Compact 3-manifolds via 4-colored graphs

# Compact 3-manifolds via 4-colored graphs

Paola CRISTOFORI Dipartimento di Scienze Fisiche, Informatiche e Matematiche, Università di Modena e Reggio Emilia Michele MULAZZANI Dipartimento di Matematica, Università di Bologna
###### Abstract

We introduce a representation of compact 3-manifolds without spherical boundary components via (regular) 4-colored graphs, which turns out to be very convenient for computer aided study and tabulation. Our construction is a direct generalization of the one given in the eighties by S. Lins for closed 3-manifolds, which is in turn dual to the earlier construction introduced by Pezzana’s school in Modena.

In this context we establish some results concerning fundamental groups, connected sums, moves between graphs representing the same manifold, Heegaard genus and complexity, as well as an enumeration and classification of compact 3-manifolds representable by graphs with few vertices ( in the non-orientable case and in the orientable one).

2010 Mathematics Subject Classification: Primary 57M27, 57N10. Secondary 57M15.

Key words and phrases: 3-manifolds, Heegaard splittings, Heegaard diagrams, colored graphs, complexity

## 1 Introduction and preliminaries

The representation of closed 3-manifolds by 4-colored graphs has been independently introduced by S. Lins and by Pezzana’s research group in Modena (see [18] and [28]), by using dual constructions. A 4-colored graph is a regular edge-colored graph of valence 4, which represents a closed 3-manifold iff it satisfies certain combinatorial conditions.

The extension of the representation to 3-manifolds with boundary was performed by C. Gagliardi in [19] by using a slightly different class of colored graphs satisfying a notion of regularity weaker than the one required in the closed case. The study of this kind of representation has yielded several results especially with regard to the definition of combinatorial invariants and their relations with topological invariants of the represented manifolds (see [22], [15], [13], [9]). Unfortunately, Gagliardi’s representation is not suitable for a satisfactory computer tabulation of non-closed 3-manifolds.

In this paper we show that any 4-colored graph, with no additional conditions, can represent a compact 3-manifold without spherical boundary components, and the whole class of such manifolds admits a representation of this type. As a consequence, an efficient computer aided tabulation of 3-manifolds with boundary can be performed by this tool.

The construction is described in Section 2 and a set of moves connecting graphs representing the same manifold is given in Section 3. In the closed case, these moves have been proved to be sufficient to connect any two graphs representing the same manifolds (see [5]). In the more general case of manifolds with boundary, this result is no longer true, at least when the boundary is not connected.

Examples of 4-colored graphs representing relevant classes of compact 3-manifolds, such as handlebodies (both orientable and non-orientable) and products of closed surfaces with the compact interval , are given in Section 5.

In Section 4 we establish the relation between the connected sum of graphs and the (possibly boundary) connected sum of the represented 3-manifolds. In Section 6 we associate to any 4-colored graph a group which is strictly related to the fundamental group of the associated manifold and, therefore, it is a convenient tool for its direct computation (in many cases the two groups are in fact isomorphic). Combinatorial invariants are defined and their relations with (generalized) Heegaard genus and Matveev complexity are discussed in Sections 7 and 8 respectively.

The last section presents some computational results, obtained by means of a C++ program, in terms of the number of non-isomorphic graphs representing compact 3-manifolds with non-empty boundary, up to 12 vertices, and the number of compact 3-manifolds admitting a graph representation up to 8 vertices in the orientable case and up to 6 vertices in the non-orientable one.

In the following we fix some notations and recall some definitions and results about Heegaard splittings/diagrams which will be largely used throughout the paper.

Let be a closed, connected surface of genus either orientable (with ) or non-orientable (with ). A system of curves on is a (possibly empty) set of simple closed orientation-preserving111This means that each curve has an annular regular neighborhood, as it always happens if is an orientable surface. curves on such that , for . Moreover, we denote by the set of connected components of the surface obtained by cutting along the curves of . The system is said to be proper if all elements of have genus zero, and reduced if either or no element of has genus zero.

Note that a proper reduced system of curves on contains exactly curves in the orientable case and curves in the non-orientable one, with even. Of course, in the non-orientable case no proper system can exists when is odd. In the following will be also considered as a subspace of in the obvious sense.

