Compact n-manifolds via (n+1)-colored graphs: a new approach
CCompact n -manifolds via ( n + 1)-colored graphs:a new approach Luigi GRASSELLI * and Michele MULAZZANI *** Dipartimento di Scienze e Metodi dell’Ingegneria, Universit`a di Modena e Reggio Emilia (Italy) ** Dipartimento di Matematica and ARCES, Universit`a di Bologna (Italy)
November 21, 2018
Abstract
We introduce a representation via ( n + 1)-colored graphs of compact n -manifoldswith (possibly empty) boundary, which appears to be very convenient for computeraided study and tabulation. Our construction is a generalization to arbitrary dimen-sion of the one recently given by Cristofori and Mulazzani in dimension three, andit is dual to the one given by Pezzana in the seventies. In this context we establishsome results concerning the topology of the represented manifolds: suspension, fun-damental groups, connected sums and moves between graphs representing the samemanifold. Classification results of compact orientable 4-manifolds representable bygraphs up to six vertices are obtained, together with some properties of the G-degreeof 5-colored graphs relating this approach to tensor models theory. Primary 57M27, 57N10. Secondary 57M15.
Key words and phrases: compact manifolds, colored graphs, fundamental groups, dipole moves.
1. Introduction
The representation of closed n -manifolds by means of ( n +1)-colored graphs has been introduced inthe seventies by M. Pezzana’s research group in Modena (see [18]). In this type of representation,an ( n + 1)-colored graph (which is an ( n + 1)-regular multigraph with a proper ( n + 1)-edge-coloration) represents a closed n -manifold if certain conditions on its subgraphs are satisfied.The study of this kind of representation has yielded several results, especially with regard tothe definition of combinatorial invariants and their relations with topological properties of therepresented manifolds (see [4], [11], [14] and [23]). a r X i v : . [ m a t h . G T ] N ov uring the eighties S. Lins introduced a representation of closed 3-manifolds via 4-coloredgraphs, with an alternative construction which is dual to Pezzana’s one (see [30]). The extensionof this representation to 3-manifolds with boundary has been performed in [15], where a compact 3-manifold with (possibly empty) boundary is associated to any 4-colored graph, this correspondencebeing surjective on the class of such manifolds without spherical boundary components.As a consequence, an efficient computer aided catalogation/classification of 3-manifolds withboundary, up to some value of the order of the representing graphs, can be performed by thistool. For example, the complete classification of orientable 3-manifolds with toric boundary rep-resentable by graphs of order ≤
14 is given in [12] and [13].In this paper we generalize the above construction to the whole family of colored graphs ofarbitrary degree n + 1, showing how they represent compact n -manifolds with (possibly empty)boundary. This opens the possibility to introduce an efficient algorithm for computer aided tab-ulation of n -manifolds with boundary, for any n ≥ n +1)–colored graph a group which is strictly related tothe fundamental group of the associated space and, therefore, it is a convenient tool for its directcomputation (in many cases the two groups are in fact isomorphic). In Section 7 we establish therelation between the connected sum of graphs and the (possibly boundary) connected sum of therepresented manifolds. In Section 8 a classification of all orientable 4-manifolds representable bycolored graphs of order ≤ n + 1)-colored graphs makes significant the theme of enumeration and classificationof all quasi-manifolds (or compact manifolds with boundary) represented by graphs of a givenG-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 explicitelystated.For n ≥
1, an n -pseudomanifold is a simplicial complex K such that: (i) any h -simplex isthe face of at least one n -simplex, (ii) each ( n − n -simplexesand (iii) every two n -simplexes can be connected by means of a sequence of alternating n - and( n − | K | of K . A pseudomanifold K is an n -manifold out of a (possibly empty) subcomplex S K of dimension ≤ n −
2, composed bythe singular simplexes (i.e., the simplexes whose links are not spheres). We refer to S K as the singular complex of K and to | S K | as the singular set of | K | .A quasi-manifold is a pseudomanifold such that the star of any simplex verifies condition (iii)above (see [20]). When n ≥
2, an n -pseudomanifold K is a quasi-manifold if and only if the linkof any 0-simplex of K is an ( n − n -dimensional quasi-manifold has dimension ≤ n − or n ≥
2, a singular n -manifold is a quasi-manifold such that the link of any 0-simplex is aclosed connected ( n − h -simplex of a singularmanifold, with h >
0, is an ( n − h − pseudo-simplicial complex is an n -dimensional ball complex in which every h -ball, consideredwith all its faces, is abstractly isomorphic to the h -simplex complex. It is a fact that the firstbarycentric subdivision of a pseudo-simplicial complex is an abstract simplicial complex (see [28]).The notions of pseudomanifolds, quasi-manifolds and singular manifolds can be extended to thesetting of pseudo-simplicial complexes by considering their barycentric subdivisions. If K is a(pseudo-)simplicial complex we will denote by | K | its underlying space.Let n be a positive integer and Γ = ( V (Γ) , E (Γ)) be a finite graph which is ( n + 1)-regular(i.e., any vertex has degree n + 1), possibly with multiple edges but without loops. An ( n + 1) -edge-coloration of Γ is any map γ : E (Γ) → C , where | C | = n + 1. The coloration is called proper if adjacent edges have different colors. An edge e of Γ such that γ ( e ) = c is also called a c -edge .Usually we set C = ∆ n = { , , . . . , n } .An ( n + 1) -colored graph is a connected ( n + 1)-regular graph equipped with a proper ( n + 1)-coloration on the edges. It is easy to see that any ( n + 1)-colored graph has even order and it iswell known that any bipartite ( n + 1)-regular graph admits a proper ( n + 1)-coloration (see [19]).If ∆ ⊂ ∆ n , denote by Γ ∆ the subgraph of Γ obtained by dropping out from Γ all c -edges, forany c ∈ (cid:98) ∆ = ∆ n − ∆. Each connected component Λ of Γ ∆ is called a ∆ -residue - as well as a | ∆ | -residue - of Γ, indicated by Λ ≺ Γ. 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 h -residue ofΓ is called essential if 2 ≤ h ≤ n . The number of ∆-residues of Γ will be denoted by g ∆ . An( n + 1)-colored graph Γ is called supercontracted if g (cid:98) c = 1, for any c ∈ ∆ n , where (cid:98) c = ∆ n − { c } .If Λ and Λ (cid:48) are residues of Γ and Λ (cid:48) is a proper subgraph of Λ, then Λ (cid:48) is also a residue of thecolored graph Λ.The set of all h -residues of an ( n + 1)-colored graph Γ is denoted by R h (Γ) and the set R (Γ) = (cid:91) ≤ h ≤ n R h (Γ)results to be partially ordered by the relation (cid:22) (as usual (cid:22) means either ≺ or =) and will playa central role in our discussion.For elementary notions about graphs we refer to [32] and for general PL-topology we refer to[24], [28] and [29]. For ∆ = { i, j } we use the simplified notation g i,j instead of g { i,j } . Such type of graphs were called contracted in [20] and in related subsequent papers, but in [15] theterm contracted refers to a more general class of colored graphs. . The construction (cid:99) M Γ Given an ( n + 1)-colored graph Γ, we associate to it an n -dimensional complex C Γ , as well as itsunderlying space (cid:99) M Γ = |C Γ | , obtained by attaching to Γ “cone-like” cells in one-to-one correspon-dence with the essential residues of Γ, where the dimension of each cell is the number of colors ofthe associated residue. For the sake of conciseness, the h -skeleton of C Γ will be denoted by Γ ( h ) ,for any h = 0 , , . . . , n .First of all we consider vertices and edges of Γ as 0-dimensional and 1-dimensional cells re-spectively, with the natural incidence structure. So the 0-skeleton Γ (0) of C Γ is V (Γ) and the1-skeleton Γ (1) is the graph Γ, considered (as well as its essential residues) as a 1-dimensionalcellular complex in the usual way. Moreover define Λ (1) = Λ, for any essential residue Λ of Γ.If n = 1 then Γ is just a bigon and it has no essential residues. In this case C Γ = V (Γ) ∪ E (Γ)and (cid:99) M Γ = Γ ∼ = S .If n ≥
2, we proceed by induction via a sequence of cone attachings Y → Y ∪ C ( X ), where X is a subspace of Y and C ( X ) = ( X × [0 , / ( x, ∼ ( x (cid:48) ,
