtained from G by removing the interior of f has a geometric realization. Moreover, we show that if G is 5-connected, then G−f has a geometric realization for any face f.