A compression body of genus is a 3-manifold with boundary obtained from , where , by attaching a finite set of 2-handles along a system of curves (called attaching circles) on and filling in with balls all the spherical boundary components of the resulting manifold, except for when The set is called the positive boundary of , while is the negative boundary of . Notice that a compression body is a handlebody if an only if (i.e., the system of the attaching circles on is proper). Obviously homeomorphic compression bodies can be obtained via (infinitely many) non isotopic systems of attaching circles. Moreover, any non-reduced system of curves properly contains at least a reduced one inducing the same compression body. Operations of reduction correspond to elimination of complementary 2- and 3-handles.

Let be a compact, connected 3-manifold without spherical boundary components. A Heegaard surface for is a closed surface embedded in such that consists of two components whose closures and are homeomorphic to genus compression bodies. The triple is called a generalized Heegaard splitting of genus of . It is a well-known fact that each compact connected 3-manifold without spherical boundary components admits a Heegaard splitting, and at least one of the two compression bodies can be assumed to be a handlebody (in this case the splitting is simply called Heegaard splitting).

Since two compact 3-manifolds are homeomorphic if and only if (i) they have the same number of spherical boundary components and (ii) they are homeomorphic after capping off by balls these components, there is no loss of generality in studying compact 3-manifolds without spherical boundary components.

On the other hand, a triple , where and are two systems of curves on , such that they intersect transversally, uniquely determines a 3-manifold corresponding to the Heegaard splitting , where and are respectively the compression bodies whose attaching circles correspond to the curves in the two systems. Such a triple is called a generalized Heegaard diagram for . In the case of a generalized Heegaard diagram of a closed 3-manifold, both systems of curves are obviously proper; if they are also reduced, is simply a Heegaard diagram in the classical sense (see [25]).

The minimum such that a manifold admits a generalized Heegaard splitting (resp. a Heegaard splitting) of genus is called the generalized Heegaard genus (resp. the Heegaard genus of ), denoted by (resp. by ). Of course the two notions coincide in the case of connected boundary. The only 3-manifold of (generalized) Heegaard genus zero is the 3-sphere, possibly with some deleted balls. Examples of compact non-closed 3-manifold of generalized Heegaard genus one are and the orientable handlebody of genus one . Of course , for every manifold and it is easy to find examples where the two genera differ: for example by construction but since the Heegaard genus of an orientable manifold can not be less than the sum of the genera of its boundary components.

For general PL-topology and elementary notions about graphs and embeddings, we refer to [27] and [36] respectively.

## 2 Construction

Let be a finite connected graph which is 4-regular (i.e., any vertex has valence four), possibly with multiple edges but with no loops. A map is called a -coloration of if adjacent edges have different colors. An edge of colored by is also called a -edge.

A -colored graph is a 4-regular graph equipped with a 4-coloration. It is easy to see that 4-colored graphs have even order, but not all 4-regular graphs of even order can be 4-colored. Easy examples of 4-colored graphs are the graph of order two and the complete bipartite graph . Two 4-colored graphs and , with coloration and respectively, are (color-)isomorphic if there exist a graph isomorphism between and and a permutation of such that . If , any connected component of the subgraph of containing exactly all -edges, for each , is called a -residue as well as a -residue. Of course 0-residues are vertices, 1-residues are edges and 2-residues are bicolored cycles with an even number of edges.

We can associate a compact connected 3-manifold without spherical boundary components to any 4-colored graph via the following construction.

First of all consider as a 1-dimensional cellular complex. By attaching to a disk for each 2-residue we obtain a 2-dimensional polyhedron , which is special in the sense of [32]. The vertices (resp. points of the edges) of have links in homeomorphic to a circle with three radii (resp. with two radii). Each 3-residue of , with the relative associated disks is a closed connected surface . If is a 2-sphere the residue is called ordinary and otherwise singular. By attaching a 3-ball to when the residue is ordinary, and just thickening it by attaching along when the residue is singular, we obtain a compact connected 3-manifold with non-spherical boundary components. We will say that represents . Obviously isomorphic 4-colored graphs represent homeomorphic 3-manifolds.