1) is the cone over X which is attachedto Y via the map ( x, ∈ C ( X ) (cid:55)→ x ∈ Y . At each step h = 2 , . . . , n these attachings are inone-to-one correspondence with the elements of R h (Γ), according to the following algorithm.By induction on h = 2 , . . . , n letΓ ( h ) = Γ ( h − (cid:91) Λ ∈R h (Γ) C (Λ ( h − )and, for any essential h (cid:48) -residue Θ of Γ with h (cid:48) > h , define Θ ( h ) = Θ ( h − ∪ Λ ∈R h (Θ) C (Λ ( h − ),which is obviously a subspace of Γ ( h ) .The final result of this process, namely Γ ( n ) , is the space (cid:99) M Γ , and we say that Γ represents (cid:99) M Γ .For any h = 2 , . . . , n the h -cells of C Γ are the cones c Λ = C (Λ ( h − ), for any Λ ∈ R h (Γ), andthe vertex of the cone is denoted by V Λ . Moreover, we can consider any edge e ∈ E (Γ) as theresult of a cone on its endpoints with vertex V e , as well as any v ∈ V (Γ) can be considered asthe result of a cone on the empty set with vertex V v = v . As a consequence, each cell of C Γ isassociated to a residue of Γ and it is a cone over the union of suitable cells of lower dimension.The set C Γ = { c Λ | Λ ∈ R (Γ) } is called the cone-complex associated to Γ and we have (cid:99) M Γ = |C Γ | = (cid:91) Λ ∈R (Γ) c Λ = (cid:91) Λ ∈R n (Γ) c Λ . For any Λ ∈ R h (Γ) the space (cid:99) M Λ = |C Λ | is an ( h − (cid:99) M Γ . In particular, (cid:99) M Λ = S if Λ is a 1-residue and (cid:99) M Λ = S − = ∅ if Λ is a 0-residue. Observe that, with thisnotation, c Λ = C ( (cid:99) M Λ ). In the following ¯ c Λ will denote the cone-complex C Λ ∪ { c Λ } .The set Υ = { V Λ | Λ ∈ R (Γ) } of the cone-vertices is a 0-dimensional subspace of (cid:99) M Γ . A cell c Λ (cid:48) is a proper face of a cell c Λ (written as usual c Λ (cid:48) < c Λ ) when c Λ (cid:48) ⊂ c Λ . Therefore, c Λ (cid:48) < c Λ ifand only if Λ (cid:48) ≺ Λ. When we want to stress the presence of the cone vertex V = ( X × { } ) / ∼ we will use the notation C V ( X ) instead of C ( X ). t is worth noting that the cells of C Γ are not in general balls. In fact, an h -cell c Λ is a ball ifand only if (cid:99) M Λ is an ( h − h -residue Λ of Γ is called ordinary if (cid:99) M Λ is an ( h − singular .Of course, all 0-, 1- and 2-residues are ordinary. As proved later (see Corollary 3.2), (cid:99) M Γ is a closed n -manifold if and only if all residues of Γ are ordinary and in this case the cone-complex C Γ resultsto be a genuine (regular) CW-complex.If n = 2 the above construction just reduces to the attaching of a disk along its boundary toany bigon of Γ, and therefore (cid:99) M Γ is a closed surface.If n = 3 the construction, which was introduced for closed 3-manifolds in [31], has an additionalstep consisting in performing the cone over any 3-residue of Γ, considered together with the diskspreviously attached to its bigons. As shown in [31], (cid:99) M Γ is a closed 3-manifold if and only ifall 3-residues of Γ are ordinary. On the contrary, if some 3-residue is singular then (cid:99) M Γ is a3-dimensional singular manifold whose singularities are the cone points of the cells correspondingto the singular 3-residues. Note that it is easy to check whether a 3-residue Λ is ordinary or notby Euler characteristic arguments: if v and b are the number of vertices and bigons of Λ, then Λis ordinary if and only if b − v/ C Γ admits a natural “barycentric” subdivision C (cid:48) Γ , which results to be asimplicial complex, as follows. The 0-simplexes of C (cid:48) Γ are the cone vertices (i.e., the elements ofΥ), so they are in one-to-one correspondence with the elements of R (Γ). The set of h -simplexesof C (cid:48) Γ is in one-to-one correspondence with the sequences (Λ , Λ , . . . , Λ h ) of residues of Γ, suchthat Λ ≺ Λ ≺ · · · ≺ Λ h . Namely, the h -simplex σ h of C (cid:48) Γ associated to (Λ , Λ , . . . , Λ h ) hasvertices V Λ , V Λ , . . . , V Λ h and it is defined by applying to V Λ the sequence of cone constructionscorresponding to the residues Λ , . . . , Λ h , in this order. Therefore σ h = (cid:104) V Λ , V Λ , . . . , V Λ h (cid:105) = C V Λ h ( C V Λ h − ( · · · ( C V Λ1 ( V Λ )) · · · )) . It is interesting to note that C (cid:48) Γ is isomorphic to the order complex of the poset R (Γ) (see Section9 of [1]). In the following with the notation (cid:104) V Λ , V Λ , . . . , V Λ h (cid:105) we always mean that Λ ≺ Λ ≺· · · ≺ Λ h .If Λ is an h -residue of Γ, we denote by c (cid:48) Λ the subcomplex of C (cid:48) Γ obtained by restrict-ing the barycentric subdivision to the cell c Λ . Therefore a k -simplex σ k = (cid:104) V Λ , V Λ , . . . , V Λ k (cid:105) is a simplex of c (cid:48) Λ if and only if Λ k (cid:22) Λ. It is a standard fact that c (cid:48) Λ = V Λ (cid:63) ˙ c (cid:48) Λ , where˙ c (cid:48) Λ = {(cid:104) V Λ , V Λ , . . . , V Λ k (cid:105) ∈ c (cid:48) Λ | Λ k ≺ Λ } . Note that | ˙ c (cid:48) Λ | = (cid:99) M Λ .For n ≥
0, we denote by ¯ s n the complex composed by the standard n -simplex s n and all itsfaces and by ˙ s n its boundary complex. Therefore ¯ s (cid:48) n and ˙ s (cid:48) n denote their barycentric subdivisions,respectively. Observe that | ˙ s n | = | ˙ s (cid:48) n | = S n − , for any n ≥
0. In the following we set, as usual,
A (cid:63) ∅ = A , for any simplicial complex A . Proposition 3.1
Let σ h = (cid:104) V Λ , V Λ , . . . , V Λ h (cid:105) be an h -simplex of C (cid:48) Γ . If Λ i is a d i -residue of Γ ,for i = 0 , , . . . , h , then Link ( σ h , C (cid:48) Γ ) is isomorphic to the complex ˙ c (cid:48) Λ (cid:63) ˙ s (cid:48) d − d − (cid:63) · · · (cid:63) ˙ s (cid:48) d h − d h − − (cid:63) ˙ s (cid:48) n − d h − . Hence, | Link ( σ h , C (cid:48) Γ ) | is homeomorphic to (cid:99) M Λ (cid:63) S n − h − d − . Proof.
For i = 1 , . . . , h suppose Λ i is a D i -residue of Γ, with | D i | = d i , and set D h +1 =∆ n . A simplex τ s = (cid:104) V Ω , V Ω , . . . , V Ω s (cid:105) belongs to Link ( σ h , C (cid:48) Γ ) if there exist j , j , . . . , j h with The condition is equivalent to the arithmetic one: g + g = g , where g , g and g are respectivelythe number of vertices, 3-residues and bigons of Γ (see [31]). ≤ j ≤ j ≤ . . . ≤ j h < n , such that Ω ≺ . . . ≺ Ω j ≺ Λ , Λ ≺ Ω j +1 ≺ . . . ≺ Ω j ≺ Λ , . . . , Λ h − ≺ Ω j h − +1 ≺ . . . ≺ Ω j h ≺ Λ h , Λ h ≺ Ω j h +1 ≺ . . . ≺ Ω s . Of course, the simplex τ = (cid:104) V Ω , . . . , V Ω j (cid:105) is a generic element of ˙ c (cid:48) Λ . On the other hand, for i = 1 , . . . , h + 1 the chainΩ j i − +1 ≺ . . . ≺ Ω j i is a generic element of the order complex of the subposet R i of R (Γ) definedby R i = { Ω ∈ R (Γ) | Λ i − ≺ Ω ≺ Λ i } . It is easy to see that the poset R i is isomorphic to theposet of the proper subsets of D i − D i − , via the correspondence Ω (cid:55)→ D (cid:48) − D i − , where D (cid:48) is theset of colors of Ω. Since the order complex of the proper subsets of a finite set X is isomorphic to˙ s (cid:48)| X |− , we obtain the proof. Corollary 3.2
Let Γ be an ( n + 1) -colored graph, then (cid:99) M Γ is a closed manifold if and only if all n -residues of Γ are ordinary. Proof.
Since (cid:99) M Γ is a closed manifold if and only if the link of any 0-simplex of C (cid:48) Γ is an( n − As a consequence, all residues of a ordinary residue are ordinary.The singular complex of C (cid:48) Γ is denoted by S Γ , and therefore |S Γ | indicates the singular set of (cid:99) M Γ . By Proposition 3.1, an h -simplex σ h = (cid:104) V Λ , V Λ , . . . , V Λ h (cid:105) of C (cid:48) Γ belongs to S Γ if and onlyif Λ is a singular residue of Γ (and therefore any Λ i is a singular residue, for i = 1 , . . . , h ). Asmentioned before, the set |S Γ | is empty when n = 2 and finite when n = 3.The set of ordinary (resp. singular) residues of Γ will be denoted by R (cid:48) (Γ) (resp. R (cid:48)(cid:48) (Γ))and let R (cid:48) h (Γ) = R h (Γ) ∩ R (cid:48) (Γ) (resp. R (cid:48)(cid:48) h (Γ) = R h (Γ) ∩ R (cid:48)(cid:48) (Γ)), for any 0 ≤ h ≤ n . Of course, R (cid:48)(cid:48) h (Γ) = ∅ for all h ≤
2. If S is a connected component of |S Γ | then define R S ⊆ R (cid:48)(cid:48) n (Γ) by setting R S = { Λ ∈ R (cid:48)(cid:48) n (Γ) | V Λ ∈ S } . As a consequence, S is a single point if and only if | R S | = 1. It iseasy to see that dim( S ) ≤ | R S | − Lemma 3.3
Let Γ be an ( n + 1) -colored graph. If S Γ is non-empty then it is a full subcomplex of C (cid:48) Γ and dim( S Γ ) = n − min { h | R (cid:48)(cid:48) h (Γ) (cid:54) = ∅} ≤ n − . Proof.
Let σ k = (cid:104) V Λ , V Λ , . . . , V Λ k (cid:105) be a simplex of C (cid:48) Γ with all vertices belonging to S Γ . Asa consequence, Λ i is a singular residue for all i = 0 , , . . . , k and therefore σ k is a simplex of S Γ .This prove that S Γ is a full subcomplex of C (cid:48) Γ . Since all 2-residues are ordinary, any simplex of S Γ has dimension < n −
2. Moreover, if Λ is a singular n -residue, and (Λ = Λ , Λ , . . . , Λ n − h ) isa maximal chain in R (Γ) then (cid:104) V Λ , V Λ , . . . , V Λ n − h (cid:105) is an ( n − h )-simplex of S Γ . This concludesthe proof.The above construction of (cid:99) M Γ is dual to the one given by Pezzana in the seventies (see forexample the survey paper [18]), which associates an n -dimensional pseudo-simplicial complex K Γ to the ( n + 1)-colored graph Γ in the following way:(i) take an n -simplex s v for each v ∈ V (Γ) and color its vertices injectively by ∆ n ;(ii) if v, w ∈ V (Γ) are joined by a c -edge of Γ, glue the ( n − s v and s w opposite tothe vertices colored by c , in such a way that equally colored vertices are identified together. Notice that a proof of Corollary 3.2 is given in [16] using the dual construction described later on. s a consequence, K Γ inherits a coloration on its vertices, thus becoming a balanced pseudo-simplicial complex. This construction yields a one-to-one inclusion reversing correspondence Λ ↔ s Λ between the residues of Γ and the simplexes of K Γ , in such a way that the h -simplex s Λ of K Γ having vertices colored by ∆ ⊂ ∆ n is associated to the (cid:98) ∆-residue of Γ having verticescorresponding via (i) to the n -simplexes of K Γ containing s Λ . Proposition 3.4
The simplicial complexes C (cid:48) Γ and K (cid:48) Γ are isomorphic. Therefore |K Γ | ∼ = |C Γ | = (cid:99) M Γ . Proof.
