Flag Bicolorings, Pseudo-Orientations, and Double Covers of Maps

This paper discusses consistent flag bicolorings of maps and maniplexes, in their own right and as generalizations of orientations and pseudo-orientations. Furthermore, a related doubling concept is introduced, and relationships between these ideas are explored.


Introduction
The main goal of this paper is to develop a general theory of flag bicolorings and the related concepts of coverings and pseudo-orientations.
The idea of consistent colorings of the flags of a map with two colors has appeared previously in literature in different contexts. For instance, when the automorphism group of a map which is a polytope has two orbits on the flags, we may color the flags with two colors in such a way that flags in different orbits have different colors. These are called 2-orbit maps in [5], but this investigation requires the extra property that the maps be abstract. Specific k-colorings of flags are equivalent to the concept of T -compatible maps introduced in [9], where T is a class of k-orbit maps. This concept, generalized to hypermaps is called T -conservative in [1]. In all these instances, the colorings are used as a tool to work with automorphisms of maps.
Although some of the motivations behind the bicoloring of flags come from maps admitting symmetries, the ideas can be applied to general maps. In this work we shall not make assumptions on the automorphism groups of the maps in consideration.
Maps admitting a bicoloring of flags in which adjacent flags have different colors are precisely maps on orientable surfaces (see Proposition 5.1 ). Although the other types of bicolorings of flags that we consider in this paper do not have a known topological equivalence, some results on orientability do translate directly to analogous results on bicolorability. In this sense we may think of bicolorings of flags as a generalization of orientability of maps.
The paper is organized as follows. Sections 2, 3, 4 introduce the concepts of maps, pseudo-orientations and bicolorings, respectively. A relationship between pseudoorientability and bicolorings is provided in Section 5. Section 6 explores the impact on bicolorability of operations on maps. In Section 7, we discuss a natural double cover of every map which is not bicolorable. The set of bicolorings of any map can be given a natural group structure. In Section 8 we determine, for each possible group, which surfaces admit maps with the given group of bicolorings. Finally, in Section 9, we generalize some of the results about maps to higher dimensional structures.

Maps
A map M is, first and foremost, an embedding of a graph (or pseudograph) on a (compact, connected) surface so that the components of the complement of the embedding (called faces) are topologically open disks. We can, for example, regard the cube as an embedding of the graph Q 3 on the sphere. The graph Q 3 is an example of a graph which is bipartite, i.e., its vertices can be colored with two colors so that every edge joins vertices of opposite colors. When speaking about maps, we will use the term vertex-bipartite to describe a map M whose underlying graph is bipartite. Similarly, we will call M face-bipartite provided its faces can be colored with two colors so that each edge separates faces of opposite colors. We will call M edge-bipartite provided that the edges can be colored with two colors so that edges which are consecutive around a face (and hence around a vertex) have different colors.
To look more closely at the structure of a map, we find the following subdivision useful: choose a point in the interior of each face to call its center and a point in the relative interior of each edge to be its midpoint. Draw dotted lines to connect each face-center with each incidence with the surrounding vertices and edge-midpoints. The original edges and these dotted lines divide the surface into triangles called flags. Figure 1 shows the subdivision of the cube into flags. If a face meets a vertex (or an edge) more than once, we emphasize that the face center is connected to each incidence-each appearance-of the vertex or midpoint. For instance, consider the map M 4 shown in the left of Figure 2. This map has only one face, an octagon, four edges and only one vertex. Nonetheless, the dissection into flags draws 16 dotted lines, dividing the octagon, and the map, into 16 flags, as shown on the right. Each flag corresponds to a mutual incidence of face, edge, and vertex, though different flags may correspond to the same triple.
Let Ω be the set of flags. Then let r 0 , r 1 , r 2 be the permutations on Ω which match each flag f with its three immediate neighbors, as in Figure 3. In that figure, we see that f and f r 0 are adjacent along a face-center-to-edgemidpoint line. Thus f and f r 0 differ only in their incidences to a vertex, a 0dimensional face of M. Similarly, f and f r 1 differ only in their incidences to an edge, a 1-dimensional face, while f and f r 2 differ only in their incidences to a 2-dimensional face. Notice from Figure 3 that the flag r 2 -adjacent to f r 0 is also r 0 -adjacent to f r 2 . In other words, as permutations on Ω, r 0 and r 2 commute.
We can take a slightly more abstract point of view by defining a map to be a pair (Ω, [r 0 , r 1 , r 2 ]) where Ω is a set of things called flags, the r i 's and r 0 r 2 are fixed point free permutations of order 2 on Ω, the connection group C(M) = r 0 , r 1 , r 2 is transitive on Ω, and r 0 and r 2 commute. This C(M) is often called the monodromy group of the map. We can then think of vertices in M as orbits of r 1 , r 2 in C(M). Similarly, edges correspond to orbits of r 0 , r 2 and faces to orbits of r 0 , r 1 .
We will use the word kite or corner for the area within a face where two consecutive edges meet. More formally, a kite is the union of two flags which are r 1 -adjacent.
If M and N are maps on surfaces S M and S N , a projection from N to M is a function φ mapping S N to S M which is locally a homeomorphism at all points of N except perhaps at vertices and/or face-centers, and which sends faces to faces, edges to edges, and vertices to vertices.
In combinatorial terms, if M = (Ω, [r 0 , r 1 , r 2 ]) and N = (Ω , [s 0 , s 1 , s 2 ]), a projection of N to M is a function φ mapping Ω to Ω such that s i φ = φr i for all i ∈ N .
We call such an N a cover of M. Notice that if N is a cover of M, and the pre-image of some one flag in M has size k, then the projection φ is k-to-1 onto every flag of M. We say then that N is a k-fold cover of M.

