Self-dual maps on the sphere

We show how to recursively construct all self–dual maps on the sphere together with their self–dualities, and classify them according to their edge–permutations. Although several well known classes of self–dual graphs, e.g., the wheels, have been known since the last century, [7], the general characteristics of self–dual graphs have only recently begun to be explored. In [10] two constructions are given to produce examples of large minimally self–dual graphs. In [2] self–dual polyhedra are constructed and classified. Given a self–dual object, Grünbaum and Shephard [5] considered the self–dual correspondence as a permutation on the elements of the object itself, and asked if every self–dual object admitted a self–duality permuation of order 2. The question was answered negatively for polyhedra by Jendrǒl [6] and by McCanna [8] and prompted a re–examination of self–dual polyhedra, [3]. In this article, we examine the more general setting of self–dual maps on the sphere, making no assumptions of higher connectivity on the underlying graphs, allowing a clear and unified approach. 1. Automorphisms of maps on the sphere Let Γ = (V,E) be a finite connected planar graph, so there exists a tame embedding ρ of Γ into the sphere, S. We regard two such embeddings, ρ and ρ′, as equivalent if there is a homeomorphism f of S such that ρ′ = fρ. The graph Γ may have parallel edges and loops, in which case there will be several inequivalent ways to place Γ in S. On the other hand, if Γ is 3–connected, then all embeddings of Γ are equivalent up to orientation. Unless there is danger of confusion, we will hereafter suppress mention of ρ. S − Γ consists of a disjoint union of open cells whose closures in S are the faces of a realization of S as a finite CW–complex, G, called a map on the sphere, or more briefly, just a map. An isomorphism of maps will be understood to be an isomorphism of cell complexes and we note that the CW–complex arising from an embedded graph will not in general be regular. By straightforward subdivision arguments one can show the following two propositions. Proposition 1. Every non-trivial orientation preserving map automorphism σ has exactly two fixed cells. Moreover, the map can be drawn so that σ is a rotation of S. Proposition 2. Suppose σ is an orientation reversing map automorphism. If σ is the identity, and some cell is sent into itself by σ, then the map can be drawn so that σ is a reflection of S about an equator. If σ is the identity and σ fixes no cell, then the map may be drawn such that σ is the antipodal map. If σ is not the identity, then the map may be drawn so that σ is a rotatory reflection. Partially supported by NSF grant DMS-9009336. 1 2 BRIGITTE SERVATIUS AND HERMAN SERVATIUS Note that when a map is drawn, as in the above propositions, to reflect the geometry of some map automorphism, the edges cannot in general be chosen to be geodesics. Any map G determines a dual map, G∗, obtained by placing a vertex f∗ in the interior of each face f and, if two faces f and f ′ meet along an edge e, then an edge e∗ is drawn connecting f∗ and f ′∗ such that e∗ intersects G only once transversely in the interior of the edge e. Each vertex v will then lie in the interior of a face v∗ of G∗. A map G is said to be self–dual if G and G∗ are map isomorphic. A planar graph Γ is said to be self–dual if there is a map G of Γ in S such that the 2–skeleton of G∗ is isomorphic to the graph Γ. The example in Figure 1 shows that not all self– dual graphs arise in this manner. The graph and its dual are pictured. There is no