On graph – theoretical invariants of combinatorial manifolds

The goal of this paper is to give some theorems which relate to the problem of classifying combinatorial (resp. smooth) closed manifolds up to piecewise-linear (PL) homeomorphism. For this, we use the combinatorial approach to the topology of PL manifolds by means of a special kind of edge–colored graphs, called crystallizations. Within this representation theory, Bracho and Montejano introduced in 1987 a nonnegative numerical invariant, called the reduced complexity, for any closed n–dimensional PL manifold. Here we consider this invariant, and extend in this context the concept of average order first introduced by Luo and Stong in 1993, and successively investigated by Tamura in 1996 and 1998. Then we obtain some classification results for closed connected smooth low–dimensional manifolds according to reduced complexity and average order. Finally, we answer to a question posed by Trout in 2013. Mathematics Subject Classifications: 57N15, 57Q15, 05C10 1 Colored Graphs and Crystallizations All spaces and maps will be considered in the PL category, for which we refer to [16]. The definitions and main results of Graph Theory can be found in [10]. For the representation of PL manifolds by means of edge–colored graphs and crystallizations see the survey papers [1, 2, 7, 9, 21]. Here we recall the necessary definitions to explain the statements of our main theorem. An (n+1)–colored graph (G, c) is a multigraph G = (V (G), E(G)), regular of degree n+1 (possibly with multiple edges, but without loops), together with a proper edge-coloring c : E(G) → ∆n = {i ∈ Z : 0 6 i 6 n}. This means that any two adjacent edges in G are the electronic journal of combinatorics 26(3) (2019), #P3.10 1 differently colored. As usual, V (G) and E(G) denote the vertex set and the edge set of G, respectively; ∆n will be called the color set, and its elements the colors. The cellular complex K = K(G) associated to G is constructed as follows: (1) for each vertex v of G, consider a standard n–simplex σ(v), and label its n + 1 vertices by the colors of ∆n; (2) if v and w are joined in G by an i–colored edge, then identify the (n− 1)–faces of σ(v) and σ(w) opposite the vertex labelled by i ∈ ∆n, so that equally labelled vertices coincide. The complex K(G) is not a classical simplicial complex for two simplexes may meet in more than a single face. On the other hand, it is a pseudocomplex in the sense of [11], p.49. This means that any simplex of K(G) is canonically isomorphic to a standard one, and the intersection of two simplexes can be either empty or a union of common faces. By construction, the graph G can be thought as the 1–skeleton of the dual cellular complex of K(G). Let M be a closed connected PL (or, smooth) n–manifold. We say that (G, c) represents M if M is PL homeomorphic to the space underlying K(G). A crystallization of M is an (n + 1)–colored graph (G, c) representing M such that K(G) has exactly n + 1 vertices (which we shall always assume to be colored by the elements of ∆n). In this case, K(G) is called a contracted triangulation of M . The following is a famous theorem of Pezzana (for the proof see, for example, [7]): Theorem 1. Every closed connected PL (or, smooth) n–manifold admits a crystallization, that is, it has a contracted triangulation. Let M be a closed connected PL n–manifold, (G, c) a crystallization of M (with color set ∆n), and K = K(G) the associated contracted triangulation of M . If Γ ⊂ ∆n, then gΓ represents the number of connected components of the partial subgraph GΓ = (V (G), c−1(Γ)). If Γ = {i, j} (resp. {r, s, t} and {h, k, r, s}), then gΓ will be simply written as gij (resp. grst and ghkrs). Let p denote the order of G, i.e., the number of vertices in the graph. We always assume that {vi : i ∈ ∆n} is the vertex set of K, and that vi corresponds to Gî, where î = ∆n \ {i}. Theorem 2. An (n + 1)-colored graph (G, c) is a crystallization of a closed connected PL n-manifold if and only if every partial subgraph Gî is connected and represents the (n− 1)-sphere, for every i ∈ ∆n. Let qh(K) denote the number of h–simplexes in K, for any h ∈ ∆n. For any Γ ⊂ ∆n with cardinality h, gΓ is also the number of (n−h)–simplexes of K = K(G) whose vertices are labelled by colors in ∆n \ Γ. Using crystallizations, we can associate some numerical invariants to any closed connected PL manifold. See, for example, [2, 3, 4, 7]. Here we are interested in two of them, called