Orientations and Pseudo-Orientations
We call a map orientable provided that the surface on which it is embedded is itself orientable. We check that a map is orientable by giving it a face orientation; this is an assignment of a circular arrow to each face of M such that at every edge, the arrows on the faces joined by the edge point along the edge in opposite directions, as in Figure 4. We can define a vertex orientation similarly, and it is clear that M has a vertex-orientation if and only if it has a face-orientation, and this happens if and only if M is orientable. The word pseudo-orientation has been used in two different ways. We will use the term vertex pseudo-orientation (VPSO for short) for what is called in [12] simply a pseudo-orientation. Here, we mean an assignment of one circular arrow to each vertex so that at each edge the two arrows cross in the same direction, as in Figure 5. If we visualize a gear wheel at each vertex so that cogs on the wheels of adjacent vertices mesh, the map is VPSO if we can turn one wheel, causing all wheels to turn at the same time.
In [12], the idea is used to make an important distinction about k-fold rotary covers of a rotary map M for which the branching is totally ramified at vertices. If M is non-orientable, k can be larger than 2 only if M is vertex pseudo-orientable.
Similarly, a face pseudo-orientation (or FPSO) is an assignment of one circular arrow to each face, as in Figure 5, so that at each edge, the arrows in adjacent faces flow along the edge in the same direction. We can simplify this by orienting each edge so that the cycle of edges around each face is consistently oriented.
In [3] this is called a pseudo-orientation, and the result in that paper is that the Dart Graph of M is connected if and only if M is not face pseudo-orientable.
Finally, we may define an edge pseudo-orientation (or EPSO) to be an orientation of the edges such that in every face the direction of the arrows in adjacent edges flows in the same direction (either into or out of the face) as in Figure 5.

Colorings
In this paper, the word 'coloring' will be used to describe what might be more fully notated as a 'consistent flag 2-coloring'. If I is any subset of R = {0, 1, 2}, then an I-coloring of a map M is a function a : Ω → Z 2 such that for every flag f , if j ∈ I, then a(f r j ) = a(f ), while if j / ∈ I, then a(f r j ) = a(f ). To say that in another way,  Each flag f is adjacent to flag f r i along their mutual sub-edges of type i. If M is I-colorable, therefore, the sub-edges bounding each color are the sub-edges of types in I.
Note that when considering maps in class 2 I as defined in [5], the subset I is used to indicate that i-adjacent flags belong to the same orbit. This means that maps in class 2 I admit an (R \ I)-coloring. In the context of colorings it turns out to be more practical to define I-coloring requiring i-adjacent flags to be a different color rather than requiring i-adjacent flags to be the same color if and only if i ∈ I.