For closed 3-manifolds the construction reduces to the one introduced by Lins (see [28]), and it is dual to the one introduced by Pezzana and others (see [18]). Pezzana’s construction was also studied by Bracho and Montejano [4] in the more general context of “colored complexes”, but still within the closed case. So, the novelty of our construction is that it works also in case of 3-manifolds with (non-spherical) boundary. Actually, a graph representation for manifolds with boundary has been introduced by Gagliardi in [19], by using colored graphs which are not 4-regular. So, our idea is to give, for the whole class of compact 3-manifolds, a unitary representation by 4-colored graphs, which seems to be more efficient than Gagliardi’s for a computer aided tabulation and classification of 3-manifolds with boundary.

###### Remark 1

Let be different colors and let , then by removing from the interior of any disk bounded either by an -residue or by an -residue, we obtain a closed connected surface which is a Heegaard surface for . Moreover, both the -residues and the -residues are two systems of curves and on such that the triple is a generalized Heegaard diagram of . So, any 4-colored graph defines three different generalized Heegaard diagrams for .

###### Example 1

Let us consider the 4-regular graphs and of Figure 1. By Remark 1, if we consider the pairs of colors and , we obtain the generalized Heegaard diagrams of and of whose planar realizations are depicted in Figure 2(a) and Figure 2(b), respectively. Each diagram consists of the two systems of curves and on the genus one surface , which is orientable in the case of , non-orientable in the case of .

Let be the genus one orientable (resp. non-orientable) handlebody whose 1-handles are attached along the curves and of (resp. ). Then (resp. ) is simply obtained from by adding a trivial 2-handle whose attaching curve is , i.e it is still homeomorphic to .

Therefore (resp. ) is a 4-colored graph representing the genus one orientable (resp. non-orientable) handlebody.

A 4-coloured graph, with 8 vertices, representing the genus one orientable handlebody was already constructed in [17]. Actually, it can also be obtained from the above graph , which is minimal with regard to the number of vertices, by “adding a 2-dipole” (see Section 3).

The following result shows that 4-colored graphs are a representation tool for all compact 3-manifolds without spherical boundary components.

###### Proposition 1

Any compact connected 3-manifold without spherical boundary components can be represented by a 4-colored graph.

Proof. Let us consider a handle decomposition of : the union of the 0-handles and 1-handles gives a genus handlebody , which can be either orientable or non-orientable. Let be a system of pairwise disjoint disks properly embedded in , such that, by cutting along them, splits into two balls and . If , for , then is a system of curves on .222Note that in the orientable case and in the non-orientable one. Let , for , be the attaching circles of the 2-handles. is a system of curves on , too. Possibly by adding a trivial 2-handle, we can suppose that . Moreover, up to isotopy we can always suppose that each curve of intersects transversally the curves of ; the graph is connected and cellularly embedded in (i.e., the regions of the embedding are disks). Now, let us take a set of pairwise disjoint regular (closed) neighborhoods in of all curves of , such that intersects transversally the curves of in such a way that any component of contains exactly one point of . So, is a system of curves on , and the graph is a finite connected 4-regular graph embedded on . Let us color the arcs of in the following way: the arcs of the curves of are colored by 2 if they belong to and by 3 if they belong to . Furthermore, we color the arcs of the curves of by 0 if they belong to some and by 1 otherwise. It is easy to see that the resulting coloration is proper, and the 3-manifold associated to via the previous construction is homeomorphic to .

Some properties of the manifold correspond to properties of the representing graph . For example:

###### Proposition 2

is orientable if and only if is bipartite.

Proof. Of course is orientable if and only if is orientable, for arbitrarily fixed , since is obtained from by adding 2-handles and possibly 3-handles. First of all let be orientable, and consider the induced orientation on its 2-cells, which are disks bounded by -residues, with and . These orientations defines for each a cyclic permutation of which is or its inverse, corresponding to the local orientation induced on the vertices. Since permutations of adjacent vertices are inverse each other, the graph cannot have odd cycles and therefore it is bipartite. Viceversa, let us suppose that is bipartite. Then , where and , such that any connects a vertex of with a vertex of . Let be a 2-cell of , and suppose its boundary is a -residue such that . Then orient in such a way that the induced orientation on its -edges goes from vertices of to vertices of . It is easy to see that the chosen orientations on the 2-cells of define a global orientation for the whole .

Note that Propositions 1 and 2 are generalizations of analogous results obtained in [29], [33] and [4] for the closed case and different representation methods.

