2005/08/05 by Basudeb Datta, A. Upadhyay, Datta, Basudeb +2
Computer Science · Mathematics · #57M20 #57N05 #57Q15 #Advanced Combinatorial Mathematics #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Topological and Geometric Data Analysis #math.AT #math.GT #msc:57M20 #msc:57N05 #msc:57Q15
paper · pdf · doi:10.48550/arxiv.math/0508106
13 pages. To appear in `Forum Mathematicum'
arxiv created 2005/08/05 · openalex publication_date 2005/08/05 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A connected combinatorial 2-manifold is called degree-regular if each of its vertices have the same degree. A connected combinatorial 2-manifold is called weakly regular if it has a vertex-transitive automorphism group. Clearly, a weakly regular combinatorial 2-manifold is degree-regular and a degree-regular combinatorial 2-manifold of Euler characteristic - 2 must contain 12 vertices. In 1982, McMullen et al. constructed a 12-vertex geometrically realized triangulation of the double-torus in \RR3. As an abstract simplicial complex, this triangulation is a weakly regular combinatorial 2-manifold. In 1999, Lutz showed that there are exactly three weakly regular orientable combinatorial 2-manifolds of Euler characteristic - 2. In this article, we classify all the orientable degree-regular combinatorial 2-manifolds of Euler characteristic - 2. There are exactly six such combinatorial 2-manifolds. This classifies all the orientable equivelar polyhedral maps of Euler characteristic - 2.