If we denote by (cid:98) s the barycenter of the simplex s of K Γ , then the bijective map d : S ( K (cid:48) Γ ) → S ( C (cid:48) Γ ) between the 0-skeletons of the two complexes defined by d ( (cid:99) s Λ ) = V Λ , forany Λ ∈ R (Γ), induces an isomorphism between the simplicial complexes K (cid:48) Γ and C (cid:48) Γ . Remark 3.5
The duality between the complexes C Γ and K Γ is given by the fact that C Γ is thecomplex K ∗ Γ dual to K Γ (i.e., composed by the dual cells of the simplexes of K Γ ). The definitionof dual complex is analogous to the one in the simplicial case (see for example page 29 of [29]),and it is well defined also in the pseudo-simplicial case since the barycentric subdivision K (cid:48) Γ of K Γ is a simplicial complex: if s Λ is an h -simplex of K Γ with vertex set V , then its dual cell s ∗ Λ is the ( n − h ) -dimensional subcomplex of K (cid:48) Γ given by s ∗ Λ = ∩ v ∈ V star( v, K (cid:48) Γ ) . Therefore the cone-complex is exactly the complex C Γ = {| s ∗ Λ | | s Λ ∈ K Γ } . Moreover, the singular set of (cid:99) M Γ is also theunderlying space of the subcomplex ¯ S Γ of K Γ composed by the simplexes s Λ such that Λ is singular.It is easy to see that S Γ = ¯ S (cid:48) Γ . Note that maximal simplexes of ¯ S Γ correspond to minimal singularresidues of Γ (i.e., singular residues having no singular subresidues). It is not difficult to see that C (cid:48) Γ = K (cid:48) Γ is an n -dimensional quasi-manifold. Viceversa, any n -dimensional quasi-manifold admits a representation by ( n + 1)-colored graphs: Proposition 3.6 [20] Let K be an n -dimensional (pseudo-)simplicial complex. Then there existsan ( n + 1) -colored graph Γ such that (cid:99) M Γ ∼ = | K | if and only if K is a quasi-manifold. The following result is straightforward:
Lemma 3.7
Let Γ be an ( n + 1) -colored graph, then (cid:99) M Γ is a singular manifold if and only if R (cid:48)(cid:48) h (Γ) = ∅ for any h < n . Proof. If (cid:99) M Γ is a singular manifold then dim( S Γ ) ≤ R (cid:48)(cid:48) h (Γ) = ∅ for any h < n . On the contrary, if R (cid:48)(cid:48) h (Γ) = ∅ for any h < n then (cid:99) M Γ is a singular manifold byProposition 3.1, since (cid:99) M Λ is a closed manifold when Λ is an n -residue of Γ. An n -dimensional pseudo-simplicial complex is called balanced if its vertices are labelled by a set of n + 1 colors in such a way that any 1-simplex has vertices labelled with different colors. .2. The manifold M Γ As previously noticed, (cid:99) M Γ is not always a manifold since it may contain singular points. In orderto obtain a manifold we remove the interior of a regular neighborhood of the singular set |S Γ | in (cid:99) M Γ . Lemma 3.8
Let N ( S Γ ) be a regular neighborhood of |S Γ | in (cid:99) M Γ , then (cid:99) M Γ − int ( N ( S Γ )) is acompact n -manifold with (possibly empty) boundary. Proof. If S Γ = ∅ the result is trivial since (cid:99) M Γ is a closed n -manifold. If S Γ (cid:54) = ∅ thenTheorem 5.3 of [10], used with Y = ∅ , says that the topological boundary bd( N ( S Γ )) of N ( S Γ )in (cid:99) M Γ is bicollared in the (non-compact) n -manifold (cid:99) M Γ − |S Γ | . It immediately follows that M Γ = (cid:99) M Γ − int( N ( S Γ )) is a compact n -manifold with boundary ∂M Γ = bd( N ( S Γ )).We define M Γ = (cid:99) M Γ − int( N ( S Γ )) and say that the ( n + 1)-colored graph Γ also represents M Γ . If S Γ is empty then M Γ = (cid:99) M Γ is a closed n -manifold. Otherwise, by the previous lemma M Γ is a compact n -manifold with non-empty boundary.The next proposition gives a combinatorial description of M Γ . Recall that, for a subcomplex H of a simplicial complex K , the simplicial neighborhood of H in K is the subcomplex N ( H, K ) of K containing all simplexes of K not disjoint from H , and their faces. Moreover, the complementof H in K is the subcomplex C ( H, K ) of K containing all simplexes of K disjoint from H andset ˙ N ( H, K ) = N ( H, K ) ∩ C ( H, K ). Proposition 3.9 If Γ is an ( n + 1) -colored graph then M Γ = | C ( S (cid:48) Γ , C (cid:48)(cid:48) Γ ) | = (cid:91) Λ ∈R (cid:48) (Γ) | Star ( V Λ , C (cid:48)(cid:48) Γ ) | . Moreover, ∂M Γ = | ˙ N ( S (cid:48) Γ , C (cid:48)(cid:48) Γ ) | . Proof.
Since S Γ is a full subcomplex of C (cid:48) Γ , we can choose N ( S Γ ) in the first barycentricsubdivision C (cid:48)(cid:48) Γ of the complex C (cid:48) Γ , by defining N ( S Γ ) = | N ( S (cid:48) Γ , C (cid:48)(cid:48) Γ ) | (see [10]), and thereforewe have M Γ = | C ( S (cid:48) Γ , C (cid:48)(cid:48) Γ ) | and ∂M Γ = | ˙ N ( S (cid:48) Γ , C (cid:48)(cid:48) Γ ) | . It is easy to realize that N ( S (cid:48) Γ , C (cid:48)(cid:48) Γ ) = ∪ v ∈ S ( S Γ ) Star( v, C (cid:48)(cid:48) Γ ) = ∪ Λ ∈R (cid:48)(cid:48) (Γ) Star( V Λ , C (cid:48)(cid:48) Γ ) and C ( S (cid:48) Γ , C (cid:48)(cid:48) Γ ) = ∪ v ∈ ( S ( C (cid:48) Γ ) − S ( S Γ )) Star( v, C (cid:48)(cid:48) Γ ) = ∪ Λ ∈R (cid:48) (Γ) Star( V Λ , C (cid:48)(cid:48) Γ ).As an immediate consequence of the fact that (cid:99) M Γ is a quasi-manifold it follows that M Γ isconnected. Moreover, the connected components of ∂M Γ are in one-to-one correpondence withthe connected components of |S Γ | , since the simplexes of S (cid:48) Γ have connected links in C (cid:48)(cid:48) Γ . Remark 3.10
The compact n -manifold M Γ can be obtained from Γ by an alternative algorithm,which differs from the one producing (cid:99) M Γ only in correspondence of singular residues, where thecone constructions are replaced by cylinder ones. Namely, for any singular h -residue Λ , instead ofattaching to Γ ( h − the cone C (Λ ( h − ) along the base, we attach the cylinder Cyl (Λ ( h − ) along oneof the two bases. In order to prove that, it suffices to show that M Γ ∩ c Λ = M Λ × I , for any singularresidue Λ of Γ . This can be achieved by applying Lemma 1.22 of [29], with V = V Λ , K = C ( S (cid:48) Λ , C (cid:48)(cid:48) Λ ) and L = ( V Λ (cid:63)D ) ∪ C (( V Λ (cid:63) S Λ ) (cid:48) , ( V Λ (cid:63) C (cid:48) Λ ) (cid:48) ) , where D = C (( V Λ (cid:63) S Λ ) (cid:48) , ( V Λ (cid:63) C (cid:48) Λ ) (cid:48) ) ∩ Star ( V Λ , ( V Λ (cid:63) C (cid:48) Λ ) (cid:48) ) . ll interesting compact connected n -manifolds can be represented by ( n + 1)-colored graphs,as stated by the next proposition. Proposition 3.11 If M is a compact connected n -manifold with a (possibly empty) boundarywithout spherical components, then there exists an ( n + 1) -colored graph Γ such that M = M Γ . Proof.
Let (cid:99) M be the space obtained from M by performing a cone over any component of ∂M , then (cid:99) M is an n -dimensional singular manifold. By Theorem 1 of [7], there exists an ( n + 1)-colored graph Γ such that (cid:99) M = (cid:99) M Γ . The singular set |S Γ | is the set of vertices of the cones, andthe union of the cones is a regular neighborhood of |S Γ | in (cid:99) M Γ . As a consequence, M Γ = M .Since two compact n -manifolds are homeomorphic if and only if (i) they have the same numberof spherical boundary components and (ii) they are homeomorphic after capping off by balls thesecomponents, there is no loss of generality in studying compact n -manifolds without sphericalboundary components.When (cid:99) M Γ is a singular manifold, the boundary of M Γ admits a simple characterization interms of the spaces represented by the singular residues. Lemma 3.12
Let Γ be an ( n + 1) -colored graph such that (cid:99) M Γ is a singular manifold and let Λ bean n -singular residue of Γ . Then the component of ∂M Γ corresponding to Λ is homeomorphic to M Λ = (cid:99) M Λ . Proof.
