3-dimensional Euclidean manifolds represented by locally regular coloured graphs (Q2702749)

A map on a surface is regular if its automorphism group acts flag transitively. Less restrictive, a map on a surface is locally regular, of type \({p,q}\), if each polygonal face has exactly \(p\) edges and there are exactly \(q\) faces incident with each vertex. For example, the Klein bottle admits no regular map, but does admit a locally regular map. These two notions have natural generalizations to cell complexes of higher dimension; they are easy to define in terms of a colored graph that encodes the complex. NEWLINENEWLINENEWLINEIn the nice paper under review the author proves that every 3-dimensional Euclidean manifold except one can be represented by a locally regular colored graph. Each of the nine that can be encoded by a locally regular colored graph can be obtained by identification of the faces of a cube.