The group T (M)
Let ∆ stand for the symmetric difference operation: I∆J = (I ∪ J)\(I ∩ J). Let P be the power set of R. It is well-known that P is a group under ∆ and it is an elementary Abelian group of order eight whose identity element is ∅. Proof: It is easy to verify that the sum of an I-coloring and a J-coloring is an I∆J-coloring From this proposition we see that, for each map M, the sets I for which M has an I-coloring form a set T (M) which is a subgroup of P.

Flag-colorings and map-colorings
Whenever I is a subset of size 1, I-colorable maps have a nice characterization. Proof: Given such a coloring, all the flags incident with one vertex will be the same color, while those at an adjacent vertex will be the other color. This is a bipartite coloring of the vertices. Conversely, given a bipartition of the vertices, color all flags incident with black vertices with color 1 and all those incident with white vertices with color 0. This is then a {0}-coloring of M.
Similar proofs lead us to these:

Colorings and words
Consider a cycle f, f r We call a cycle (f, W ) I-consistent if the number of occurrences in W of indices which are in I is even. If (f, W ) is I-consistent, this says that, at least along the cycle, we can color flags from two colors so that flags which are i-connected for i / ∈ I are the same color and those for which i ∈ I are not. A cycle which is not I-consistent is I-inconsistent. It follows, then that M is I-colorable if and only if every cycle in M is I-consistent.
Consider this quite general fact: for any function F from a set X to Z 2 , extend F to the power set of X by F (A) = Σ x∈A F (x) for each subset A of X. Then for any subsets A, B of X, it is clear that F (A∆B) = F (A) + F (B). Proof: Fix a cycle (f, W ) of M, and define F on R with F (i) being the parity of the number of occurrences in W of r i ; furthermore, consider F extended to P. The cycle is I-consistent, then, provided that F (I) = 0 . Now consider F (I) + F (J) = F (I∆J) = F (K). These are elements of Z 2 , and so we can rephrase that in this form: F (I) + F (J) + F (K) = 0. The number of zeros in {F (I), F (J), F (K)} must be 1 or 3; the conclusion follows directly.
We will have use for this lemma in Section 7.

Connected sum of maps
A natural operator in topology is the connected sum of two surfaces, which is defined to be the surface formed by cutting a small disk from each of the two surfaces and attaching them along the newly created border. We extend this definition to maps by causing the attachment sets to be the boundaries of two suitably chosen faces.
Formally, let M = (Ω, [r 0 , r 1 , r 2 ]) and N = (Ω , [s 0 , s 1 , s 2 ]) be maps and suppose f M and f N are flags of these maps for which their respective faces F M and F N have the same number k of sides. Furthermore, suppose that no flag of F M is r 2 -connected to any of the flags of F M , and similarly for F N . Then we define the connected sum M ⊕ N with respect to (f M , f N ) to be the map whose flag set is (Ω\F M ) ∪ (Ω \F N ), with connections t 0 , t 1 , t 2 , where t 0 and t 1 are the restrictions of r 0 ∪s 0 , r 1 ∪s 1 to this set, and t 2 is the same except that, for each j, Because of the restriction on the faces, these flags are both in the new flag set, and it is easy to check that t 0 t 2 = t 2 t 0 .
The following Figure

Colorings and Orientations
We now offer and prove a series of propositions connecting the ideas of orientations and colorings. First, we show a collection of easy facts to embody our belief that colorings generalize orientations. We then generalize these facts in Theorem 5.4.

