Le Schéma de carte et ses applications

Abstract

In this article, we define a new combinatorial concept, the sketch of map. Intuitively, the sketch of map of a rooted map is a map (in a generalized way) which describes how the map “rolls” itself up round the holes of the surface. Thus the sketch of map is associated with an “infinite cone” of maps, obtained from the given sketch by “enriching” itself in all the possible ways. The first part presents the topological operation of extraction of a sketch of map, its properties and the translation of this operation in terms of operators (derivation, composition) on generating series. In the second part, we apply this new topological operation to obtain new classifications and associated functional equations for rooted maps, respectively, on the projective plane, the Klein bottle and the two-hole torus.