The component of ∂M Γ corresponding to Λ is B = | Link( V Λ , C (cid:48)(cid:48) Γ ) | = | Link( V Λ , ( V Λ (cid:63) C (cid:48) Λ ) (cid:48) ) | . An h -simplex of ( V Λ (cid:63) C (cid:48) Λ ) (cid:48) is σ h = (cid:104) (cid:98) L , (cid:98) L , . . . (cid:98) L h (cid:105) , where L i is a chain Λ i, ≺ Λ i, ≺ · · · ≺ Λ i,s i in R (Λ) ∪ { Λ } , for i = 0 , , . . . , h , such that L < L < · · · < L h and (cid:98) L i is the barycenter ofthe simplex (cid:104) V Λ i, , V Λ i, , . . . , V Λ i,si (cid:105) . The simplex σ h belongs to Link( V Λ , ( V Λ (cid:63) C (cid:48) Λ ) (cid:48) ) (resp. to C (cid:48)(cid:48) Λ )if and only if Λ (cid:54) = Λ , is the last element of the chain L (resp. Λ is not an element of the chain L h ). Then the map ι : S ( C (cid:48)(cid:48) Λ ) → S (Link( V Λ , ( V Λ (cid:63) C (cid:48) Λ ) (cid:48) )) defined by ι ( (cid:98) L ) = (cid:98) L (cid:48) , where L is thechain Λ ≺ Λ ≺ · · · ≺ Λ k , with Λ k ≺ Λ, and L (cid:48) is the chain Λ ≺ Λ ≺ · · · ≺ Λ k ≺ Λ, inducesan isomorphism between C (cid:48)(cid:48) Λ and Link( V Λ , ( V Λ (cid:63) C (cid:48) Λ ) (cid:48) ). As a consequence, B is homeomorphic to |C (cid:48)(cid:48) Λ | = (cid:99) M Λ (= M Λ since all residues of Λ are ordinary). Corollary 3.13 If (cid:99) M Γ is a singular manifold then ∂M Γ has no spherical components. If dim( S Γ ) = 1, the graph Γ has no singular ( n −
2) residues, and any singular ( n − n -residues Λ and Λ (cid:48) . Using the isomorphism ι defined in the proofof Lemma 3.12, we can suppose that (cid:99) M Ω = M Ω is a boundary component of both M Λ and M Λ (cid:48) .Therefore, we can define the space M Λ ∪ ∂ M Λ (cid:48) by gluing M Λ with M Λ (cid:48) along their commonboundary components (corresponding to common singular ( n − M Γ can be described as gluings of the manifolds with boundary corresponding to thesingular n -residues of Γ. Proposition 3.14
Let Γ be an ( n + 1) -colored graph such that dim( S Γ ) = 1 and let S be aconnected component of |S Γ | . Then the component of ∂M Γ corresponding to S is homeomorphicto (cid:83) ∂ ≤ i ≤ s M Λ i , where Λ , . . . , Λ s are the n -residues of R S . roof. Let Λ be a singular n -residue of Γ belonging to R S , and let Ω , Ω , . . . , Ω m be thesingular ( n − | ˙ N ( S (cid:48) Γ , C (cid:48)(cid:48) Γ ) ∩ ¯ c (cid:48)(cid:48) Λ | is homeomorphic to M Λ and (ii) | ˙ N ( S (cid:48) Γ , C (cid:48)(cid:48) Γ ) ∩ ¯ c (cid:48)(cid:48) Ω i | is homeomorphic to M Ω i and is a boundary component of | ˙ N ( S (cid:48) Γ , C (cid:48)(cid:48) Γ ) ∩ ¯ c (cid:48)(cid:48) Λ | ,for i = 1 , . . . , m .Referring to the proof of Lemma 3.12, the simplex σ h = (cid:104) (cid:98) L , (cid:98) L , . . . (cid:98) L h (cid:105) of ( V Λ (cid:63) C (cid:48) Λ ) (cid:48) belongsto ˙ N ( S (cid:48) Γ , C (cid:48)(cid:48) Γ ) if and only if either (i) Λ is the last element of the chain L , and Λ (cid:54) = Λ , (cid:54) = Ω j ,for j = 1 , . . . , m , or (ii) there exists j such that Ω j (cid:54) = Λ , is the last element of the chain L .Let K be the complex containing the simplexes satisfying (i) and, for i = 1 , . . . , m , let K i be thecomplex containing the simplexes satisfying (ii) and such that Λ , is a residue of Ω i . Moreover, σ h belongs to C ( S (cid:48) Λ , C (cid:48)(cid:48) Λ ) if and only if Λ is not an element of the chain L h and Λ , (cid:54) = Ω j , for i = 1 , . . . , m . In particular, σ h belongs to C ( S (cid:48) Ω i , C (cid:48)(cid:48) Ω i ) if and only if Λ is not an element of thechain L h and Λ , is a residue of Ω i .The map κ : S ( C ( S (cid:48) Λ , C (cid:48)(cid:48) Λ )) → S ( K ) defined by κ ( (cid:98) L ) = (cid:98) L (cid:48) , where L is the chain Λ ≺ Λ ≺· · · ≺ Λ k , with Λ k ≺ Λ, and L (cid:48) is the chain Λ ≺ Λ ≺ · · · ≺ Λ k ≺ Λ, induces an isomorphismbetween C ( S (cid:48) Λ , C (cid:48)(cid:48) Λ ) and K . Therefore, | K | is homeomorphic to M Λ = | C ( S (cid:48) Λ , C (cid:48)(cid:48) Λ ) | . Moreover, for i = 1 , . . . , m , the map κ i : S ( C ( S (cid:48) Ω i , C (cid:48)(cid:48) Ω i ) → S ( K i ) defined in the same way of κ with L (cid:48) beingthe chain Λ ≺ Λ ≺ · · · ≺ Λ k ≺ Λ (resp. Λ ≺ Λ ≺ · · · ≺ Λ k ≺ Ω i ) if Λ k = Ω i (resp. ifΛ k ≺ Ω i ) induces an isomorphism between C ( S (cid:48) Ω i , C (cid:48)(cid:48) Ω i ) and K i . Therefore, | K i | is homeomorphicto | C ( S (cid:48) Ω i , C (cid:48)(cid:48) Ω i ) | , which is homeomorphic to M Ω i × I by Lemma 1.22 of [29]. Since K ∩ K i isisomorphic to ˙ N ( S (cid:48) Ω i , C (cid:48)(cid:48) Ω i ) = ˙ N ( S (cid:48) Γ , C (cid:48)(cid:48) Γ ) ∩ ¯ c (cid:48)(cid:48) Ω i via κ i and | ˙ N ( S (cid:48) Ω i , C (cid:48)(cid:48) Ω i ) | = M Ω i , then | ˙ N ( S (cid:48) Γ , C (cid:48)(cid:48) Γ ) ∩ ¯ c (cid:48)(cid:48) Λ | is homeomorphic to M Λ . This concludes the proof.When dim( S Γ ) > M Γ is in general rather more involved.Note that Lemma 3.12 (resp. Proposition 3.14) always applies when dim( M Γ ) = 3 (resp. whendim( M Γ ) = 4).Some properties of M Γ and (cid:99) M Γ correspond to properties of the representing graph Γ. Forexample the following result generalizes Theorem 4 of [18]: Proposition 3.15
The quasi-manifold (cid:99) M Γ , as well as the manifold M Γ , is orientable if and onlyif Γ is bipartite. Proof.
Obviously any orientation on (cid:99) M Γ restricts to an orientation on M Γ . Furthermore, anyorientation on M Γ extends to an orientation on (cid:99) M Γ , since | ¯ σ (cid:48)(cid:48) ∩ C ( ¯ S (cid:48)(cid:48) Γ , K (cid:48)(cid:48) Γ ) | is an n -ball (resp. an( n − n -simplex (resp. ( n − σ of K Γ . Therefore it suffices to prove thestatement for (cid:99) M Γ . By construction any global orientation on (cid:99) M Γ induces a local orientation forany maximal simplexes of K Γ , which corresponds to one of the two classes of total orderings of theset ∆ n , up to even permutations, in such a way that two n -simplexes having a common ( n − V (Γ) = V (cid:48) ∪ V (cid:48)(cid:48) , thenorient any n -simplex associated to elements of V (cid:48) according with one class of orderings of ∆ n andany n -simplex associated to elements of V (cid:48)(cid:48) according with the other class. This choice providesa global orientation for (cid:99) M Γ .The computation of the Euler characteristic of M Γ , (cid:99) M Γ and |S Γ | is a routine fact: Lemma 3.16
Let Γ be an ( n + 1) -colored graph. Then χ ( M Γ ) = n (cid:88) h =0 ( − h |R (cid:48) h (Γ) | , χ ( (cid:99) M Γ ) = n (cid:88) h =0 ( − n − h |R h (Γ) | and χ ( |S Γ | ) = n (cid:88) h =3 ( − n − h |R (cid:48)(cid:48) h (Γ) | . roof. The manifold M Γ retracts to the space X = ∪ Λ ∈R (cid:48) (Γ) c Λ by Remark 3.10, and therefore χ ( M Γ ) = χ ( X ) can be obtained by the usual formula for the CW-complexes. On the other hand,both χ ( (cid:99) M Γ ) and χ ( |S Γ | ) can be obtained from the pseudo-simplicial complex K Γ by consideringthat any h -residue of the graph corresponds to an ( n − h )-simplex of K Γ .
4. Graph suspension
If Γ is an n -colored graph, for a fixed c ∈ ∆ n − define Σ c (Γ) as the ( n + 1)-colored graph ob-tained from Γ by adding a set of n -edges parallel to the c -edges of Γ. We refer to Σ c (Γ) as the c -suspension of Γ. The following result states that this construction is strictly related to thesuspension construction of PL spaces (in the following Σ( · ) denotes the suspension of either a PLspace or a simplicial complex). Theorem 4.1
Let Γ be an n -colored graph and c be any color in ∆ n − , then:(i) (cid:99) M Σ c (Γ) = Σ( (cid:99) M Γ ) ;(ii) |S Σ c (Γ) | = | Σ( S Γ ) | , if (cid:99) M Γ is not a sphere;(iii) M Σ c (Γ) = M Γ × I , if (cid:99) M Γ is not a sphere. Proof.