Proposition 5.1. A map is orientable if and only if it has an R-coloring.
Proof: If M has an R-coloring, orient each face so that, along each edge, the orientation points from the flag with color 0 to the flag with color 1. This is consistent within the face and faces that meet along an edge have orientations which meet correctly. Conversely, given the orientation, assign colors so that along each edge within a face, the orientation points from the flag with color 0 to the flag with color 1. This causes colors to alternate within each face. Because the orientation opposes the orientation in each adjacent face, the colors alternate at each edge, giving an R-coloring.
Similar arguments give us the following two propositions:

I-Pseudo-Orientability
Our aim in this subsection is to generalize the definition of pseudo-orientability and the previous results. For any subset I of R (except ∅), we form the map X = X(M, I) whose faces are the regions formed by conjoining flags which are connected by each r i for which i /  We call M I-pseudo-orientable provided that X(M, I) is face pseudo-orientable. If this happens, we can orient the edges of X so that around each face of X, the arrows all point the same way. We say a map is ∅-pseudo-orientable when it is orientable.
It is clear that X Furthermore, let us think of the sub-edges as directed. Call the directions above the 'standard' orientations for the sub-edges, and the reverse of these are the nonstandard orientations.
In order to form X(M, I), we delete or ignore sub-edges of each type i / ∈ I, so we only consider sub-edges of type in I. Suppose M is already (R\I)-colored. Then in X(M, I), both flags around each sub-edge have the same color. Then we may assign this as the color of the sub-edge. Now give every sub-edge with color 0 the standard orientation, and each sub-edge with color 1 the non-standard direction.
If I has 2 elements, then consecutive sub-edges around a face of X of different types are the same color, while those of the same type are opposite colors. This causes the directions to be consistent about a face. On the other hand, if I has only one element, then the sub-edges around a face of X alternate in color and are all of the same type; again, this forces the orientation to be consistent.
The following examples show how this works: 1. If I = ∅, then by definition the map is I-pseudo-orientable if and only if it is orientable, and this happens if and only if it is R-colorable.
2. If I = {2}, then X(M, I) = M, which is face pseudo-orientable if and only if it is {0, 1}-colorable. In this case, we orient black type 2 sub-edges from vertex to midpoint-of-edge but white type 2 sub-edges from midpoint-of-edge to vertex.
3. If I = {1, 2}, we orient black type 1 sub-edges from vertex to face-center and black type 2 sub-edges from midpoint-of-edge to vertex. We orient the opposite colors in the opposite direction: 4. If I = R, the ∅-coloration colors all sub-edges the same and so the faces of X, the flags of M, are all oriented consistently. Every map is ∅-colorable and every map is R-PSO.
For the converse we may use the orientation to color in the way prescribed to match the previous list.

The Cheat Sheet
We summarize here the results of previous sections relating to the question of which maps have I-colorings for a given I.

Operators
In this section we shall see how colorings and orientations interact with common map operators such as dual, Petrie, opposite and medial.
This corresponds exactly to the classical geometric dual of a polyhedron. The maps M and D(M) lie on the same surface.
The Petrie of M, P (M), is the map (Ω, [p 0 , p 1 , p 2 ]), where p 0 = r 0 r 2 , p 1 = r 1 , and p 2 = r 2 . This is a less familiar operator on maps. The faces of P (M) are the Petrie paths (see [11] ) of M and vice-versa. Note that the vertices and the edges are preserved by the operation.
We call a map formed from M by any composition of the operation D and P a direct derivate of M. Of special interest among the direct derivates of M is opp(M) = P DP (M) = DP D(M). Formally, this is (Ω, [s 0 , s 1 , s 2 ]) where s 0 = r 0 , s 1 = r 1 and s 2 = r 0 r 2 . More intuitively, opp(M) is formed from M by cutting along the edges and then re-attaching along the same matching edges but with the reverse local orientation. See [11] for more information about these operators.
The medial of a map M is drawn on the same surface as M. The vertices of the medial are the edge-midpoints of M and two are connected by an edge diagonally across each kite to which both belong.