For , with , we denote by the number of -residues of . Moreover, for each , the number of 3-residues corresponding to the colors of will be denoted by . We say that is -contracted if either or all -residue are singular. The graph is said to be contracted if it is -contracted for all . We will see in Section 3 that any 3-manifold can be represented by a contracted graph.

Moreover, boundary components of correspond to colors such that there exists at least a singular -residue. We call them singular colors. By the previous construction, which produces the graph from a handle decomposition of the manifold , it is always possible to represent a manifold by a 4-colored graph with at most one singular color. In fact, colors 2 and 3 are not singular by construction and color 0 is non-singular since -residues are attaching boundaries of the 2-handles.

A vertex of is called a boundary vertex if it belongs to at least one singular 3-residue, otherwise it is called internal. A boundary vertex is called of order if it belongs to exactly singular 3-residues. So and, as a convention, an internal vertex is considered as a boundary vertex of order zero. Next section shows that any 3-manifold can be represented by a graph with at least one internal vertex and, when the boundary is not empty, it can also be represented by a graph with at least a boundary vertex of order one.

## 3 Moves

Given a 4-colored graph , an -dipole () involving colors is a subgraph of consisting of two vertices and joined by edges, colored by , such that and belong to different -residues of , where .

By cancelling from , we mean to remove and to paste together the hanging edges according to their colors, thus obtaining a new 4-colored graph . Conversely, is said to be obtained from by adding . An -dipole is called proper if and only if and represent the same manifold.

###### Proposition 3

An -dipole of a -colored graph is proper if and only if one of the following conditions holds:

• and at least one of the -residues containing and is ordinary.

Proof. Let be an -dipole involving colors .

If , let (resp. ) be the -residue of containing (resp. ). Cancelling corresponds to removing a tunnel in connecting the two surfaces represented by and respectively. The boundary of the tunnel is a cylinder composed by three bands , whose sides are portions of the -edges () involved in the dipole. It is obvious that, if the -residue containing a vertex of , say , is ordinary, then represents the boundary of a 3-ball and the cancellation of yields a new 4-colored graph still representing .

If , then is a 2-component (i.e., a 2-cell) of the complex which is a special spine of ; then our claim follows from observing that the cancellation of is the inverse of the lune move defined by Matveev and Piergallini on spines of 3-manifolds (see [30] and [34]). Since is a dipole, the two -residues containing and respectively are different 2-components of , so the cancellation of transforms into another special spine of .

If , then there exist two proper 1-dipoles, both involving the only color of , adjacent to and respectively. The cancellation of is equivalent to the cancellation of one of these 1-dipoles.

As a consequence of the above proposition, we can obtain some useful properties.

###### Corollary 4

Let be a -manifold without spherical boundary components, then:

(i) can be represented by a contracted -colored graph;

(ii) can be represented by a -colored graph with at least an internal vertex;

(iii) can be represented by a -colored graph with at least a boundary vertex of order one, if .

Proof. Let be a 4-colored graph representing .

(i) If is not contracted with respect to a color , then there exists a 1-dipole , involving color , such that at least one of the two -residues containing and is ordinary. Hence is proper and by cancelling it we obtain a new 4-colored graph which still represents . A finite sequence of such cancellations of 1-dipoles obviously yields a contracted 4-colored graph representing .

(ii, iii) If there is nothing to prove. Let be a boundary vertex with minimal order and let be such that the -residue containing is singular. By adding a -dipole along the -edge containing we obtain two new vertices, and , which are both singular of order . In fact, the -residue containing them is obviously a 2-sphere, and for each any -residue containing them is singular if and only if the -residue containing in is singular. So by induction on we can obtain an internal vertex (resp. a boundary vertex of order one) in not more than four steps (resp. three steps).

In the closed case all dipoles are proper and Casali proved in [5] that dipole moves are sufficient to connect different 4-colored graphs representing the same manifold. A similar fact is not generally true in our context; in fact dipole moves do not change the singular colors of the involved graph, and for any 3-manifold with disconnected boundary it is easy to find two different graphs representing it, the first one with only a singular color and the second one with (at least) two singular colors. In Section 7 another move will be introduced, the bisection, which changes the coloration but not the represented manifold. The problem whether, by adding bisection, it is possible to extend Casali’s result, is currently under investigation.