Let Ω = Σ c (Γ) and denote by R c (Γ) the set of all residues of Γ containing c -edges.Each ∆-residue Λ of Γ can be considered as a ∆-residue of Ω, as well as Γ. So Ω has a unique (cid:98) n -residue Γ, and a unique (cid:98) c -residue Γ (cid:48)(cid:48) , which is isomorphic to Γ. Moreover, if Λ ∈ R c (Γ) let uscall Λ (cid:48) (resp. Λ (cid:48)(cid:48) ) the (∆ ∪ { n } )-residue of Ω (resp. the (∆ ∪ { n } − { c } )-residue of Ω) having thesame vertices of Λ.(i) Consider the n -dimensional simplicial complex K = Σ( C (cid:48) Γ ) = { P (cid:48) , P (cid:48)(cid:48) } (cid:63) C (cid:48) Γ , then S ( K ) = { P (cid:48) , P (cid:48)(cid:48) } ∪ n − h =0 { V Λ | Λ ∈ R h (Γ) } . We will show by induction that a suitable subdivision of K isisomorphic to C (cid:48) Ω .The induction is performed on the h -residues of R c (Γ), for h = 1 , . . . , n −
1. The first step of theinduction is to obtain a stellar subdivision K of K , by starring the 1-simplex (cid:104) P (cid:48) , V e (cid:105) at an internalpoint V (cid:48) e and the 1-simplex (cid:104) P (cid:48)(cid:48) , V e (cid:105) at an internal point V (cid:48)(cid:48) e , for any c -edge e of Γ. It is obvious that K does not depend on the order of the above starrings. After this step the 1-simplex (cid:104) P (cid:48) , V e (cid:105) (resp. (cid:104) P (cid:48)(cid:48) , V e (cid:105) ) is subdivided into the two 1-simplexes (cid:104) P (cid:48) , V (cid:48) e (cid:105) and (cid:104) V e , V (cid:48) e (cid:105) (resp. (cid:104) P (cid:48)(cid:48) , V (cid:48)(cid:48) e (cid:105) and (cid:104) V e , V (cid:48)(cid:48) e (cid:105) )and the maximal simplex (cid:104) V v , V e , V Λ , . . . , V Λ n − , P (cid:48) (cid:105) (resp. (cid:104) V v , V e , V Λ , . . . , V Λ n − , P (cid:48)(cid:48) (cid:105) ) of K issubdivided into the two maximal simplexes (cid:104) V v , V (cid:48) e , V Λ , . . . , V Λ n − , P (cid:48) (cid:105) and (cid:104) V v , V (cid:48) e , V e , V Λ , . . . , V Λ n − (cid:105) (resp. (cid:104) V v , V (cid:48)(cid:48) e , V Λ , . . . , V Λ n − , P (cid:48)(cid:48) (cid:105) and (cid:104) V v , V (cid:48)(cid:48) e , V e , V Λ , . . . , V Λ n − (cid:105) ).Suppose the first h − K h − of K . Wedefine the induction step h as follows: produce the complex K h as the stellar subdivision of K h − obtained by starring both the 1-simplex (cid:104) P (cid:48) , V Λ h (cid:105) at an internal point V (cid:48) Λ h and the 1-simplex (cid:104) P (cid:48)(cid:48) , V Λ h (cid:105) at an internal point V (cid:48)(cid:48) Λ h , for any h -residue Λ h of R c (Γ). Again the result of thesestarrings do not depend on their order. After this step the 1-simplex (cid:104) P (cid:48) , V Λ h (cid:105) (resp. (cid:104) P (cid:48)(cid:48) , V Λ h (cid:105) )is subdivided into the two 1-simplexes (cid:104) P (cid:48) , V (cid:48) Λ h (cid:105) and (cid:104) V Λ h , V (cid:48) Λ h (cid:105) , (resp. (cid:104) P (cid:48)(cid:48) , V (cid:48)(cid:48) Λ h (cid:105) and (cid:104) V Λ h , V (cid:48)(cid:48) Λ h (cid:105) ).The maximal simplex (cid:104) V Λ , . . . , V Λ j , V (cid:48) Λ j +1 , . . . , V (cid:48) Λ h − , V Λ h , V Λ h +1 , . . . , V Λ n − , P (cid:48) (cid:105) of K h − is subdi-vided into the two maximal simplexes (cid:104) V Λ , . . . , V Λ j , V (cid:48) Λ j +1 , . . . , V (cid:48) Λ h − , V (cid:48) Λ h , V Λ h +1 , . . . , V Λ n − , P (cid:48) (cid:105) nd (cid:104) V Λ , . . . , V Λ j , V (cid:48) Λ j +1 , . . . , V (cid:48) Λ h − , V (cid:48) Λ h , V Λ h , V Λ h +1 , . . . , V Λ n − (cid:105) , where 0 ≤ j ≤ h −
1. Analo-gously, the maximal simplex (cid:104) V Λ , . . . , V Λ j , V (cid:48)(cid:48) Λ j +1 , . . . , V (cid:48)(cid:48) Λ h − , V Λ h , V Λ h +1 , . . . , V Λ n − , P (cid:48)(cid:48) (cid:105) is subdi-vided into the two maximal simplexes (cid:104) V Λ , . . . , V Λ j , V (cid:48)(cid:48) Λ j +1 , . . . , V (cid:48)(cid:48) Λ h − , V (cid:48)(cid:48) Λ h , V Λ h +1 , . . . , V Λ n − , P (cid:48)(cid:48) (cid:105) and (cid:104) V Λ , . . . , V Λ j , V (cid:48)(cid:48) Λ j +1 , . . . , V (cid:48)(cid:48) Λ h − , V (cid:48)(cid:48) Λ h , V Λ h , V Λ h +1 , . . . , V Λ n − (cid:105) .At the end of the inductive process we obtain a complex K n − , which is a subdivision of K ,having vertex set S ( K n − ) = S ( K ) ∪ { V (cid:48) Λ | Λ ∈ R c (Γ) } ∪ { V (cid:48)(cid:48) Λ | Λ ∈ R c (Γ) } .Let φ : S ( K n − ) → S (Ω) be the map defined by φ ( P (cid:48) ) = V Γ , φ ( P (cid:48)(cid:48) ) = V Γ (cid:48)(cid:48) , φ ( V Λ ) = V Λ (cid:48) ifΛ ∈ R c (Γ) and φ ( V Λ ) = V Λ if Λ (cid:54)∈ R c (Γ), φ ( V (cid:48) Λ ) = V Λ and φ ( V (cid:48)(cid:48) Λ ) = V Λ (cid:48)(cid:48) , for any Λ ∈ R c (Γ). Then φ is clearly a bijection and induces an isomorphism between K n − and C (cid:48) Ω .In order to prove that, let σ = (cid:104) V Λ , V Λ , . . . , V Λ n − , V Λ n (cid:105) be a maximal simplex of C (cid:48) Ω , whereΛ i is a D i -residue of Ω such that | D i | = i , for i = 0 , . . . , n . If D n = (cid:98) n and h = min { i | c ∈ D i } (resp. D n = (cid:98) c and h = min { i | n ∈ D i } ), then σ is the image via φ of the maximal sim-plex (cid:104) V Λ , . . . , V Λ h − , V (cid:48) Λ h , V (cid:48) Λ h +1 , . . . , V (cid:48) Λ n − , P (cid:48) (cid:105) (resp. (cid:104) V Λ , . . . , V Λ h − , V (cid:48)(cid:48) ˜Λ h , V (cid:48)(cid:48) ˜Λ h +1 , . . . , V (cid:48)(cid:48) ˜Λ n − , P (cid:48)(cid:48) (cid:105) ,where ˜Λ i is the residue of Γ such that ˜Λ (cid:48)(cid:48) i = Λ i , for i = h, . . . , n − D n = (cid:98) j with j (cid:54) = c, n . Define h (cid:48) = min { i | c ∈ D i } and h (cid:48)(cid:48) = min { i | n ∈ D i } . If h (cid:48) < h (cid:48)(cid:48) then σ is the image via φ ofthe maximal simplex (cid:104) V Λ , . . . , V Λ h (cid:48)− , V (cid:48) Λ h (cid:48) , V (cid:48) Λ h (cid:48) +1 , . . . , V (cid:48) Λ h (cid:48)(cid:48)− , V ˜Λ h (cid:48)(cid:48)− , V ˜Λ h (cid:48)(cid:48) , . . . , V ˜Λ n − (cid:105) , where ˜Λ i is the residue of Γ such that ˜Λ (cid:48) i = Λ i +1 for i = h (cid:48)(cid:48) − , . . . , n −
1. If h (cid:48)(cid:48) < h (cid:48) then σ is the image via φ of the maximal simplex (cid:104) V Λ , . . . , V Λ h (cid:48)(cid:48)− , V (cid:48)(cid:48) Λ h (cid:48)(cid:48) , V (cid:48)(cid:48) Λ h (cid:48)(cid:48) +1 , . . . , V (cid:48)(cid:48) Λ h (cid:48)− , V ˜Λ h (cid:48)− , V ˜Λ h (cid:48) , . . . , V ˜Λ n − (cid:105) , where˜Λ i is the residue of Γ such that ˜Λ (cid:48)(cid:48) i = Λ i for i = h (cid:48)(cid:48) , . . . , h (cid:48) −
1, and ˜Λ (cid:48) i = Λ i +1 for i = h (cid:48) − , . . . , n − σ ∈ C (cid:48) Γ (possibly σ = ∅ ) then Link( P (cid:48) (cid:63) σ, K ) = Link( P (cid:48)(cid:48) (cid:63) σ, K ) = Link( σ, C (cid:48) Γ ) (recallthat Link( ∅ , C (cid:48) Γ ) = C (cid:48) Γ ). Therefore σ is singular in C (cid:48) Γ if and only if both P (cid:48) (cid:63) σ and P (cid:48)(cid:48) (cid:63) σ aresingular in K . This proves that S K = { P (cid:48) , P (cid:48)(cid:48) } (cid:63) S Γ = Σ( S Γ ) if |C (cid:48) Γ | = (cid:99) M Γ is not a sphere and S K = ∅ if (cid:99) M Γ is a sphere. Now the statement follows from the fact that the singular set of acomplex is invariant under subdivisions.(iii) By the previous points we can consider M Σ c (Γ) as | K | − int( N ( S K )). The complexes H = P (cid:48) (cid:63) C (cid:48) Γ and H = P (cid:48)(cid:48) (cid:63) C (cid:48) Γ are isomorphic, as well as S H and S H . Therefore, it suffices toprove that M = | H | − int( N ( S H )) = M Γ × I .By (ii), if (cid:99) M Γ is not a sphere then S H = P (cid:48) (cid:63) S Γ , which is a full subcomplex of H containing P (cid:48) . As a consequence, N ( S H ) = | N ( S (cid:48) H , H (cid:48) ) | and M = | C ( S (cid:48) H , H (cid:48) ) | = | ∪ v ∈ ( S ( H ) − S ( S H )) Star( v, H (cid:48) ) | = | ∪ v ∈ ( S ( C (cid:48) Γ ) − S ( S Γ )) Star( v, H (cid:48) ) | . The complex D = C ( S (cid:48) H , H (cid:48) ) ∩ Star( P (cid:48) , H (cid:48) )is isomorphic to C ( S Γ , C (cid:48)(cid:48) Γ ), L = ( P (cid:48) (cid:63) D ) ∪ C ( S (cid:48) H , H (cid:48) ) is a subdivision of P (cid:48) (cid:63) C ( S (cid:48) Γ , C (cid:48)(cid:48) Γ ) andStar( P (cid:48) , L ) ∩ C ( S (cid:48) Γ , C (cid:48)(cid:48) Γ ) = ∅ . Then Lemma 1.22 of [29] applies and M = | C ( S (cid:48) H , H (cid:48) ) | = | P (cid:48) (cid:63)C ( S (cid:48) Γ , C (cid:48)(cid:48) Γ ) | − int | Star( v, L ) | is homeomorphic to | C ( S (cid:48) Γ , C (cid:48)(cid:48) Γ ) | × I = M Γ × I .It is noteworthy that Theorem 4.1 extends to the whole class of colored graphs a result obtainedin [17] for the case in which (cid:99) M Γ is a closed manifold. Actually, a graph representing Σ( (cid:99) M Γ ) iseasily obtainable from Γ by doubling it, namely taking two isomorphic copies of Γ and joining thecorresponding vertices with an n -edge. The importance of the previous result relies in the factthat the graphs Γ and Σ c (Γ) have the same order.As a relevant example of the suspension graph construction, Figure 1 shows a 6-colored graphΓ which is the double suspension of a 4-colored graph representing the Poincar`e homology spheredepicted in [31]. By a remarkable result by J. W. Cannon (see [2]), (cid:99) M Γ is topologically homeo-morphic to S despite that, with the structure induced by C (cid:48) Γ , it is not PL-homeomorphic to S .In fact, |S Γ | is not empty and it is homeomorphic to S . Remark 4.2
From the proof of Proposition 3.11 it follows that any compact connected n -manifoldadmits a representation by an ( n + 1) -colored graph where all the singular residues (if any) are n -residues. Nevertheless, the representation with general ( n + 1) -colored graphs may be moreeconomical in terms of the order of the representing graph. For example, the 4-manifold M = S × S × B can be represented by the order six 5-colored graph Γ (cid:48) = Σ (Σ (Γ)) depicted inFigure 2, where Γ is the standard order six graph representing S × S . But it is easy to realize that M does not admit any representation by 5-colored graphs of order less than 24 and without singular3-residues, since in this case it would admit a 4-residue representing ∂ ( S × S × B ) = S × S × S ,and it is well known that S × S × S does not admits representation by 4-colored graphs withless than 24 vertices (see [30]). It is easy to see that an ( n + 1)-colored graph of order two always represents an n -sphere,since it is the ( n − S . Also for graphsof order 4 the characterization is easy. Proposition 4.3
Let Γ be an ( n + 1) -colored graph of order 4, then:(i) (cid:99) M Γ = M Γ = S n if Γ is bipartite;(ii) (cid:99) M Γ = Σ n − ( RP ) and M Γ = RP × B n − if Γ is non-bipartite. S × S × B . Proof.