Operators and colorings
In this subsection, we describe the ways in which the colorings of a map come from the colorings of maps related to it by operators.
Proof: Suppose M is I-colored. We shall take opposites but keep the coloring of each flag. Let f be a flag. Then the r 0 and r 1 -adjacent flags in opp(M) are the same as in M, and the r 2 adjacent flag is f r 2 r 0 . If 0 / ∈ I, then f r 2 is the same color as f r 2 r 0 , which means the coloring doesn't change when taking opposites. Analogously, if 0 ∈ I, f r 2 is the opposite color as f r 2 r 0 , so taking opposite changes the color of the flag r 2 -adjacent to f .
Proof: This follows from the previous theorem applying Lemma 6.1.
Here, we write "f i " for (f, i).
Notice that the function sending f i to f is a projection of N onto M, and so I M is a covering of M. Also notice that the function which sends f i to i ∈ Z 2 , is an I-coloring of I M. is I-inconsistent, then f 1 s W = f 0 , and f 0 s W = f 1 . Then the cycle (f, W ) is covered by a single cycle of twice the length of (f, W ). In that case the covering cycle is (f 0 , W 2 ), which is consistent for any subset of R.
We have remarked above that the map I M always has an I-coloring. Moreover, we claim, it is universal minimal in the sense of the following proposition: As a final example, the reader might like to verify that if T is the tetrahedron then {0, 2} T is the orientable map with hexagonal faces, six of them around each vertex, shown in Figure 10. We say that a map N is the Sherk double cover of a non-vertex-bipartite map M whenever N can be obtained from M by the above procedure used by Sherk to obtain the chiral maps with hexagonal faces.  ∈ T (M), there is some cycle (f, W ) which is J-inconsistent. If any such cycle is I-consistent, then in N , the cycle (f 0 , W ) has the same length as (f, W ), and so is also J-inconsistent, contradicting J ∈ T (N ). Thus every cycle which is J-inconsistent is also I-inconsistent. A similar argument in N shows that every cycle in M which is Iinconsistent is also J-inconsistent. Therefore every cycle is either consistent for both, or inconsistent for both. By Lemma 4.5, every cycle in M is K = I∆J-consistent, and then K ∈ T (M), as required.

Which maps are I-double covers?
We want to generalize the result from [2] which says, loosely speaking, that a reflexible map is a 2-fold orientable cover of a non-orientable reflexible map if and only if its rotation group contains an involutory element which conjugates each of the generators to its inverse.
To re-phrase that in our context, a reflexible map N which has an R-coloring is an R-double of some reflexible map M which does not have such a coloring if and only if the subgroup of Aut(N ) which preserves colors in the coloring contains an involutory element which conjugates each of the generators (of the color-preserving group) to its inverse.
If N is I-colorable, let a be either of the I-colorings of N and define C + (N , I) to be the subgroup of C(N ) consisting of all w such that a(f ) = a(f w) for all f ∈ Ω. Because N is I-colorable, this group has index 2, and so is normal, in C(N ).
Note that Theorem 7.8 generalizes the result from [2] not only to all I-colorings, but also to maps with no assumption on their automorphism group. Because M is not I-colorable, I is not empty. So consider any i ∈ I and let w = f −1 (s i ). Note that w is an involution since f is an isomorphism. This w is not r i , since r i / ∈ C + ; it is, instead, expressible as some longer product of r j 's. Then for any c ∈ C + (M) we have f (wcw) = s i f (c)s i , while f (r i cr i ) = s i f (c)s i , since r i cr i is in C + , and f acts there as letter-to-letter projection. Since f is an isomorphism and hence one-to-one, we conclude that wcw = r i cr i . Now assume we have w ∈ C + (N , I) of order 2 and i ∈ I such that for every element c ∈ C + (N , I), wcw = r i cr i . In particular, with c = w, we have w = www = r i wr i . From this we see that wr i = r i w. We then construct M by identifying flags in the following consistent way: let u = wr i , so that u is an involution. We shall identify flag f ∈ Ω with f u. Because u is an involution, this identification is unambiguous, and each flag is identified with exactly one other. Since w preserves coloring and r i changes it, we are identifying a white flag with a black flag. These equivalence classes are the flags of M.
Then In order to use this theorem we would, in theory, need to prove the existence of w and observe that under conjugation, w acts in the same way as some r i on every element of C + (N ). But in practice, we only need to find w that acts in such a way for the generators of C + (N ). The element r i must not be in C + and so we will refer to it as an external generator. Table I below summarizes information about each subset I of R. The second column (adapted from [5] ) gives generators for C + (N ) as a subgroup of C(N ). The third column gives the r i s which are not in C + , and the last column shows the actions on the generators of C + under conjugation by this external generator.
8 Which subgroups appear on which surfaces?
The purpose of this section is to prove that every subgroup H of P occurs on every surface on which it can appear. To clarify that statement, we call a group which includes R an orientable subgroup of P, and subgroups which do not contain R are non-orientable. So we want H to appear on S when they are both orientable or both non-orientable. This is almost true, as explained by the following Theorem: The proof of this claim is contained in and scattered through the remaining parts of this section. We will consider the 16 possible groups subgroups of P individually and in dual pairs. Such a proof would appear to require 16 constructions, but we can use Lemma 6.1 and Proposition 7.2 for some simplification. Moreover, the insertion techniques that we will introduce and the connected sum construction of Subsection 4.4 allow even more simplification. We consider the non-orientable cases first; then a simple argument, using the results of Section 7, takes care of the orientable ones.