The proof of Proposition 3, case , shows the effect of the cancellation of a non-proper 1-dipole. More precisely, we have the following result:

###### Proposition 5

Given a -colored graph , let be a non-proper 1-dipole involving color . If is the graph obtained from by cancelling , then is the manifold obtained from by removing a tunnel along connecting the boundary components of which correspond to the -residues involved in the dipole cancellation.

## 4 Connected sums

Suppose that and are two 4-colored graphs and let and . We can construct a new 4-colored graph , called the connected sum of and along and , and denoted by , by removing the vertices and and by welding the resulting hanging edges with the same color.

Obviously, the connected sum of two 4-colored graphs depends on the choice of the cancelled vertices. But when both vertices are internal or they are boundary vertices of the same order with respect to the same colors (the latter condition always holds, up to color permutation in one of the two graphs), then the connected sum of the graphs is strictly connected with the connected sum of the represented manifolds.

###### Proposition 6

Let be -colored graphs and .

(i) if and are both internal vertices, then

(ii) if and are both boundary vertices of order one each belonging to a singular -residue, then , where the boundary connected sum of the manifolds is performed along the boundaries corresponding to the singular residues.

Proof. (i) Since and are internal vertices we can perform the connected sum between and by erasing two balls and containing only the vertices and respectively and such that and are both homeomorphic to the 1-skeleton of a tetrahedron, where each edge belongs to a certain bicolored residue containing either or . By gluing the two spheres and in such a way that is glued with coherently with the above 2-residues, we obtain a 4-colored graph embedded in , and representing it in accordance with the main construction.

(ii) With regard to the case of boundary connected sum, if the singular -residue containing (resp. ) is the surface (resp. ), then we can retract to (resp. to ) obtaining a new 3-manifold homeomorphic to (resp. homeomorphic to ) such that (resp. ). Now we can perform the boundary connected sum between and by erasing two balls and containing only the vertices and respectively and such that (resp. ) is homeomorphic to the 1-skeleton of a tetrahedron, with three edges belonging to (resp. ) and corresponding to a -residue, , containing (resp. ), and the other three edges not containing (resp. ) and corresponding to the other -residues, . By gluing the two emispheres and in such a way that is glued with coherently with the above 2-residues, we obtain a 4-colored graph embedded in . By gluing to the -residue we obtain the manifold which is obviously homeomorphic to .

When the order of the boundary vertices involved in the connected sum is greater than one, more complicated topological facts occur. If are two manifolds with at least two boundary components and , we define the double boundary connected sum of and as the manifold obtained by removing a tunnel from and connecting with and with respectively, and adding a ”holed” 1-handle , in such a way that is attached to and is attached to as in Figure 3. An example of such operation is the double boundary connected sum of with , performed by choosing and respectively as tunnels, for any and . It is easy to see that, if both and are orientable, the sum is homeomorphic to .

Of course, the double boundary connected sum in general depends on the choice of tunnels, but in the next proposition tunnels are trivial, and the sum is uniquely defined, up to homeomorphism.

###### Proposition 7

Let be -colored graphs and . If and are both boundary vertices of order two such that they belong to a singular -residue and to a singular -residue, with , then , where the double boundary connected sum of the manifolds is performed between the boundaries corresponding to the singular residues.

Proof. Without loss of generality we can suppose and . Add a -dipole to the -edge containing , as well as to the -edge containing , obtaining two new 4-colored graphs called again and . The new vertices and are boundary vertices of order one. Performing we obtain a graph representing , where the boundary connected sum is made by connecting by a 1-handle the boundaries corresponding to the -residues containing and respectively. After that we remove the -dipole containing and , obtaining a new graph still representing . Now the vertices and are boundary vertices of order two (with respect to colors 0 and 1), and they are connected by a -edge, which is a non-proper 1-dipole, and which is the core of the 1-handle . The result is achieved by applying Lemma 5, using as tunnel a regular neighborhood of , where , , where and are respectively the -residue and -residue containing (see Figure 3).

###### Remark 2

Note that the graph appearing in the proof of the above proposition can be simply obtained directly from and by ”switching” the two 0-colored edges containing and . More precisely, if we call (resp. ) the vertex of (resp. ) 0-adjacent to (resp. ), we join with and with by a 0-colored edge. More generally, we can obtain a graph representing the boundary connected sum performed along the boundary components corresponding to two given -residues and , by switching two -colored edges and belonging to and respectively and such that for each , and belong to ordinary -residues.