Since Γ is connected it contains at least a bigon of order 4, and we can suppose, upto isomorphism, that it is a { , } -residue. If n = 1 the graph Γ is a bigon and (cid:99) M Γ = M Γ = S .If n > c -edges, for c = 2 , . . . , n is parallel to either the pairof 0-edges or the pair of 1-edges. By Theorem 4.1 we obtain (i). On the contrary, if Γ is notbipartite then it contains a cycle of order three. As a consequence, there is at least a 3-residue Λof Γ (say a { , , } -residue) which is isomorphic to the complete graph of order 4, and therefore (cid:99) M Λ = M Λ = RP . Then any pair of c -edges, for c = 3 , . . . , n is parallel to either the pair of0-edges or the pair of 1-edges or the pair of 2-edges, and the statement follows from Theorem 4.1.
5. Dipole moves
Given an ( n + 1)-colored graph Γ, an h -dipole (1 ≤ h ≤ n ) involving colors c , . . . , c h ∈ ∆ n ofΓ is a subgraph θ of Γ consisting of two vertices v (cid:48) and v (cid:48)(cid:48) joined by exactly h edges, colored by c , . . . , c h , such that v (cid:48) and v (cid:48)(cid:48) belong to different (cid:92) { c , . . . , c h } -residues of Γ.By cancelling θ from Γ, we mean to remove θ and to paste together the hanging edges accordingto their colors, thus obtaining a new ( n + 1)-colored graph Γ (cid:48) . Conversely, Γ is said to be obtainedfrom Γ (cid:48) by adding θ . A dipole move is either the cancellation or the addition of a dipole.An h -dipole θ of Γ is called proper when (cid:99) M Γ is homeomorphic to (cid:99) M Γ (cid:48) (see [22]). As aconsequence, if two colored graphs Γ and Γ are connected by a sequence of proper dipole movesthen (cid:99) M Γ = (cid:99) M Γ , as well as M Γ = M Γ . Furthermore, θ is called singular if the two (cid:92) { c , . . . , c h } -residues containing the vertices of θ are both singular. Otherwise the dipole is called ordinary . Proposition 5.1 [22] Any ordinary dipole of an ( n + 1) -colored graph Γ is proper. As a conse-quence, all n - and ( n − -dipoles of Γ are proper. f Γ represents a closed n -manifold all dipoles of Γ are proper and Casali proved in [3] thatdipole moves are sufficient to connect different 4-colored graphs representing the same closed 3-manifold. This result is no longer true in the case with boundary, even in dimension three (see[15]).It seems natural to argue that the converse of Proposition 5.1 also holds, but we are able toprove it only for singular manifolds (see Corollary 5.4). Lemma 5.2 If Γ (cid:48) is obtained from Γ by cancelling a dipole θ , then |S Γ (cid:48) | is homeomorphic to aquotient of |S Γ | . Proof.
Topologically, the effect of the cancellation of the h -dipole θ involving colors c , . . . , c h is the following: (cid:99) M Γ (cid:48) is obtained from (cid:99) M Γ by removing the open n -ball int( B ) where B = | Star( v (cid:48) , C (cid:48) Γ ) ∪ Star( v (cid:48)(cid:48) , C (cid:48) Γ ) | and afterthat attaching the ( n − ∂B ∩ ∂ | Star( v (cid:48) , C (cid:48) Γ ) | with the( n − ∂B ∩ ∂ | Star( v (cid:48)(cid:48) , C (cid:48) Γ ) | via the identification of any simplex σ (cid:48) = (cid:104) V Λ (cid:48) , V Λ (cid:48) , . . . , V Λ (cid:48) k (cid:105) withthe simplex σ (cid:48)(cid:48) = (cid:104) V Λ (cid:48)(cid:48) , V Λ (cid:48)(cid:48) , . . . , V Λ (cid:48)(cid:48) k (cid:105) , where Λ (cid:48) i and Λ (cid:48)(cid:48) i are ∆ ( i ) -residues containing v (cid:48) and v (cid:48)(cid:48) respectively, with ∅ (cid:54) = ∆ (0) (cid:54)⊆ { c , . . . , c h } , for i = 0 , , . . . , k . The only vertices of C (cid:48) Γ belonging toint( B ) are the cone vertices associated either to θ or to a residue of θ . Since (cid:99) M θ is a sphere, thesevertices do not belong to S Γ . As a consequence, int( B ) ∩ |S Γ | = ∅ and |S Γ (cid:48) | is obtained from |S Γ | just by considering the attachings of σ (cid:48) with σ (cid:48)(cid:48) when they are both simplexes of S Γ . Lemma 5.3
Let θ be a singular h -dipole involving colors c , . . . , c h of an ( n + 1) -colored graph Γ . If for any ∆ ⊂ (cid:92) { c , . . . , c h } at least one of the two ∆ -residues containing the vertices of θ isordinary, then θ is not proper. Proof.
Let v (cid:48) , v (cid:48)(cid:48) be the endpoints of θ and let Γ (cid:48) be the ( n + 1)-colored graph obtained fromΓ by cancelling θ . Then S Γ (cid:48) is obtained from S Γ by attaching each simplex (cid:104) V Λ (cid:48) , V Λ (cid:48) , . . . , V Λ (cid:48) k (cid:105) with the simplex (cid:104) V Λ (cid:48)(cid:48) , V Λ (cid:48)(cid:48) , . . . , V Λ (cid:48)(cid:48) k (cid:105) , where Λ (cid:48) and Λ (cid:48)(cid:48) are the (cid:92) { c , . . . , c h } -residues containing v (cid:48) and v (cid:48)(cid:48) respectively. Topologically, the operation consists in the attaching of two ( h − χ ( |S Γ (cid:48) | ) = χ ( |S Γ | ) + ( − h and therefore |S Γ (cid:48) | is not homeomorphicto |S Γ | . As a consequence, (cid:99) M Γ (cid:48) is not homeomorphic to (cid:99) M Γ and the dipole is not proper. Corollary 5.4
Let Γ be an ( n + 1) -colored graph such that (cid:99) M Γ is a singular manifold, then adipole of Γ is proper if and only if it is ordinary. Proof.
By Lemma 3.7 any singular residue of Γ is an n -residue and contains no singularresidues, therefore Lemma 5.3 holds.For the manifold with boundary M Γ we have the following consequence of the previous results. Proposition 5.5
Let Γ be an ( n + 1) -colored graph such that (cid:99) M Γ is a singular manifold, and let Γ (cid:48) be the graph obtained from Γ by cancelling a dipole θ . Then M Γ (cid:48) = M Γ if and only if θ isordinary. Proof. If θ is ordinary then (cid:99) M Γ (cid:48) = (cid:99) M Γ by Proposition 5.1 and therefore M Γ (cid:48) = M Γ . If θ issingular then it is a 1-dipole and consequently |R (cid:48)(cid:48) n (Γ (cid:48) ) | < |R (cid:48)(cid:48) n (Γ) | . Since in this case |R (cid:48)(cid:48) n (Γ) | (resp. R (cid:48)(cid:48) n (Γ (cid:48) ) | ) is exactly the number of boundary components of M Γ (resp. of M Γ (cid:48) ), then M Γ (cid:48) (cid:54) = M Γ .A vertex v ∈ V (Γ) is called an internal vertex if all n -residues containing v are ordinary,otherwise it is called a boundary vertex . The index of v is the number of singular n -residuesof Γ containing v . So an internal vertex has index 0 and a boundary vertex has index r , with1 ≤ r ≤ n + 1.Some useful properties follow from the previous results. Lemma 5.6
Let M be a compact connected n -manifold without spherical boundary components,then: (i) M can be represented by an ( n + 1) -colored graph with no ordinary dipoles;(ii) M can be represented by an ( n + 1) -colored graph with at least one internal vertex.(iii) if ∂M (cid:54) = ∅ then M can be represented by an ( n + 1) -colored graph with at least oneboundary vertex of index one. Proof.
Let Γ be an ( n + 1)-colored graph representing M .(i) If Γ has an ordinary dipole θ , then the dipole is proper and by cancelling it we obtain anew ( n + 1)-colored graph still representing M . A finite sequence of such cancellations of ordinarydipoles obviously yields an ( n + 1)-colored graph representing M and without ordinary dipoles.(ii, iii) If ∂M = ∅ there is nothing to prove. Otherwise, let v be a boundary vertex withminimal index r > c ∈ ∆ n be such that the (cid:98) c -residue containing v is singular. By addingan n -dipole along the c -edge containing v we obtain two new vertices, v (cid:48) and v (cid:48)(cid:48) , which are bothsingular of order r −
1. In fact, the (cid:98) c -residue containing them is obviously ordinary (it is thestandard n -colored graph representing S n − ), and for each d ∈ (cid:98) c any (cid:98) d -residue containing themis singular if and only if the (cid:98) d -residue containing v in Γ is singular. So by induction on r we canobtain an internal vertex (resp. a boundary vertex of index one) in not more than r steps (resp. r −
6. Fundamental group
If Γ is an ( n + 1)-colored graph, with n >
1, then the fundamental group of the manifold M Γ coincides with the fundamental group of the associated 2-dimensional polyhedron Γ (2) , since M Γ isobtained by attaching to Γ (2) pieces which are retractable (in virtue of Remark 3.10) and h -balls,for 3 ≤ h ≤ n . Therefore, the computation of π ( M Γ ) is a routine fact: a finite presentationfor it has generators corresponding to the edges which are not in a fixed spanning tree of Γ andrelators corresponding to all 2-residues of Γ. The fundamental group of the quasi-manifold (cid:99) M Γ is a quotient of the one of M Γ , since retractable pieces are replaced by cones which kill someelements of π ( M Γ ).In several cases the two groups can be obtained by selecting a particular class of edges and2-residues, as follows. If c ∈ ∆ n , define the c -group of Γ as the group π (Γ , c ) generated by all c -edges (with a fixed arbitrary orientation) and with relators corresponding to all { i, c } -residues, or any i ∈ (cid:98) c , obtained in the following way: give an orientation to each involved 2-residue, choosea starting vertex and follow the bigon according to the chosen orientation. The relator is obtainedby taking the c -edges of the bigon in the order they are reached in the path, with the exponent+1 or − π (Γ , c ) depends on c , but when c is an ordinary color the group is strictly connectedwith the fundamental group of M Γ (see [30] for closed 3-manifolds and [25] for closed n -manifolds). Proposition 6.1
Let Γ be an ( n +1) -colored graph and c be an ordinary color for Γ . Then π ( M Γ ) is the quotient of π (Γ , c ) , obtained by adding to the relators a minimal set of c -edges which connectthe graph Γ (cid:98) c . Proof.
