Degree-regular triangulations of the double-torus

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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 $\RR^3$. 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.
13 pages. To appear in `Forum Mathematicum'

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Algebraic Topology, Geometric Topology, 57Q15, 57M20, 57N05

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http://salesiana.dossiersoluciones.com/handle/123456789/75983

Consulte el texto completo en el siguiente enlace:

https://arxiv.org/abs/math/0508106
 http://arxiv.org/abs/math/0508106

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arXiv