H = {∅}
Consider any triangulation of a non-orientable surface S. Refine the triangulation slightly by placing one new vertex inside some triangular face and connect it to the three vertices on its boundary so as to have one vertex of degree 3. Call this map M. Because of the triangular faces and the the degree-3 vertex, the Cheat Sheet of Section 5 shows us that none of the sets of size 1 or 2 are in the group, and because S is non-orientable, it cannot contain R either. Thus T (M) must be the trivial group.

Insertion
In this subsection, we introduce the technique of insertion. We shall prove that any map M can be easily modified to produce a map which is face or vertex bipartite and also face or vertex pseudo-orientable.
Let us assume, for example, that we wish to modify M to construct a vertexbipartite map. Color the vertices of M at random. For the edges for which both vertices are the same color, add a vertex (of degree 2) to split the edge in two.
The same idea can be applied if we wish vertex pseudo-orientability. We may assign orientations to the vertices at random and then split the edges whose endpoints don't match into two edges. Call this process vertex-insertion. This can also be used to turn a map which is vertex-bipartite or VPSO into one that is not. Note that vertex-insertion does not modify the status of either face-bipartiteness of FPSO of M.
Dually, we may replace an edge by a face bounded by two edges to get either facebipartition or FPSO. The following picture illustrates this for face pseudo-orientation: Figure 11: Adjusting one edge We begin with an arbitrary map on a surface S, and an arbitrary orientation of each face. If there is an edge, as in Figure 11a, where the assigned arrows flow in opposite directions along the edge, replace the edge with a face bounded by a pair of edges as shown in 11b. Orient the resulting lune with an arrow as shown. This gives a map on the same surface with one fewer edge for which the given arrows fail to give a face pseudo-orientation. Continuing this process eventually gives us a map on S admitting a face pseudo-orientation.
Let's call this process face-insertion. Here, we used it to produce a FPSO map on any surface. We can use it equally well to turn any map into a face-bipartite map on the same surface. As with vertex insertion, this same process can also be used to convert a map which is face bipartite or FPSO to one which is not.
For many groups we can use these two operations to produce maps with that group on any non-orientable surface. We will show one of these in detail and summarize the rest.
Consider the non-orientable group H = {∅, {0, 1}}. For any non-orientable surface S, we choose any map on S, and, using face-insertion as above, construct a map which is FPSO. Now we must be certain that the group of the map is no bigger than Proof: Let S be an arbitrary non-orientable surface. To construct a map on S with the desired properties, first consider any map for which no face shares a vertex with itself. Because the surface is non-orientable, we may choose an arbitrarily fine subdivision for which there is a Möbius band of faces which does not use all the faces or edges. Then apply the operations described in subsection 8.2 to make the map h-colorable for each h in H. This is possible since any group H with 4 elements with R / ∈ H must contain two of the following: vertex-bipartite, face-bipartite, FPSO or VPSO, with the other non trivial element being the symmetric difference of the other two.
Now consider an edge that is not in any of the faces of the chosen Möbius band and convert it into three edges, as with face-insertion but with three edges instead of two. This operation does not change the status of vertex-bipartition, face-bipartition, VPSO or FPSO as required. Now pick either of the two newly created faces and call it F . Since we still have a Möbius strip, R / ∈ T (M \ F ), but every I ∈ T (M) is in T (M \ F ). Therefore, T (M \ F ) = H. We give explicit constructions for these last two groups: Let H = {∅, {1}}, and consider the map G = G(m, n, k) where k, m, n > 0. This map has mn faces, each one a square. They are arranged in an m × n rectangle. The top m edges are identified with the bottom m directly and orientably. Each vertical edge is identified with the one diametrically across from it; the first k orientably, the rest non-orientably. Figure 12 shows G(5, 7, 3). In general we have a rectangle one unit wide, having any height. The maps shown in Figure 13 have characteristic 0 and 1 respectively. In the left picture, by making the strip long enough, we may switch enough pairs of adjacent labels on the right side (for example, switch labels 3 and 4 on the right) to make any even genus non-orientable surface. Similarly, for the picture on the right we may switch enough pairs of adjacent labels in a sufficiently long strip to construct any odd genus non-orientable surface.
To see that the group is indeed {∅, {0, 2}}, we see that neither map is face-bipartite, since some faces are glued to themselves, or face pseudo-orientable, since both contain faces that are glued to themselves in an orientable way. Similarly, neither are vertexbipartite or vertex pseudo-orientable.
This concludes the proof for the non-orientable case.

