In particular, we obtained a homeomorphism [M.sub.5]([pi]/2) [congruent to] [[summation].sub.5], where [[summation].sub.5] denotes a connected closed orientable surface
of genus 5.
For i = 0,1, 2, ..., let [g.sub.i](G) be the number of topologically distinct cellular embeddings of the graph G in the orientable surface
[S.sub.i] of genus i.
A map is a connected graph embedded in a compact connected orientable surface
in such a way that the regions delimited by the graph, called faces, are homeomorphic to open discs.
Let M be a 3-manifold which is constructed as a fiber bundle over a closed orientable surface
with fiber a circle.
Throughout this paper, S will be an orientable surface
from [R.sup.3.sub.1] and n its unit normal vector field.
Certainly, on an orientable surface
, an equivalence preserve the orientation on this surface.
The proof in the general case is essentially the same, we glue [[summation].sub.1], ..., [[summation].sub.k] cylinders to the surface [M.sub.1] to obtain an orientable surface
that is homeomorphic to M.
Henceforth, by surface we shall always mean an orientable surface
, equivalently, one which is embeddable in [R.sup.3].
A map M is an embedding of a connected graph G, with possibly multi-edges or loops, into a closed, connected and orientable surface
S such that all faces, i.e.
This graph is embedded in a locally orientable surface
such that if we cut the graph from the surface, the remaining part consists of connected components called faces or cells, each homeomorphic to an open disk.
A unicellular map is a graph embedded on a compact orientable surface
, in such a way that its complement is a topological polygon.
Bender and Canfield ([BC86]) obtained the asymptotic number of maps on a given orientable surface
. Gao ([Gao93]) obtained formulas for the asymptotic number of 21-angulations on orientable surfaces
, and conjectured a formula for more general families (namely maps where the degrees of the faces are restricted to lie in a given finite subset of 2N).