The group π ( M Γ ) = π (Γ (2) ) is isomorphic to the fundamental group of the space X obtained by adding to Γ (2) the n -balls corresponding to the (cid:98) c -residues. The space X has the samehomotopy type of a 2-complex with 0-cells corresponding to the (cid:98) c -residues, 1-cells correspondingto the c -edges of Γ and 2-cells corresponding to the { c, i } -residues of Γ, for i ∈ (cid:98) c . So the result isstraightforward. Corollary 6.2
Let Γ be an ( n +1) -colored graph and c be an ordinary color for Γ such that g (cid:98) c = 1 ,then π ( M Γ ) ∼ = π (Γ , c ) . When Γ has no more than a singular color, we have the following characterization of thefundamental group of (cid:99) M Γ . Proposition 6.3
Let Γ be an ( n + 1) -colored graph and let c ∈ ∆ n be such that any color differentfrom c is ordinary. Then π ( (cid:99) M Γ ) is the quotient of π (Γ , c ) , obtained by adding to the relators aminimal set of c -edges which connect the graph Γ (cid:98) c . Proof.
The group π ( (cid:99) M Γ ) is isomorphic to the fundamental group of the space X obtainedfrom Γ (2) by performing cone constructions corresponding to the (cid:98) c -residues and all their residues.Since c Λ is a cone over Λ ( n − , for any (cid:98) c -residue Λ, the space X has the same homotopy type of a2-complex with 0-cells corresponding to the (cid:98) c -residues, 1-cells corresponding to the c -edges of Γand 2-cells corresponding to the { c, i } -residues of Γ, for any i ∈ (cid:98) c . So the result is straightforward. Corollary 6.4
Let Γ be an ( n + 1) -colored graph and c ∈ ∆ n . If g (cid:98) c = 1 and any color differentfrom c is ordinary, then π ( (cid:99) M Γ ) ∼ = π (Γ , c ) .
7. Connected sums
Suppose that Γ (cid:48) and Γ (cid:48)(cid:48) are two ( n + 1)-colored graphs and let v (cid:48) ∈ V (Γ (cid:48) ) and v (cid:48)(cid:48) ∈ V (Γ (cid:48)(cid:48) ). Wecan construct a new ( n + 1)-colored graph Γ, called connected sum of Γ (cid:48) and Γ (cid:48)(cid:48) (along v (cid:48) and v (cid:48)(cid:48) ),and denoted by Γ = Γ (cid:48) v (cid:48) v (cid:48)(cid:48) Γ (cid:48)(cid:48) , by removing the vertices v (cid:48) and v (cid:48)(cid:48) and by welding the resultinghanging edges with the same color. A color c ∈ ∆ n is called ordinary if Γ has no singular (cid:98) c -residues, otherwise c is called singular . The result was first proved in [9] by using the dual construction. n general, the connected sum of two ( n + 1)-colored graphs depends on the choice of thecancelled vertices. But when these vertices are either internal or boundary vertices of index onewith respect to n -residues of the same colors (the latter condition always holds, up to a colorpermutation in one of the two graphs), then the connected sum of the graphs is strictly connectedwith the connected sum of the represented manifolds. Proposition 7.1
Let Γ (cid:48) , Γ (cid:48)(cid:48) be ( n + 1) -colored graphs and v (cid:48) ∈ V (Γ (cid:48) ) , v (cid:48)(cid:48) ∈ V (Γ (cid:48)(cid:48) ) .(i) if v (cid:48) and v (cid:48)(cid:48) are both internal vertices, then M Γ (cid:48) v (cid:48) v (cid:48)(cid:48) Γ (cid:48)(cid:48) = M Γ (cid:48) M Γ (cid:48)(cid:48) ; (ii) if v (cid:48) and v (cid:48)(cid:48) are both boundary vertices of index one, each belonging to a singular (cid:98) c -residue, then M Γ (cid:48) v (cid:48) v (cid:48)(cid:48) Γ (cid:48)(cid:48) = M Γ (cid:48) ∂ M Γ (cid:48)(cid:48) , where the boundary connected sum of the manifoldsis performed along the boundary components corresponding to the involved (cid:98) c -residues. Proof.
Let Γ = Γ (cid:48) v (cid:48) v (cid:48)(cid:48) Γ (cid:48)(cid:48) . Then the complex C (cid:48) Γ is obtained from ( C (cid:48) Γ (cid:48) − Star( v (cid:48) , C (cid:48) Γ (cid:48) )) ∪ ( C (cid:48) Γ (cid:48)(cid:48) − Star( v (cid:48)(cid:48) , C (cid:48) Γ (cid:48)(cid:48) )) ∪ Link( v (cid:48) , C (cid:48) Γ (cid:48) ) ∪ Link( v (cid:48)(cid:48) , C (cid:48) Γ (cid:48)(cid:48) ) by attaching any simplex (cid:104) V Λ (cid:48) , V Λ (cid:48) , . . . , V Λ (cid:48) h (cid:105) of Link( v (cid:48) , C (cid:48) Γ (cid:48) ) with the simplex (cid:104) V Λ (cid:48)(cid:48) , V Λ (cid:48)(cid:48) , . . . , V Λ (cid:48)(cid:48) h (cid:105) of Link( v (cid:48)(cid:48) , C (cid:48) Γ (cid:48)(cid:48) ), where Λ (cid:48) i and Λ (cid:48)(cid:48) i are both D i -residues, the first one of Γ (cid:48) containing v (cid:48) and the second one of Γ (cid:48)(cid:48) containing v (cid:48)(cid:48) . Therefore, (cid:99) M Γ is obtained by removing from (cid:99) M Γ (cid:48) and (cid:99) M Γ (cid:48)(cid:48) the n -balls int( | Star( v (cid:48) , C (cid:48) Γ (cid:48) ) | ) and int( | Star( v (cid:48)(cid:48) , C (cid:48) Γ (cid:48)(cid:48) ) | )and attaching by a homeomorphism the ( n − | Link( v (cid:48) , C (cid:48) Γ (cid:48) ) | and | Link( v (cid:48)(cid:48) , C (cid:48) Γ (cid:48)(cid:48) ) | .(i) In this case | Star( v (cid:48) , C (cid:48) Γ (cid:48) ) | ∩ N ( S Γ (cid:48) ) = | Star( v (cid:48)(cid:48) , C (cid:48) Γ (cid:48)(cid:48) ) | ∩ N ( S Γ (cid:48)(cid:48) ) = ∅ , and therefore M Γ = M Γ (cid:48) M Γ (cid:48)(cid:48) . (ii) In this case | Star( v (cid:48) , C (cid:48) Γ (cid:48) ) | ∩ N ( S Γ (cid:48) ) (cid:54) = ∅ (cid:54) = | Star( v (cid:48)(cid:48) , C (cid:48) Γ (cid:48)(cid:48) ) | ∩ N ( S Γ (cid:48)(cid:48) ) and both | Star( v (cid:48) , C (cid:48) Γ (cid:48) ) | ∩| C ( S (cid:48) Γ (cid:48) , C (cid:48)(cid:48) Γ (cid:48) ) | and | Star( v (cid:48)(cid:48) , C (cid:48) Γ (cid:48)(cid:48) ) | ∩ | C ( S (cid:48) Γ (cid:48)(cid:48) , C (cid:48)(cid:48) Γ (cid:48)(cid:48) ) | are n -balls. In fact, they are homeomorphic to s n − int( | Star( P, ¯ s (cid:48)(cid:48) n ) | ), where P is any vertex of the standard n -simplex s n . Moreover, B (cid:48) = | Star( v (cid:48) , C (cid:48) Γ (cid:48) ) | ∩ ∂ ( M Γ (cid:48) ) and B (cid:48)(cid:48) = | Star( v (cid:48)(cid:48) , C (cid:48) Γ (cid:48)(cid:48) ) | ∩ ∂ ( M Γ (cid:48)(cid:48) ) are both ( n − | Link( P, ¯ s (cid:48)(cid:48) n ) | , as well as A (cid:48) = | Link( v (cid:48) , C (cid:48) Γ (cid:48) ) | ∩ | C ( S (cid:48) Γ (cid:48) , C (cid:48)(cid:48) Γ (cid:48) ) | and A (cid:48)(cid:48) = | Link( v (cid:48)(cid:48) , C (cid:48) Γ (cid:48)(cid:48) ) | ∩ | C ( S (cid:48) Γ (cid:48)(cid:48) , C (cid:48)(cid:48) Γ (cid:48)(cid:48) ) | , since they are both the complement of an ( n − n − M (cid:48) = M Γ (cid:48) − int( | Star( v (cid:48) , C (cid:48) Γ (cid:48) ) | ) and M (cid:48)(cid:48) = M Γ (cid:48)(cid:48) − int( | Star( v (cid:48)(cid:48) , C (cid:48) Γ (cid:48)(cid:48) ) | ), then M (cid:48) (resp. M (cid:48)(cid:48) ) is homeomorphic to M Γ (cid:48) (resp. to M Γ (cid:48)(cid:48) ) and A (cid:48) ⊂ ∂M (cid:48) (resp. A (cid:48)(cid:48) ⊂ ∂M (cid:48)(cid:48) ). Since M Γ isobtained by attaching M (cid:48) with M (cid:48)(cid:48) via a homeomorphism from A (cid:48) to A (cid:48)(cid:48) , the proof is achieved.
8. Results in dimension four
The dimension four is the smallest one where quasi-manifolds which are not singular manifoldsappear. In this context we have the following characterization. Lemma 8.1
Let Γ be a -colored graph of order p . Then:(i) |R (Γ) | − |R (Γ) | + 10 p ≥ ;(ii) (cid:99) M Γ is a singular manifold if and only if |R (Γ) | = 3 |R (Γ) | − p . Compare Lemma 21 of [6]. roof. For each 3-residue Λ of Γ we have the relation 2 − ρ Λ = b − v/ , where v (resp. b ) is the number of vertices (resp. of bigons) of Λ, and ρ Λ ≥ (cid:99) M Λ . By summing over all 3-residuesof Γ we obtain 2 |R (Γ) | − (cid:80) Λ ∈R (Γ) ρ Λ = 3 |R (Γ) | − p. So we have 0 ≤ (cid:80) Λ ∈R (Γ) ρ Λ =2 |R (Γ) | − |R (Γ) | + 10 p , which proves (i). Since all 3-residues of a singular 4-manifold areordinary, we obtain (ii).Various classification results of 3-manifolds with boundary representable by 4-colored graphsof small order are contained in [15], [12] and [13].In the 4-dimensional case, as pointed out in Section 4, the 5-colored graph of order tworepresents S and a 5-colored graph of order four represents S if it is bipartite and RP × B ifit is non-bipartite. For order six 5-colored bipartite graphs we have the following result. Proposition 8.2
Let Γ be a -colored bipartite graph of order six. Then M Γ is one of the following4-manifolds: S , B , S × B , S × S × B . Proof.
