Compact -manifolds via -colored graphs:
a new approach
We introduce a representation via -colored graphs of compact -manifolds with (possibly empty) boundary, which appears to be very convenient for computer aided study and tabulation. Our construction is a generalization to arbitrary dimension of the one recently given by Cristofori and Mulazzani in dimension three, and it is dual to the one given by Pezzana in the seventies. In this context we establish some results concerning the topology of the represented manifolds: suspension, fundamental groups, connected sums and moves between graphs representing the same manifold. Classification results of compact orientable -manifolds representable by graphs up to six vertices are obtained, together with some properties of the G-degree of 5-colored graphs relating this approach to tensor models theory.
2010 Mathematics Subject Classification: Primary 57M27, 57N10. Secondary 57M15.
Key words and phrases: compact manifolds, colored graphs, fundamental groups, dipole moves.
1 . Introduction
The representation of closed -manifolds by means of -colored graphs has been introduced in the seventies by M. Pezzana’s research group in Modena (see ). In this type of representation, an -colored graph (which is an -regular multigraph with a proper -edge-coloration) represents a closed -manifold if certain conditions on its subgraphs are satisfied.
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 properties of the represented manifolds (see , ,  and ).
During the eighties S. Lins introduced a representation of closed 3-manifolds via 4-colored graphs, with an alternative construction which is dual to Pezzana’s one (see ). The extension of this representation to -manifolds with boundary has been performed in , where a compact -manifold with (possibly empty) boundary is associated to any -colored graph, this correspondence being surjective on the class of such manifolds without spherical boundary components.
As a consequence, an efficient computer aided catalogation/classification of -manifolds with boundary, up to some value of the order of the representing graphs, can be performed by this tool. For example, the complete classification of orientable -manifolds with toric boundary representable by graphs of order is given in  and .
In this paper we generalize the above construction to the whole family of colored graphs of arbitrary degree , showing how they represent compact -manifolds with (possibly empty) boundary. This opens the possibility to introduce an efficient algorithm for computer aided tabulation of -manifolds with boundary, for any .
The construction is described in Section 3, while Section 4 deals with graph suspensions and their connections with the topological/simplicial suspension of the represented spaces. A set of moves connecting graphs representing the same manifold is given in Section 5. However, these moves are not sufficient to ensure the equivalence of any two graphs representing the same manifold. In Section 6 we associate to any –colored graph a group which is strictly related to the fundamental group of the associated space and, therefore, it is a convenient tool for its direct computation (in many cases the two groups are in fact isomorphic). In Section 7 we establish the relation between the connected sum of graphs and the (possibly boundary) connected sum of the represented manifolds. In Section 8 a classification of all orientable 4-manifolds representable by colored graphs of order is presented. Certain properties related with tensor models theory (see for example  and ) involving the G-degree in dimension four are also obtained. More generally, as pointed out in , the strong interaction between random tensor models and the topology of -colored graphs makes significant the theme of enumeration and classification of all quasi-manifolds (or compact manifolds with boundary) represented by graphs of a given G-degree and the arguments presented in this paper might be a useful tool for this purpose.
2 . Basic notions
Throghout this paper all spaces and maps are considered in the PL-category, unless explicitely stated.
For , an -pseudomanifold is a simplicial complex such that: (i) any -simplex is the face of at least one -simplex, (ii) each -simplex is the face of exactly two -simplexes and (iii) every two -simplexes can be connected by means of a sequence of alternating - and -simplexes, each simplex being incident with the next one in the sequence. The notion of pseudomanifold naturally transfers to the underlying polyhedron of . A pseudomanifold is an -manifold out of a (possibly empty) subcomplex of dimension , composed by the singular simplexes (i.e., the simplexes whose links are not spheres). We refer to as the singular complex of and to as the singular set of .
A quasi-manifold is a pseudomanifold such that the star of any simplex verifies condition (iii) above (see ). When , an -pseudomanifold is a quasi-manifold if and only if the link of any 0-simplex of is an -quasi-manifold. It is easy to prove that the singular complex of an -dimensional quasi-manifold has dimension .
For , a singular -manifold is a quasi-manifold such that the link of any 0-simplex is a closed connected -manifold. It easily follows that the link of any -simplex of a singular manifold, with , is an -sphere. So the singular set of a singular manifold is a (possibly empty) finite set of points, and this property caracterizes singular manifolds among quasi-manifolds. Note that in dimension three (resp. dimension two) any quasi-manifold is a singular manifold (resp. is a closed surface).