## 5 Basic examples

Any color of a 4-colored graph can be interpreted as a fixed point free involution on . When is bipartite with vertex bipartition and , a color can also be interpreted as a bijection , and the maps , and are permutations of which completely determine , up to isomorphism.

### 5.1 4-colored graphs representing Sg×I

Let be a closed orientable (resp. non-orientable) surface of genus and let be the 4-colored bipartite (resp. non-bipartite) graph with (resp. ) vertices obtained from the standard 3-colored graph representing described in [23], by adding 3-edges parallel (i.e. having the same endpoints) to the 0-edges (see Figures 4 and 5).

Note that the singular colors of are and , while and are not singular. Moreover, is contracted.

Let us consider the generalized Heegaard diagram for associated to and the pair . The Euler characteristic of the Heegaard surface can be computed via the cellular decomposition induced by on it. More precisely, we have:

 χ(S0,2)=p−2p+g0,1+g1,2+g2,3+g0,3=−p+1+1+1+p2=3−p2 .

Hence, has genus and therefore it is homeomorphic to .

The system of curves (resp. ) on consisting of the -residues (resp. -residues) contains a single curve (resp. ), and it is neither proper nor reduced. In fact, (resp. ) bounds a disk on ; therefore it can be removed from (resp. ) without changing the related compression body, which is homeomorphic to . As a consequence, we obtain a reduced generalized Heegaard diagram for , where both systems of curves on are empty. Hence represents .

Note that the above 4-colored graph for the case can be also obtained from the one representing by performing iterated connected sums of this graph with itself, which correspond to double boundary connected sums as described in the previous section.

### 5.2 4-colored graphs representing handlebodies

A representation of the genus handlebody can be easily obtained starting from the one of given in Section 2 and then performing boundary connected sums.

In the orientable case, we start with the solid torus , which can be represented by the bipartite 4-colored graph depicted in Figure 1: it has 6 vertices and its coloring can be described by , and the three permutations of : , and . All vertices of are boundary vertices of order one, corresponding to a singular -residue, which is a torus.

By performing the connected sum of two copies of along any pair of vertices, we obtain a bipartite graph representing with 10 vertices and defined by the permutations , and (see Figure 6).

In order to get a 4-colored graph representing , we iterate the boundary connected sum, taking care, for each , to perform it with respect to the vertex of and the “rightmost” vertex of . As a consequence we obtain (see Figure 7):

###### Proposition 8

The genus orientable handlebody is represented by a bipartite -colored graph with vertices , defined by the permutations

 φ1 =(a1a2…a2g+1),φ2=(a1a2g+1a2a2g…aia2g+2−i…agag+2ag+1), φ3 =(a1a2…ag+1).

In the non-orientable case, we start with the solid Klein bottle , which can be represented by the (non-bipartite) 4-colored graph of Figure 1: it has 6 vertices , the same -edges () as , and the 2-edges connecting with , with and with . All vertices are boundary vertices of order one, corresponding to a singular -residue which is a Klein bottle.

By performing the connected sum of two copies of along any pair of vertices, we obtain a graph representing with 10 vertices with the same -edges () as and 2-edges connecting with , with , with , with and with (see Figure 8).

As in the orientable case, for each , we perform the boundary connected sum with respect to the vertex of and the “rightmost” vertex of , thus obtaining (see Figure 9):

###### Proposition 9

The genus non-orientable handlebody is represented by a -colored graph with vertices , with the same -edges, , as the graph of Proposition 8 and -edges connecting with , with , for , and with , for .

## 6 Fundamental group

If is a 4-colored graph, then the fundamental group of the represented manifold coincides with the fundamental group of the associated 2-dimensional polyhedron , since is obtained from by adding to 3-balls and pieces which are retractable to . Therefore, the computation of is a routine algebraic topology exercise: a finite presentation has generators corresponding to edges which are not in a fixed spanning tree of and relators corresponding to all 2-residues of .

In several cases the group can be obtained by selecting a particular class of edges and 2-residues, as follows. Let be any color, we define the -group of as the group generated by all -edges (with a fixed arbitrary orientation) and whose relators correspond to all -residues, for each . Just give an orientation to any involved 2-residue, choose a starting vertex and follow the cycle according to orientation. The relator is obtained by taking the -edges of the cycle in the order they are reached in the path and with the exponent or according to whether the orientation of the edge is coherent or not with the one of the cycle.