reduced complexity and average order, which will be presented in the next two sections together with new results about characterizations of certain PL manifolds, up to PL homeomorphisms. the electronic journal of combinatorics 26(3) (2019), #P3.10 2 2 Reduced complexity Let M be a closed connected PL n–manifold. Following [2], we define the complexity c(M) of M as the minimum number of n–simplexes which a contracted triangulation of M must have. In other words, c(M) is the minimum order of a crystallization which represents M . Since any crystallization has at least two vertices, it was defined in [2] the reduced complexity of M as c̃(M) = c(M)− 2. This combinatorial invariant gives a finite–to–one map from the class of closed connected PL n–manifolds to the set of nonnegative even integers. Of course, the only n–manifold of reduced complexity zero is the standard n– sphere S. For any closed connected surface M , we have c̃(M) = 4−2χ(M), where χ(M) is the Euler characteristic of M (see Theorem 3.13 of [2]). Thus the reduced complexity can be regarded as a generalization of the Euler characteristic. Moreover, it has the nice property of classifying manifolds up to a finite ambiguity. More precisely, if we know a closed connected manifold M has a specific value of reduced complexity, then there are only finitely many topological types possible for M . The classification of all closed connected 3–manifolds with reduced complexity less than or equal to 28 was given in [5] and [12], §5, by using computer algorithms. There are exactly sixty-nine of such manifolds. Among them, there are S, S×S, twenty–eight lens spaces, the six Euclidean orientable 3–manifolds, and sixteen quotients of S by the action of their finite (non-cyclic) fundamental groups. The complete classification of all closed connected PL 4–manifolds up to reduced complexity 14 was obtained in [6]. To clarify the next statement, we first explain the twisted bundle notation. Let S1×Sn−1 (resp. S× ∼ Sn−1) denote the orientable (resp. non orientable or twisted) Sn−1-bundle over S. Then the main theorem of [6] is the following: Theorem 3. (a) There are no closed connected 4–manifolds M of reduced complexity 0 < c̃(M) < 6. The unique closed connected 4–manifold of reduced complexity 6 is the complex projective plane CP . (b) M be a closed connected 4–manifold. If c̃(M) = 8, then M is PL homeomorphic to either S × S or S × ∼ S. There are no closed connected 4–manifolds of reduced complexity 10. (c) The unique closed connected prime 4–manifold of reduced complexity 12 is the topological product S × S. (d) The unique closed connected prime 4–manifold of reduced complexity 14 is the real projective 4–space RP . In [8], it was given the classification of the closed connected PL (or, smooth) 5– manifolds up to reduced complexity 20. This gives combinatorial characterizations of S × S, S × ∼ S and S × S among closed PL 5–manifolds. More precisely, the main result of [8] is the following: the electronic journal of combinatorics 26(3) (2019), #P3.10 3 Theorem 4. (a) The only reduced complexity zero 5–manifold is S, and there are no closed connected 5–manifolds M of reduced complexity 0 < c̃(M) < 10. The only closed connected 5–manifolds of reduced complexity 10 are S × S and S × ∼ S. (b) There are no closed connected 5–manifolds M of reduced complexity 10 < c̃(M) < 20. The only closed connected spin 5–manifolds of reduced complexity 20 are S × S and the connected sums N1 #N2, where each Ni, i = 1, 2, is either S × S or S × ∼ S. Further results and conjectures concerning with the reduced complexity of triangulated manifolds can be found in the quoted papers.


Colored Graphs and Crystallizations
All spaces and maps will be considered in the PL category, for which we refer to [16].The definitions and main results of Graph Theory can be found in [10].For the representation of PL manifolds by means of edge-colored graphs and crystallizations see the survey papers [1,2,7,9,21].
Here we recall the necessary definitions to explain the statements of our main theorem.An (n+1)-colored graph (G, c) is a multigraph G = (V (G), E(G)), regular of degree n+1 (possibly with multiple edges, but without loops), together with a proper edge-coloring c : E(G) → ∆ n = {i ∈ Z : 0 i n}.This means that any two adjacent edges in G are differently colored.As usual, V (G) and E(G) denote the vertex set and the edge set of G, respectively; ∆ n will be called the color set, and its elements the colors.