Orientable Groups
If the subgroup H contains R, then it is generated by R and some subgroup H not containing R. Each orientable surface S is the orientable twofold cover of some nonorientable surface S . By the preceding subsections, S has a map M whose group is H (unless S is the sphere, S is the projective plane and H is {∅, {1}, {0, 2}, {0, 1, 2}}). Then R M is a map on S whose group is H. By Proposition 7.7, T (M) is exactly H. Thus, with the one exception, for each subgroup H containing R, every orientable surface admits a map whose group is H.
All that is left to prove now is that the three exceptions mentioned in the theorem are in fact exceptions -namely that there is no map on the specified surfaces with the specified group. We shall prove this first for the group {∅, {1}, {0, 2}, R} on the sphere. Proof: Suppose M is one such map. Since {0} / ∈ T (M), the underlying graph of M is not bipartite, which means it has an odd cycle C. By Jordan's Closed Curve Theorem, this divides the sphere into two connected components. Choose either and erase every edge and vertex exterior to it in order to form the deleted part into a single face whose boundary is C. This gives a new map M , also on the sphere. Since {1} ∈ T (M), every face of M must have an even number of sides. Therefore M has exactly one face with an odd number of sides. Then the underlying graph of D(M ) has exactly one vertex of odd degree, a well-known impossibility. Proof: If such a map were to exist, its orientable double cover would violate Lemma 8.4.
The proof of Theorem 8.1 is now complete.
Corollary 8.6. No map on a non-orientable surface has an R-coloring. In all other cases, for any subset I of R, and any surface S, there is a map M on S which is I-colorable.