First of all, it is well-known that the only closed orientable 4-manifold admittinga representation with six vertices is S (see [5]). Moreover, the compact orientable 3-manifoldsadmitting an order six representation are S , S × B and S × S × I (see [30] and [15]). FromTheorem 4.1 it follows that the three 4-manifolds S , S × B , S × S × B admit order sixrepresentation which are the suspension of the 4-colored graphs of order six representing theabove 3-manifolds and no other compact 4-manifold can be obtained via suspension process froma 4-colored graph. So we can restrict our attention to order six bipartite 5-colored graphs whichare not the suspension of a 4-colored graph. It is easy to see that, up to isomorphism, there isonly a graph of such type (depicted on the right of Figure 3), which represents a simply connected4-manifold M (by Corollary 6.2) with connected spherical boundary. The represented manifoldturns out to be a 4-ball by simple homology arguments. In fact, by Lemma 3.16 we have χ ( M ) = 1and, if M (cid:48) is the closed 4-manifold obtained from M by capping off its boundary by a 4-ball, then χ ( M (cid:48) ) = 2. Since M (cid:48) is simply connected and orientable we have β ( M (cid:48) ) = β ( M (cid:48) ) = 0 and β ( M (cid:48) ) = β ( M (cid:48) ) = 1, therefore β ( M (cid:48) ) = 0 and M (cid:48) is a 4-sphere. As a consequence, M ishomeomorphic to B .The previous result shows that in dimension > M Γ may have sphericalcomponents. Order six graphs representing S × S × B , S × B and B are depicted inFigures 2 and 3. As sketched in the introduction, a strong interaction is known to exist between edge-coloredgraphs, representing quasi-manifolds of arbitrary dimension, and random tensor models (see forexample [26], [27] and [6]). In this framework, colored graphs naturally arise as Feynman graphsencoding tensor trace invariants. The key tool for this relashionship is the so-called
G-degree From Proposition 3.14 the boundary of M results to be a 3-manifold admitting a genus one Heegaardsplitting and simple Van Kampen type arguments show that it is simply connected. S × B and B , respectively. ω G (Γ) of an ( n + 1)-colored graph Γ, which drives the 1 /N expansion (cid:88) ω G ≥ N − n − ω G F ω G [ { t B } ] , (1)within the n -tensor product of the complex space C N .The 1 /N expansion of formula (1) describes the role of colored graphs (and of their G-degree ω G ) within colored tensor models theory and explains the importance of looking for cataloguesand classification results concerning all n -quasi-manifolds represented by ( n + 1)-colored graphswith a given G-degree. The representation theory via ( n + 1)-colored graphs described in thepresent paper for all n -quasi-manifolds (and their associated compact n -manifold possibly withboundary) might be a significant tool for this purpose.As previously cited, several classification results for 3-manifolds with boundary represented by4-colored graphs has been recently obtained. Nevertheless, dimension four appears to be a veryinteresting context within this approach, since it is the least dimension in which colored graphsmay represent quasi-manifolds with non-isolated singularities. In this direction, Proposition 8.2seems to be particularly significant.The G-degree arises from the existence of particular embeddings of colored graphs into closedsurfaces. Proposition 8.3 [21] Let Γ be a bipartite (resp. non-bipartite) ( n + 1) -colored graph of order p .Then for each cyclic permutation ε = ( ε ε . . . ε n ) of ∆ n , up to inversion, there exists a cellularembedding, called regular , of Γ into an orientable (resp. non-orientable) closed surface F ε (Γ) The coefficients F ω G [ { t B } ] of the formal series are generating functions of bipartite ( n + 1)-coloredgraphs with fixed G-degree ω G . A parallel tensor models theory, involving real tensor variables T ∈ ( R N ) ⊗ n , has been developed, takinginto account also non-bipartite colored graphs (see [33]): this is why both bipartite and non-bipartite coloredgraphs will be considered in this context. hose regions are bounded by the images of the { ε j , ε j +1 } -bigons, for each j ∈ Z n +1 . Moreover,the genus (resp. half the genus) ρ ε (Γ) of F ε (Γ) satisfies χ ( F ε (Γ)) = 2 − ρ ε (Γ) = (cid:88) j ∈ Z n +1 g ε j ,ε j +1 + (1 − n ) p. ( ∗ ) No regular embeddings of Γ exist into non-orientable (resp. orientable) surfaces. The G-degree of a colored graph is defined in terms of these embeddings as follows. Let Γ bean ( n + 1)-colored graph, with n ≥
2, and let Ξ n be the set of the n ! / n ,up to inversion. For any ε ∈ Ξ n , the genus ρ ε (Γ) is called the regular genus of Γ with respect to ε .Then, the Gurau degree (or
G-degree for short) of Γ is defined as ω G (Γ) = (cid:88) ε ∈ Ξ n ρ ε (Γ) . For n = 2 any bipartite (resp. non-bipartite) 3-colored graph Γ represents an orientable (resp.non-orientable) surface (cid:99) M Γ and ω G (Γ) is exactly the genus (resp. half of the genus) of (cid:99) M Γ . On theother hand, for n ≥
3, the G-degree of any ( n + 1)-colored graph is proved to be a non-negative integer , even in the non-bipartite case (see [6]).In dimension 4 the G-degree is always a multiple of 3, giving rise to the reduced G-degree ω (cid:48) G (Γ) = ω G (Γ) /
3. Moreover, from Theorems 1 and 2 of [8] we know that when a 5-colored graphis either bipartite or represents a singular 4-manifold, then ω (cid:48) G (Γ) is even. This fact implies that: • in the 4-dimensional complex contest, the only non-vanishing terms in the 1 /N expansionof (1) are the ones corresponding to even powers of 1 /N ; • in the 4-dimensional real tensor models framework, where also non-bipartite graphs are in-volved, only 5-colored graphs representing quasi-manifolds which are not singular manifoldsmay appear in the terms corresponding to even powers of 1 /N .Now we can give a characterization of supercontracted ω (cid:48) G ≤ ε c of (cid:98) c , for each c ∈ ∆ , in the followingway: ε = (1 3 4 2) , ε = (0 3 2 4) , ε = (0 3 4 1) , ε = (0 2 1 4) , ε = (0 2 3 1) . If Γ is a 5-colored graph of order 2 p , then the sum of the relations ( ∗ ) in Proposition 8.3 over all (cid:98) c -residues of Γ gives: 2 g (cid:98) c − ρ (cid:98) c = (cid:88) i ∈ Z g ε ci ,ε ci +1 − p ( ∗∗ ) , where ρ (cid:98) c denotes the sum of the regular genera of the (cid:98) c -residues with respect to ε c .By summing the five relations ( ∗∗ ) over c ∈ ∆ , we obtain2 |R (Γ) | − (cid:88) c ∈ ∆ ρ (cid:98) c = 2 |R (Γ) | − p. Lemma 5.6(i) proves that any closed n-manifold M can be represented by a supercontracted (n+1)-colored graph, called a crystallization of M (see [18]). he sum (cid:80) c ∈ ∆ ρ (cid:98) c is called the subdegree ρ G (Γ) of the 5-colored graph Γ. Therefore ρ G (Γ) = |R (Γ) | + 5 p − |R (Γ) | . This definition is motivated by the following result.
Lemma 8.4
Let ω G (Γ (cid:98) c ) be the sum of the G-degrees of the connected components of Γ (cid:98) c , then (cid:88) c ∈ ∆ ω G (Γ (cid:98) c ) = 3 ρ G (Γ) . Proof.
The relations of Proposition 7 and Lemma 13 of [6] for d = 4 give: 3(4+6 p −|R (Γ) | ) = ω G (Γ) = 3(4 + p − |R (Γ) | ) + (cid:80) c ∈ ∆ ω G (Γ (cid:98) c ) and therefore (cid:80) c ∈ ∆ ω G (Γ (cid:98) c ) = 3(5 p − |R (Γ) | + |R (Γ) | ) = 3 ρ G (Γ). Proposition 8.5
Let Γ be a supercontracted 5-colored graph, then:(i) ω (cid:48) G (Γ) (cid:54) = 1 ;(ii) ω (cid:48) G (Γ) = 0 if and only if Γ is the order two graph (representing S );(iii) ω (cid:48) G (Γ) = 2 if and only if Γ is the order four graph of Figure 4 (representing S );(iv) ω (cid:48) G (Γ) = 3 if and only if Γ is the order four graph in the left of Figure 5 (representing RP × B ). Proof.
It is straightforward that the order two graph has zero (reduced) G-degree. Let 2 p be the order of Γ. Up to isomorphism there is only a bipartite supercontracted graph (resp.two non-bipartite supercontracted graphs) of order four, namely the one of Figure 4 (resp. thetwo of Figure 5), and it has reduced G-degree = 2 (resp. they have reduced G-degree = 3 and= 4 respectively). Furthermore, up to isomorphism there are 8 bipartite (resp. 31 non-bipartite)supercontracted graphs of order six , and a direct computation says that their reduced G-degreeis always > ρ G (Γ) = ω G (Γ) / − − p + |R (Γ) | = ω (cid:48) G (Γ) + 1 − p ,since Γ is supercontracted, and therefore ω (cid:48) G (Γ) ≥ p −
1. If ω (cid:48) G (Γ) = 0 then p ≤
1, which proves(ii). If ω (cid:48) G (Γ) = 1 then p ≤
2, which proves (i). If ω (cid:48) G (Γ) = 2 then p ≤
3, which proves (iii). If ω (cid:48) G (Γ) = 3 then p ≤
4. In this case, if p = 4 then (cid:80) c ∈ ∆ ω G (Γ (cid:98) c ) = 3 ρ G (Γ) = 0. This means thatall the 4-residues of Γ represent S (see Propositions 8 and 9 of [6]) and therefore (cid:99) M Γ should be aclosed 4-manifold. By Theorem 2 of [8] the G-degree ω G (Γ) would be a multiple of six, in contrastwith the assumption ω G (Γ) = 9. This concludes the proof. Acknowledgement.
The authors have been supported by the ”National Group for Algebraicand Geometric Structures, and their Applications” (GNSAGA-INdAM) and University of Modenaand Reggio Emilia and University of Bologna, funds for selected research topics. The authors wishto thank Paola Cristofori for her help in generating the catalogue of all order six 5-colored graphs. This result has been obtained by a computer program by adapting the algorithmic procedure describedin [5]. S .Figure 5: Two graphs representing RP × B . References [1] A. Bj¨orner,
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An SYK-Lyke Model Without Disorder , preprint, 2016. arXiv:1610.09758Luigi GRASSELLIDipartimento di Scienze e Metodi dell’Ingegneria, Universit`a di Modena e Reggio EmiliaVia Giovanni Amendola, 41-43, 42122 Reggio Emilia, ITALYe-mail: [email protected]
Michele MULAZZANIDipartimento di Matematica and ARCES, Universit`a di BolognaPiazza di Porta San Donato 5, 40126 Bologna, ITALYe-mail: [email protected]@unibo.it