In general depends on , but when is a non-singular color, the group is strictly connected with the fundamental group of (see [29] and [24] for the case of closed manifolds).

###### Proposition 10

Let be a -colored graph, and be a non-singular color for . Then is a quotient of , obtained by adding to the relators a minimal set of -edges which connect .

Proof. The group is isomorphic to the fundamental group of the space obtained by adding to only the 3-balls corresponding to the -residues. The space has the same homotopy type of a 2-complex with 0-cells corresponding to the -residues, 1-cells corresponding to the -edges of and 2-cells corresponding to the -residues of , for . So the result is straightforward.

###### Corollary 11

Let be a -colored graph, and be a non-singular color for such that , then

## 7 Generalized regular genus

A cellular embedding of a 4-colored graph into a closed surface is called regular if there exists a cyclic permutation of such that any region of the embedding is bounded by a -residue of , for .

We recall the following result from [21]:

###### Proposition 12

Let be a bipartite (resp. non-bipartite) -colored graph. Then:

(i) for any cyclic permutation of , the graph regularly embeds into a closed orientable (resp. non-orientable) surface of Euler characteristic

 χ(Sε)=∑i∈Z4gεi,εi+1−p ,

where is the number of vertices of ;

(ii) up to equivalence there exist exactly three regular embeddings of into closed orientable (resp. non-orientable) surfaces, one for each cyclic permutation of , up to inversion, and there exist no regular embeddings of into non-orientable (resp. orientable) surfaces.

We denote by the minimum genus of among all cyclic permutations of .

###### Definition 1

Given a compact 3-manifold , the generalized regular genus of is:

 ¯¯¯G(M)=min {ρ(Γ) | Γ\emph{ represents }M}.

As observed in Remark 1, any 4-colored graph defines three generalized Heegaard splittings of the represented 3-manifold, which are induced by the choice of two colors of . Since any two colors define (up to inversion) a cyclic permutation of where they are non-consecutive, the following relation between regular embeddings of and generalized Heegaard splittings of can be established.

###### Proposition 13

Given a -colored graph , for each unordered pair , where is a cyclic permutation of , there exists a generalized Heegaard splitting of , such that regularly embeds into and (resp. ) is obtained from by attaching -handles on (resp. on ) along the residues (resp. residues). Moreover, is a Heegaard splitting (i.e. at least one of or is a handlebody) if and only if there are two non-singular colors which are non-consecutive in .

As a consequence we have:

###### Corollary 14

If is any compact -manifold, then .

Let us denote by the minimum where is taken among all 4-colored graphs representing and having at most one singular color. It is proved in [15] and [13] that coincides with the regular genus of , as originally defined by Gagliardi333Gagliardi’s definition of regular genus of a 3-manifold with non-empty boundary was given in [22] through a representation of the manifold by means of 4-colored graphs regular with respect to color 3 (i.e., obtained from a 4-colored graph by deleting some 3-edges). In fact, an analog of Proposition 12 holds for these graphs, by a suitable adaptation of the concept of regular embedding into surfaces with boundary., and that the regular genus equals the Heegaard genus (resp. is twice the Heegaard genus) of an orientable (resp. non-orientable) 3-manifold.

Obviously and there exist 3-manifolds for which the strict inequality holds, as proved in the following proposition.

###### Proposition 15

Let be a closed surface of genus , then

 ¯¯¯G(Sg×I)=g

Proof. The regular genus of has been proved to be in [3]. In order to prove the first equality, let be the graph described in Section 5.1 as representing (see Figure 4 or 5, according to the orientability of ). The genus Heegaard surface described in Section 5.1, is precisely the surface (with ) into which regularly embeds. Hence . On the other hand, by Remark 3 below, we have . This completes the proof.

However, if has connected boundary, then it follows easily from the construction in Section 2 that . More generally, we can state:

###### Proposition 16

For each compact 3-manifold , we have

 ¯¯¯G(M)=min {ρ(Γ) | Γ\emph{ represents }M\ \emph{and has at most two singular colors}}.

The result is a direct consequence of Lemma 17 below. In order to prove it, we first introduce a suitable transformation on 4-colored graphs (see [20]).

Given , let be a -residue, with , and