A pseudo-simplicial complex is an -dimensional ball complex in which every -ball, considered with all its faces, is abstractly isomorphic to the -simplex complex. It is a fact that the first barycentric subdivision of a pseudo-simplicial complex is an abstract simplicial complex (see ). The notions of pseudomanifolds, quasi-manifolds and singular manifolds can be extended to the setting of pseudo-simplicial complexes by considering their barycentric subdivisions. If is a (pseudo-)simplicial complex we will denote by its underlying space.
Let be a positive integer and be a finite graph which is -regular (i.e., any vertex has degree ), possibly with multiple edges but without loops. An -edge-coloration of is any map , where . The coloration is called proper if adjacent edges have different colors. An edge of such that is also called a -edge. Usually we set .
An -colored graph is a connected -regular graph equipped with a proper -coloration on the edges. It is easy to see that any -colored graph has even order and it is well known that any bipartite -regular graph admits a proper -coloration (see ).
If , denote by the subgraph of obtained by dropping out from all -edges, for any . Each connected component of is called a -residue - as well as a -residue - of , indicated by . Of course, 0-residues are vertices, 1-residues are edges, 2-residues are bicolored cycles with an even number of edges, also called bigons. An -residue of is called essential if . The number of -residues of will be denoted111For we use the simplified notation instead of . by . An -colored graph is called supercontracted 222Such type of graphs were called contracted in  and in related subsequent papers, but in  the term contracted refers to a more general class of colored graphs. if , for any , where . If and are residues of and is a proper subgraph of , then is also a residue of the colored graph .
The set of all -residues of an -colored graph is denoted by and the set
results to be partially ordered by the relation (as usual means either or ) and will play a central role in our discussion.
3 . The construction
3.1 . The quasi-manifold
Given an -colored graph , we associate to it an -dimensional complex , as well as its underlying space , obtained by attaching to “cone-like” cells in one-to-one correspondence with the essential residues of , where the dimension of each cell is the number of colors of the associated residue. For the sake of conciseness, the -skeleton of will be denoted by , for any .
First of all we consider vertices and edges of as 0-dimensional and 1-dimensional cells respectively, with the natural incidence structure. So the 0-skeleton of is and the 1-skeleton is the graph , considered (as well as its essential residues) as a 1-dimensional cellular complex in the usual way. Moreover define , for any essential residue of .
If then is just a bigon and it has no essential residues. In this case and .
If , we proceed by induction via a sequence of cone attachings , where is a subspace of and is the cone over which is attached to via the map .333When we want to stress the presence of the cone vertex we will use the notation instead of . At each step these attachings are in one-to-one correspondence with the elements of , according to the following algorithm.
By induction on let
and, for any essential -residue of with , define , which is obviously a subspace of .
The final result of this process, namely , is the space , and we say that represents .
For any the -cells of are the cones , for any , and the vertex of the cone is denoted by . Moreover, we can consider any edge as the result of a cone on its endpoints with vertex , as well as any can be considered as the result of a cone on the empty set with vertex . As a consequence, each cell of is associated to a residue of and it is a cone over the union of suitable cells of lower dimension. The set is called the cone-complex associated to and we have
For any the space is an -dimensional subspace of . In particular, if is a 1-residue and if is a 0-residue. Observe that, with this notation, . In the following will denote the cone-complex .
The set of the cone-vertices is a 0-dimensional subspace of . A cell is a proper face of a cell (written as usual ) when . Therefore, if and only if .
It is worth noting that the cells of are not in general balls. In fact, an -cell is a ball if and only if is an -sphere.
An -residue of is called ordinary if is an -sphere, otherwise it is called singular. Of course, all 0-, 1- and 2-residues are ordinary. As proved later (see Corollary 3.2), is a closed -manifold if and only if all residues of are ordinary and in this case the cone-complex results to be a genuine (regular) CW-complex.
If the above construction just reduces to the attaching of a disk along its boundary to any bigon of , and therefore is a closed surface.
If the construction, which was introduced for closed 3-manifolds in , has an additional step consisting in performing the cone over any 3-residue of , considered together with the disks previously attached to its bigons. As shown in , is a closed 3-manifold if and only if all 3-residues of are ordinary.444The condition is equivalent to the arithmetic one: , where and are respectively the number of vertices, 3-residues and bigons of (see ). On the contrary, if some 3-residue is singular then is a 3-dimensional singular manifold whose singularities are the cone points of the cells corresponding to the singular 3-residues. Note that it is easy to check whether a 3-residue is ordinary or not by Euler characteristic arguments: if and are the number of vertices and bigons of , then is ordinary if and only if .