Maniplexes and higher dimensions
To generalize the results of this paper to higher-dimensional objects, we use the idea of a maniplex, first introduced in [13]. We summarize the definitions here: an nmaniplex is a pair (Ω, [r 0 , r 1 , r 2 , . . . , r n ]), where Ω is a set of things called flags, and each r i is a fixed-point-free involution on Ω such that r 0 , r 1 , . . . , r n is transitive on Ω, and for every i, j such that 0 ≤ i < j − 1 < n, the permutations r i and r j commute and are disjoint.
Maniplexes clearly generalize maps and slightly generalize (the flag-graphs of) abstract polytopes. In particular, every map is a 2-maniplex and every 2-maniplex is a map.
The language of maniplexes is the language of polytopes: an i-face is an orbit under the group R i = r 0 , r 1 , . . . ,r i , . . . , r n , generated by all of the r j 's except r i . A 0-face is a vertex, a 1-face is an edge, an n-face is a facet.
The definitions of coloring, projection, cover generalize easily. There is no notion of surface or manifold which applies to maniplexes; nevertheless, we describe a maniplex which has an R = {0, 1, 2, . . . , n}-coloring as being orientable. In view of Theorem 5.4, one may define a maniplex to be I-pseudo-orientable if and only if it is (R \ I)colorable. We may give a definition of FPSO that is more in the spirit of our intuition: First, we need each facet to be orientable and second, orientations can be assigned to the facets so that they agree on every subfacet. It is easy to see that this is equivalent to an (R \ {n})-coloring of M.
The idea of an I-coloring, for any I ⊆ R, comes forward easily, as does the fact that the set of I's for which a given maniplex M is I-colorable forms a subgroup T (M) of the power set P of R under ∆. We can define i-face bipartiteness: An i-face is a connected component under all of the r j 's except r i . Make these the vertices of a graph, and join two of them by an edge is some flag of one is r i adjacent to some flag of the other. If this graph is bipartite, we say that M is i-face bipartite. Then, it is easy to show then that M is i-face bipartite if an only if it is {i}-colorable.
We can define connected sum of two n-maniplexes, removing isomorphic facets, one from each maniplex and adjusting the r n connections to form the sum. The theorem that the sum has any coloring common to both still holds.
The operators generalize with a little care. We form opp(M) from M by replacing r 2 with the product r 0 r 2 . We form D(M) from M by reversing the order of the generators. And then P (M) = D(opp(D(M))). Similar facts hold about how colorings of M are inherited by its direct derivates. The medial operation is more difficult to generalize.
The construction of I-doubles is straightforward to generalize, and all of the facts about the doubles do as well. The fact that maps have rank 3 plays no role in any of these proofs.
Again, because there is no notion equivalent to that of 'surface', there is no natural way to generalize the results of Section 8.

Open Questions
The constructions in Section 8 lead to quite general maps, although some of them may be somewhat degenerate in the viewpoint of other previous work. It may be the case that more exceptions in Theorem 8.1 are needed if we ask the extra requirement that the maps are polyhedral maps (that is, the intersection of the closure of two distinct faces is either empty, a single vertex, or a single edge. See [6] ). However, this is beyond the scope of this paper.
In previous sections we considered colorings of the flags of maps insisting that we use only two colors. In doing so, we ensure that if a given flag is i-adjacent to another flag with the same color then all flags are the same color as their i-adjacent flags.
Consistent colorings with k colors can be defined following the idea of [1] and [9], that is, each flag is assigned a color in {1, . . . , k} with the restriction that for every i ∈ {0, 1, 2} and j ∈ {1, . . . , k}, the color of the i-adjacent flag to a flag f colored j depends on j but not on f .
When k-coloring a map consistently for k ≥ 3 then it may not be true that a flag is i-adjacent to another flag with the same color if and only if each flag is i-adjacent to another flag with the same color. For example, we can color the flags of the pentagons of a pentagonal prism with color 1, the flags of the edges shared by two squares with color 2, and the remaining flags with color 3, as in Figure 14. The reader can easily verify that this is a consistent 3-coloring, however, flags colored 1 are 1-adjacent to flags colored 1, and flags colored 2 are 1-adjacent to flags colored 3. It is unlikely that the concept of pseudo-orientability admits a generalization that preserves the connections between 2-colorings and pseudo-orientations of maps. However, k-colorings may still keep an interesting relation with k-fold covers of maps and maniplexes as well.
Some results in [5], [9], [1] relate symmetry with I-colorings. We suspect that many more results can be obtained linking symmetry with I-colorings.