Balanced Folding Over a Polygon and Euler Numbers

Now there is a graph Kf associated to Cf in a natural way. In fact the vertices of Kf are just the n-cells of Cf, and its edges are the (n-1)-cells. If e∈Cf is (n-1)-cell, then e lies in the frontiers of exactly two n-cells σ, σ/. We then say that e is an edge in Kf with end points σ, σ/. The graph Kf can be realized as a graph  f K embedded in M as follows. For each n-cell σ, choose any point σ σ ∈  . If the n-cells σ, σ/ are end points of e, then we can join σ to σ ′  by an arc  e in M that runs from σ through σ and σ' to σ ′  , crossing e transversely at a single point. Trivially, the correspondence σ σ →  , →  e e is a graph isomorphism. Figure 1 illustrates this relationship in case n=2.


Introduction
Let K and L be cellular complexes and : → f K L a continuous map. Then f: K →L is a cellular map if (i) for each cell σ ∈ K, f (σ) is a cell in L, (ii) dim (f (σ)) ≤ dim(σ), [1].
A cellular map f: K→L is a cellular folding iff (i) for each i-cell σ i ∈K, f(σ i ) is an i-cell in L , i.e., f maps i-cells to i-cells, (ii) if σ contains n vertices, then ( ) σ f must contains n distinct vertices, [2].
A cellular folding f: K→L is neat if L n -L n-1 consists of a single n-cell, interior L. The set of all cellular folding of K into L is denoted by C(K, L) and the set of all neat foldings of K into L by Ɲ(K, L).
If f ∈ C(K, L), then x∈K is said to be a singularity of f iff f is not a local homeomorphism at x. The set of all singularities of f corresponds to the "folds" of the map. This set associates a cell decomposition C f of M. If M is a surface, then the edges and vertices of C f form a graph Γ f embedded in M [3]. Now there is a graph K f associated to C f in a natural way. In fact the vertices of K f are just the n-cells of C f , and its edges are the (n-1)-cells. If e∈C f is (n-1)-cell, then e lies in the frontiers of exactly two n-cells σ, σ/. We then say that e is an edge in K f with end points σ, σ/. The graph K f can be realized as a graph  It should be noted that the graph K f may have more than one edge joining a given pair of vertices. For instance, consider the cellular folding f of the torus into itself with the cellular subdivision shown in Figure 2. The graph K f has just two vertices but two edges, see Figure 2.

Balanced Folding
Definition 1: Let M be a compact connected surface, and P n a cell complex having n 0-cells, n 1-cells and just one 2-cell. Again a continuous map f: M→P n is called neat folding if there is a cell decomposition C f of M such that: (i) f is a cellular map of C f onto C(P n ).
is a homeomorphism of σ onto a closed cell of C(P n ).
To avoid trivial cases, we require that that each 0-cell of M is an end point of more than two 1-cells. Thus the 0-cells and 1-cells of this decomposition form a finite graph Γ f without loops (but possibly with multiple edges) and f folds M along the edges or 1-cells of Γ f [4].   Let A=σ 4 , B=σ 7 be the 2-cells shaded in Figure 3. Then there is a homeomorphism f AB : A → B given by f AB (x,y)=f(x′,y′) iff f(x,y)=f (x′,y′) where (x,y)∈A and (x,y)∈B. This homeomorphism has an extension to a homeomorphism : . This is because three 1-cells of A are interior to M, while two 1-cells of B have this property.

Example 2:
Let M be a sphere partitioned by the cells shown in Figure 4.
A cellular folding f may be defined from M to a polygon P 3 We denote the set of all balanced foldings of M into P n by ẞ(M, P n ).

Example 3:
Let M be a sphere partitioned by the cells shown in Figure 5. The valencies of the vertices of each 2-cells are 4, 6 and 8.
A neat folding f may be defined from M to a polygon P 3 . The vertices are labeled in such a way that vertices with the same image under f are labeled alike.
If we considered any 2-cells A and B of M (e.g. the shaded 2-cells in Figure 5) then, it can be checked that a homeomorphism f AB : A → B, (where A and B are the 2-cells shaded in Figure 5) can be extended to a homeomorphism of any neighborhoods , respectively. This is because the vertices of the 2-cells A and B have the same valencies. It follows that f is balanced.

The Properties of the Graph K f of Neat Folding
Let f∈ Ɲ(M, P n ), then K f has the following special features.
(a) Edge coloring: Let e 1 , e 2 ,…, e n be the 1-cells of P n , we can regard the indices i, i=1, 2,…, n "colors". Each edge of K f is mapped by f to one     (b) Sources and sinks: If A and B are 2-cells of M with a common 1-cell in their frontiers, then A and B are given opposite orientations by f. It follows that each edge of the graph K f may be oriented in such a way that every vertex is either a source or a sink (where a vertex u is a source if all the oriented edges with u as a vertex begin at u, and is a sink if all the edges end at w), see Figure 6. For such a graph, every circuit has an even number of edges (and hence of vertices).
(c) Regularity: If f∈ Ɲ(M, P n ), so that every 2-cell of M is mapped homeomorphically by f onto interior P n , then the graph K f is regular. This follows immediately from the fact that the 1-cells in the frontier of each 2-cell is 1-1 correspondence under f with those of P n . It is also worth observing that every color i occurs once in the set of colored edges at each vertex of K f . Consequently, the valency of each vertex of K f is the cardinality of the set of 1-cells of P n , that is to say, of the set of colors.
The properties of the graph K f we have already discussed suggest that in certain cases the graph K f may be a Cayley color graph. In the following we can show that this is indeed the case for a large class of balanced foldings. The existence and uniqueness of this extension are guaranteed by the fact that M is 1-connected. Now, to prove that F AB is onto, let y ∈M a non-singular point.
Then y belongs to a 2-cell σ. Let B 1 , B 2 ,…, B  We have now shown that F AB is a local homeomorphism of the simply connected manifold M onto itself. In fact, F AB is a covering map. Thus F AB is a homeomorphism.
The set of all such homeomorphisms is the required group H(f), which by its construction acts 1-transitively on the set of 2-cells. = f x y z x y z . Then f is a neat folding and the graph Γ f is a regular graph of valency 4, with 6 vertices, twelve 1-cells and eight 2-cells. The image is the positive octant P 3 where x≥0, y≥0, z≥0 see Figure 7a. Since Γ f is a regular graph, it follows that f is a balanced folding and the graph , which is a Cayley color graph, has the form given in Figure 7b. Hence H(f) is isomorphic to Z 2 ×Z 2 ×Z 2 and it acts 1-transitively on the set of eight 2-cells A 1 , A 2 ,…, A 8 .
We now explore the relationship between balanced foldings and covering maps.

General considerations
Let f∈Ɲ (M,N), where M and N are surfaces. To avoid too many complications, let us suppose that M is compact, connected and without boundary, and let N be connected.
Since M is compact the graph Γ f is a finite graph. Let Γ f divides M into k 2-cells, or faces, A 1 , A 2 ,…, A k . In this case f |A i , i=1, …, k is a homeomorphism onto the interior of N.