The cone-complex admits a natural “barycentric” subdivision , which results to be a simplicial complex, as follows. The 0-simplexes of are the cone vertices (i.e., the elements of ), so they are in one-to-one correspondence with the elements of . The set of -simplexes of is in one-to-one correspondence with the sequences of residues of , such that . Namely, the -simplex of associated to has vertices and it is defined by applying to the sequence of cone constructions corresponding to the residues , in this order. Therefore
It is interesting to note that is isomorphic to the order complex of the poset (see Section 9 of ). In the following with the notation we always mean that .
If is an -residue of , we denote by the subcomplex of obtained by restricting the barycentric subdivision to the cell . Therefore a -simplex is a simplex of if and only if . It is a standard fact that , where . Note that .
For , we denote by the complex composed by the standard -simplex and all its faces and by its boundary complex. Therefore and denote their barycentric subdivisions, respectively. Observe that , for any . In the following we set, as usual, , for any simplicial complex .
Let be an -simplex of . If is a -residue of , for , then is isomorphic to the complex . Hence, is homeomorphic to .
Proof. For suppose is a -residue of , with , and set . A simplex belongs to if there exist with , such that , , , , . Of course, the simplex is a generic element of . On the other hand, for the chain is a generic element of the order complex of the subposet of defined by . It is easy to see that the poset is isomorphic to the poset of the proper subsets of , via the correspondence , where is the set of colors of . Since the order complex of the proper subsets of a finite set is isomorphic to , we obtain the proof.
Let be an -colored graph, then is a closed manifold if and only if all -residues of are ordinary.
Proof. Since is a closed manifold if and only if the link of any 0-simplex of is an -sphere, the result immediately follows from Proposition 3.1.555Notice that a proof of Corollary 3.2 is given in  using the dual construction described later on.
As a consequence, all residues of a ordinary residue are ordinary.
The singular complex of is denoted by , and therefore indicates the singular set of . By Proposition 3.1, an -simplex of belongs to if and only if is a singular residue of (and therefore any is a singular residue, for ). As mentioned before, the set is empty when and finite when .
The set of ordinary (resp. singular) residues of will be denoted by (resp. ) and let (resp. ), for any . Of course, for all . If is a connected component of then define by setting . As a consequence, is a single point if and only if . It is easy to see that .
Let be an -colored graph. If is non-empty then it is a full subcomplex of and
Proof. Let be a simplex of with all vertices belonging to . As a consequence, is a singular residue for all and therefore is a simplex of . This prove that is a full subcomplex of . Since all 2-residues are ordinary, any simplex of has dimension . Moreover, if is a singular -residue, and is a maximal chain in then is an -simplex of . This concludes the proof.
The above construction of is dual to the one given by Pezzana in the seventies (see for example the survey paper ), which associates an -dimensional pseudo-simplicial complex to the -colored graph in the following way:
(i) take an -simplex for each and color its vertices injectively by ;
(ii) if are joined by a -edge of , glue the -faces of and opposite to the vertices colored by , in such a way that equally colored vertices are identified together.
As a consequence, inherits a coloration on its vertices, thus becoming a balanced666An -dimensional pseudo-simplicial complex is called balanced if its vertices are labelled by a set of colors in such a way that any 1-simplex has vertices labelled with different colors. pseudo-simplicial complex. This construction yields a one-to-one inclusion reversing correspondence between the residues of and the simplexes of , in such a way that the -simplex of having vertices colored by is associated to the -residue of having vertices corresponding via (i) to the -simplexes of containing .
The simplicial complexes and are isomorphic. Therefore .
Proof. If we denote by the barycenter of the simplex of , then the bijective map between the 0-skeletons of the two complexes defined by , for any , induces an isomorphism between the simplicial complexes and .
The duality between the complexes and is given by the fact that is the complex dual to (i.e., composed by the dual cells of the simplexes of ). The definition of dual complex is analogous to the one in the simplicial case (see for example page 29 of ), and it is well defined also in the pseudo-simplicial case since the barycentric subdivision of is a simplicial complex: if is an -simplex of with vertex set , then its dual cell is the -dimensional subcomplex of given by Therefore the cone-complex is exactly the complex Moreover, the singular set of is also the underlying space of the subcomplex of composed by the simplexes such that is singular. It is easy to see that . Note that maximal simplexes of correspond to minimal singular residues of (i.e., singular residues having no singular subresidues).