The cellular complex K = K(G) associated to G is constructed as follows: (1) for each vertex v of G, consider a standard n-simplex σ n (v), and label its n + 1 vertices by the colors of ∆ n ; (2) if v and w are joined in G by an i-colored edge, then identify the (n − 1)-faces of σ n (v) and σ n (w) opposite the vertex labelled by i ∈ ∆ n , so that equally labelled vertices coincide.The complex K(G) is not a classical simplicial complex for two simplexes may meet in more than a single face.On the other hand, it is a pseudocomplex in the sense of [11], p.49.This means that any simplex of K(G) is canonically isomorphic to a standard one, and the intersection of two simplexes can be either empty or a union of common faces.By construction, the graph G can be thought as the 1-skeleton of the dual cellular complex of K(G).
Let M n be a closed connected PL (or, smooth) n-manifold.We say that (G, c) represents M if M is PL homeomorphic to the space underlying K(G).A crystallization of M is an (n + 1)-colored graph (G, c) representing M such that K(G) has exactly n + 1 vertices (which we shall always assume to be colored by the elements of ∆ n ).In this case, K(G) is called a contracted triangulation of M .
The following is a famous theorem of Pezzana (for the proof see, for example, [7]): Theorem 1.Every closed connected PL (or, smooth) n-manifold admits a crystallization, that is, it has a contracted triangulation.
Let M be a closed connected PL n-manifold, (G, c) a crystallization of M (with color set ∆ n ), and K = K(G) the associated contracted triangulation of M .If Γ ⊂ ∆ n , then g Γ represents the number of connected components of the partial subgraph G Γ = (V (G), c −1 (Γ)).If Γ = {i, j} (resp.{r, s, t} and {h, k, r, s}), then g Γ will be simply written as g ij (resp.g rst and g hkrs ).Let p denote the order of G, i.e., the number of vertices in the graph.We always assume that {v i : i ∈ ∆ n } is the vertex set of K, and that v i corresponds to G i , where i = ∆ n \ {i}.
Theorem 2. An (n + 1)-colored graph (G, c) is a crystallization of a closed connected PL n-manifold if and only if every partial subgraph G i is connected and represents the (n − 1)-sphere, for every i ∈ ∆ n .
Let q h (K) denote the number of h-simplexes in K, for any h ∈ ∆ n .For any Γ ⊂ ∆ n with cardinality h, g Γ is also the number of (n−h)-simplexes of K = K(G) whose vertices are labelled by colors in ∆ n \ Γ.
Using crystallizations, we can associate some numerical invariants to any closed connected PL manifold.See, for example, [2,3,4,7].Here we are interested in two of them, called reduced complexity and average order, which will be presented in the next two sections together with new results about characterizations of certain PL manifolds, up to PL homeomorphisms.

Reduced complexity
Let M be a closed connected PL n-manifold.Following [2], we define the complexity c(M ) of M as the minimum number of n-simplexes which a contracted triangulation of M must have.In other words, c(M ) is the minimum order of a crystallization which represents M .Since any crystallization has at least two vertices, it was defined in [2] the reduced complexity of M as c(M ) = c(M ) − 2. This combinatorial invariant gives a finite-to-one map from the class of closed connected PL n-manifolds to the set of nonnegative even integers.Of course, the only n-manifold of reduced complexity zero is the standard nsphere S n .For any closed connected surface M , we have c(M ) = 4 − 2χ(M ), where χ(M ) is the Euler characteristic of M (see Theorem 3.13 of [2]).Thus the reduced complexity can be regarded as a generalization of the Euler characteristic.Moreover, it has the nice property of classifying manifolds up to a finite ambiguity.More precisely, if we know a closed connected manifold M has a specific value of reduced complexity, then there are only finitely many topological types possible for M .
The classification of all closed connected 3-manifolds with reduced complexity less than or equal to 28 was given in [5] and [12], §5, by using computer algorithms.There are exactly sixty-nine of such manifolds.Among them, there are S 3 , S 1 × S 2 , twenty-eight lens spaces, the six Euclidean orientable 3-manifolds, and sixteen quotients of S 3 by the action of their finite (non-cyclic) fundamental groups.
The complete classification of all closed connected PL 4-manifolds up to reduced complexity 14 was obtained in [6].To clarify the next statement, we first explain the twisted bundle notation.Let S 1 × S n−1 (resp.S 1 × ∼ S n−1 ) denote the orientable (resp.non orientable or twisted) S n−1 -bundle over S 1 .
Then the main theorem of [6] is the following: (d) The unique closed connected prime 4-manifold of reduced complexity 14 is the real projective 4-space RP 4 .
