Unoriented Exponents of Maps
by
M. Hužvár
Banská Bystrica

In study of maps, it is often very useful to replace topological objects by their algebraic (or combinatorial) descriptions. Every map, on an orientable or unorientable surface, can be completely described in terms of three involutory permutations acting on the flags of the embedded graph. A flag can be viewed as a couple consisting of an arc and one of the two (possibly equal) faces incident to the arc. Thus we can refer to a map as a quadruple M=(F(K);\lambda, \rho, \tau), where F(K) is the set of flags, K is the underlying graph of M, and \lambda, \rho and \tau are fixed-point free involutory permutations of F(K). This 3-involutory description of a map M also allows map automorphisms to be simply defined as permutations of flags commuting with \lambda, \rho and \tau.

We introduce the unoriented exponent as a generalization of the concept of exponent defined for oriented maps by R. Nedela and M. Skoviera. An expression t=ep^j will be called an unoriented exponent of M=(F(K);\lambda, \rho, \tau) if the map M^t=(F(K);\lambda\tau^j, (\rho\tau)^e-1\rho, \tau) is isomorphic to M, for some integer e coprime with the valency of the graph K, and for some j in Z₂. Compared with the oriented exponent, the definition of unoriented exponent includes the Petrie operation that preserves both the underlying graph K and the automorphism group of M.

The unoriented exponents of a map M (reduced modulo the valency n of M) form an Abelian group Ex(M) which is a subgroup of Z_n^*×Z₂. The mapping which realizes the isomorphism M --> M^t will be called an exomorphism (an external morphism) of M associated with the exponent t. The exomorphisms of M form a group Exo(M), which is an extension of the automorphism group Aut(M). The exponent and exomorphism groups are important for regular maps since they provide a measure of their external symmetry. However, many results of this theory can be applied to arbitrary unoriented maps.

We illustrate the use of the exponent theory to the classification of regular maps by the example of canonical double covering graphs.

Date received: May 26, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafd-12.