It is not difficult to see that is an -dimensional quasi-manifold. Viceversa, any -dimensional quasi-manifold admits a representation by -colored graphs:
 Let be an -dimensional (pseudo-)simplicial complex. Then there exists an -colored graph such that if and only if is a quasi-manifold.
The following result is straightforward:
Let be an -colored graph, then is a singular manifold if and only if for any .
3.2 . The manifold
As previously noticed, is not always a manifold since it may contain singular points. In order to obtain a manifold we remove the interior of a regular neighborhood of the singular set in .
Let be a regular neighborhood of in , then is a compact -manifold with (possibly empty) boundary.
Proof. If the result is trivial since is a closed -manifold. If then Theorem 5.3 of , used with , says that the topological boundary of in is bicollared in the (non-compact) -manifold . It immediately follows that is a compact -manifold with boundary .
We define and say that the -colored graph also represents . If is empty then is a closed -manifold. Otherwise, by the previous lemma is a compact -manifold with non-empty boundary.
The next proposition gives a combinatorial description of . Recall that, for a subcomplex of a simplicial complex , the simplicial neighborhood of in is the subcomplex of containing all simplexes of not disjoint from , and their faces. Moreover, the complement of in is the subcomplex of containing all simplexes of disjoint from and set .
If is an -colored graph then
Proof. Since is a full subcomplex of , we can choose in the first barycentric subdivision of the complex , by defining (see ), and therefore we have and . It is easy to realize that and .
As an immediate consequence of the fact that is a quasi-manifold it follows that is connected. Moreover, the connected components of are in one-to-one correpondence with the connected components of , since the simplexes of have connected links in .
The compact -manifold can be obtained from by an alternative algorithm, which differs from the one producing only in correspondence of singular residues, where the cone constructions are replaced by cylinder ones. Namely, for any singular -residue , instead of attaching to the cone along the base, we attach the cylinder along one of the two bases. In order to prove that, it suffices to show that , for any singular residue of . This can be achieved by applying Lemma 1.22 of , with , and , where .
All interesting compact connected -manifolds can be represented by -colored graphs, as stated by the next proposition.
If is a compact connected -manifold with a (possibly empty) boundary without spherical components, then there exists an -colored graph such that .
Proof. Let be the space obtained from by performing a cone over any component of , then is an -dimensional singular manifold. By Theorem 1 of , there exists an -colored graph such that . The singular set is the set of vertices of the cones, and the union of the cones is a regular neighborhood of in . As a consequence, .
Since two compact -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 -manifolds without spherical boundary components.
When is a singular manifold, the boundary of admits a simple characterization in terms of the spaces represented by the singular residues.
Let be an -colored graph such that is a singular manifold and let be an -singular residue of . Then the component of corresponding to is homeomorphic to .
Proof. The component of corresponding to is . An -simplex of is , where is a chain in , for , such that and is the barycenter of the simplex . The simplex belongs to (resp. to ) if and only if is the last element of the chain (resp. is not an element of the chain ). Then the map defined by , where is the chain , with , and is the chain , induces an isomorphism between and . As a consequence, is homeomorphic to ( since all residues of are ordinary).
If is a singular manifold then has no spherical components.
If , the graph has no singular residues, and any singular -residue is contained in exactly two -residues and . Using the isomorphism defined in the proof of Lemma 3.12, we can suppose that is a boundary component of both and . Therefore, we can define the space by gluing with along their common boundary components (corresponding to common singular -residues). Using this trick the boundary of can be described as gluings of the manifolds with boundary corresponding to the singular -residues of .
Let be an -colored graph such that and let be a connected component of . Then the component of corresponding to is homeomorphic to , where are the -residues of .
Proof. Let be a singular -residue of belonging to , and let be the singular -residues of . It suffices to prove that (i) is homeomorphic to and (ii) is homeomorphic to and is a boundary component of , for .
Referring to the proof of Lemma 3.12, the simplex of belongs to if and only if either (i) is the last element of the chain , and , for , or (ii) there exists such that is the last element of the chain . Let be the complex containing the simplexes satisfying (i) and, for , let be the complex containing the simplexes satisfying (ii) and such that is a residue of . Moreover, belongs to if and only if is not an element of the chain and , for . In particular, belongs to if and only if is not an element of the chain and is a residue of .
The map defined by , where is the chain , with , and is the chain , induces an isomorphism between and . Therefore, is homeomorphic to . Moreover, for , the map defined in the same way of with being the chain (resp. ) if (resp. if ) induces an isomorphism between and . Therefore, is homeomorphic to , which is homeomorphic to