In [8], it was given the classification of the closed connected PL (or, smooth) 5manifolds up to reduced complexity 20.This gives combinatorial characterizations of S 1 × S 4 , S 1 × ∼ S 4 and S 2 × S 3 among closed PL 5-manifolds.More precisely, the main result of [8] is the following:  The only closed connected spin 5-manifolds of reduced complexity 20 are S 2 × S 3 and the connected sums Further results and conjectures concerning with the reduced complexity of triangulated manifolds can be found in the quoted papers.

Average order
Let K be a simplicial triangulation of a closed connected 3-manifold M with E 0 (K) edges and F 0 (K) triangles.Note that we distinguish a simplicial triangulation from a pseudocomplex (or, in general, a cell decomposition) into a union of 3-simplexes, that is, such a cell decomposition is a triangulation when the intersection of any two simplexes is actually a single face of each of them.The order of an edge in K is the number of triangles incident to that edge.The average edge order of K was defined in [13] as Luo and Stong showed in [13] that for a closed 3-manifold M , µ 0 (K) being small implies that the topology of M is fairly simple and restricts the triangulation K.The relations between this quantity and the topology of M were investigated in the quoted paper, and the main result of [13] is stated as follows: Theorem 5. Let K be any simplicial triangulation of a closed connected 3-manifold M .Then (a) 3 µ 0 (K) < 6, equality holds if and only if K is the simplicial triangulation of the boundary of a 4-simplex.
(b) For any > 0 there are simplicial triangulations K 1 and K 2 of M such that Similar results for 3-manifolds with non-empty boundary were established by Tamura [17,18].For a related study see also the paper of Walkup [22].This concept was extended the electronic journal of combinatorics 26(3) (2019), #P3.10 in [4] to higher dimension, and successively investigated there (see [3] for the 3-dimensional case) for the class of colored triangulations of PL n-manifolds.
Let now K be a colored triangulation of a closed connected n-manifold M , that is, K is a pseudocomplex triangulating M , whose vertices are labeled by ∆ n so that the coloring is injective on each n-simplex of K.For such a colored triangulation K, it is natural to define the average (n − 2)-simplex order of K (see [4]) as where q k (K) is the number of k-simplexes of K, for k = 0, . . ., n.The following is the main theorem of [4] (which extends that of [3] obtained in dimension 3).
Theorem 6.Let K be any colored triangulation of a closed connected PL n-manifold M n , n 3. Then (a) 2 µ(K) < 6, equality holds if and only if K is the standard (two n-simplexes) colored triangulation of S n .
(b) For any > 0 there exists a colored triangulation K of M such that , then K is a colored triangulation of one of the following n-manifolds: S n , S 1 × S n−1 , S 1 × ∼ S n−1 or (for n = 3) the real projective space RP 3 .
Remark.Note that µ 0 in (1) and µ in (2) have essentially the same definition, but they assume in general quite different values (this is the reason to use distinct notations).In fact, the former is defined by using the class of simplicial triangulations, while the latter arises from the different class of colored pseudocomplexes.Now we recall the definition of a further combinatorial invariant of closed connected PL n-manifolds.Let K be a contracted triangulation of a closed connected PL n-manifold M , and µ(K) the average (n − 2)-simplex order of K.By [7] the (contracted) average (n − 2)simplex order of M (in short, the average order), written µ − (M ), is the smallest µ(K) for a contracted triangulation K of M .The motivation for introducing this invariant comes from the 2-dimensional case (see also [13]).Suppose we have a contracted triangulation K of a closed connected surface M .Then q 0 (K) = 3, q 1 (K) = (3p)/2 and q 2 (K) = p for p 2 even.Hence µ − (M ) = p = 6 − 2χ(M ) by a Euler characteristic calculation.
the electronic journal of combinatorics 26(3) (2019), #P3.10 Therefore, the average vertex order being less than 6, equal to 6, or greater than 6 corresponds to M having a spherical, Euclidean, or hyperbolic structure, respectively.It is also convenient to introduce a new numerical invariant for a closed connected PL n-manifold M .Let µ + (M ) denote the maximum µ(K) for a contracted triangulation In dimension 3, we are able to improve Theorem II of [3] in order to get the following result: Theorem 7. Let M be a closed connected PL 3-manifold.Then the reduced complexity and the average order of M are linked by the following formula: Furthermore, we have: (b) Suppose now M prime and orientable.Then we have: , and L(5, 2), or the quaternionic space S 3 / < 2, 2, 2 >.
