2018/04/18 by Datta, Basudeb, Maity, Dipendu · 1 citation
#51M20 #52B70 #52C20 #57M60 #Combinatorics (math.CO) #FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.1804.06692
A vertex-transitive map X is a map on a surface on which the automorphism group of X acts transitively on the set of vertices of X. If the face-cycles at all the vertices in a map are of same type then the map is called a semi-equivelar map. Clearly, a vertex-transitive map is semi-equivelar. Converse of this is not true in general. In particular, there are semi-equivelar maps on the torus, on the Klein bottle and on the surfaces of Euler characteristics -1 & -2 which are not vertex-transitive. It is known that the boundaries of Platonic solids, Archimedean solids, regular prisms and antiprisms are vertex-transitive maps on \mathbbS2. Here we show that there is exactly one semi-equivelar map on \mathbbS2 which is not vertex-transitive. More precisely, we show that a semi-equivelar map on \mathbbS2 is the boundary of a Platonic solid, an Archimedean solid, a regular prism, an antiprism or the pseudorhombicuboctahedron. As a consequence, we show that all the semi-equivelar maps on \mathbbRP2 are vertex-transitive. Moreover, every semi-equivelar map on \mathbbS2 can be geometrized, i.e., every semi-equivelar map on \mathbbS2 is isomorphic to a semi-regular tiling of \mathbbS2. In the course of the proof of our main result, we present a combinatorial characterization in terms of an inequality of all the types of semi-equivelar maps on \mathbbS2. Here, we present self-contained combinatorial proofs of all our results.