Abstract
In this paper, we develop the theory of cellular folding of compact connected surfaces onto polygons. The first question that naturally arises from the definition of cellular foldings are. For a given compact connected surface M and a given polygon P_n, is there any cellular folding of M onto P_n?. Also if there are, are their topological types finitely many or infinitely many?. This is the existence problem.We discuss this problem in the first part of this paper and we obtain a wide range of existence theorems for cellular foldings of a given surface onto a given polygon. Now, any simplicial folding decompose the surface into simplexes of dimensions 0, 1 and 2 which are called vertices, edges and faces respectively.In the second part of this paper, we classify all the possible simplicial foldings of the sphere, the connected sum of n-tori and the connected sum of n-projective planes onto a polygon〖 P〗_3. For each surface we obtain certain relations satisfied by the number of vertices, edges and faces of the simplicial decomposition of the surface to get either regular simplicial folding or just a simplicial folding.
Highlights
It is known that any surface is homeomorphic to exactly one of the following (factors) and
We denote by a closed orientable surface with genus and by a closed non-orientable surface with genus
The graph obtained by taking the barycentric subdivision of is the graph of cellular folding
Summary
Be a cellular folding and let be the cell decomposition of , the edges and vertices of embedded in This graph is the singular point set of for which every vertex has even. Proposition 1.3 [3]: Let Γ be a finite connected graph embedded in a surface , cellular folding if and only if Γ admits a cyclic n-colorings. Let and denote the set of equivalence classes of cellular n-foldings of an orientable surface of genus and a non-orientable surface of genus. Identify the resulting surfaces along the boundary of σ and τ in such a way that and are obtained This construction gives us the cellular folding of to , because the generated graph is n-colourable with the colouring inherited from of and
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