• For 5 < µ − (M ) < 5.2, there are six lens spaces, seven spherical manifolds with finite noncyclic fundamental groups, and four euclidean fibered manifolds (see the Appendix for more detailed information).
We are going to prove our main result in dimension 4. We emphasize that the useful topological invariant switches from µ − for χ(M ) < 5 to µ + for χ(M ) > 5.It is also remarkable that when χ(M ) = 5 the distribution of the µ(K) gives no topological information whatsoever.The trichotomy between χ(M ) < 5, χ(M ) = 5 and χ(M ) > 5 is striking.It should be interesting to know of other instances where this is important.Theorem 8. Let M 4 be a closed connected PL 4-manifold, and T c the class of all 4dimensional contracted triangulations.Then we have: and equality holds if and only if M is PL homeomorphic to a 4-sphere.Furthermore, for every > 0 there exists a contracted triangulation K of M such that µ(K ) > 5− .
Furthermore, for every > 0 there exists a contracted triangulation K of M such that µ(K ) < 5 + .
Proof.Let (G, c) be a crystallization of M of order p, K = K(G) the associated contracted triangulation.Then c(M ) p − 2. By Lemma (2.1)a [6] we have q 3 (K) = (5p)/2 and q 2 (K) = i<j g ij , hence the electronic journal of combinatorics 26(3) (2019), #P3.10 We now show that g ij p/2 for every i, j ∈ ∆ 4 , i = j.Fix i, j ∈ ∆ 4 , i = j, and let p α,ij be the number of vertices of a connected component C α,ij of G colored by i, j ∈ ∆ 4 , i = j, for α = 1, . . ., g ij .Then p α,ij is even ( 2) and g ij α=1 p ij = p.In fact, the set {C α,ij } α of cardinality g ij induces a partition of the vertex (resp.edge) set of G.So we obtain 2 g ij g ij α=1 p ij = p, hence g ij p/2 for arbitarily fixed distinct colors i, j ∈ ∆ 4 .This implies that i<j g ij 5p.Hence the formula in (6) gives µ(K) 2 for any contracted triangulation K of M .By Lemma (2.2) [6], we have Substituting ( 7) into (6) yields as g rst > 0 for every distinct colors r, s, t ∈ ∆ 4 .Thus for every contracted triangulation K of M , we have 2 µ(K) < 6, hence 2 µ − (M ) < 6.If µ − (M ) = 2, then there is a contracted triangulation K of M such that 5p = i<j g ij , hence g ij = p/2 for every i, j ∈ ∆ 4 and g rst = p/2 for every r, s, t ∈ ∆ 4 .Then the 5colored graph G is a dipole of type 5, that is, it consists of two vertices joined by 5 edges, one of each color in ∆ 4 .This means that the pseudocomplex K = K(G) associated to G consists of two 4-simplexes with identified boundary, so p = 2 and c(M ) = 0 (compare also with [4], p.257).Then M 4 is PL homeomorphic to a 4-sphere.
By Lemma (2.2) [6] we have Substituting ( 8) into (6) gives Thus for any contracted triangulation K of M , we get Then µ(K) goes to 5 when p goes to infinity.Note that for any closed connected PL 4-manifold M 4 there is a contracted triangulation of arbitrarily large order p = q 4 (K).Then inf K µ(K) = 5 and for every > 0 there exists a contracted triangulation K of M such that µ − (M ) µ(K ) < 5 + .Thus µ + (M ) < 6, and (c) is proved.
Combining Theorem 8 with Theorem 3, we get the following characterizations: Theorem 9. Let M 4 be a prime closed connected PL 4-manifold.Then we have: hence c(M ) 12. By Theorem 3(a-c) it follows that M is PL homeomorphic to the topological product S 2 × S 2 .
Examples.The following are explicit computations of the considered combinatorial invariants for the manifolds listed above: The following is our main result on the average order for the class of 5-dimensional manifolds (here statement (a) is new, while statements (b) and (c) follow from [4]): Proof.Let (G, c) be a crystallization of M with minimum order p = c(M ) + 2, and let K = K(G) be the associated contracted triangulation.By Lemma 2.1(a) [8] we get By Lemma 2.2 and Lemma 2.1(d) of [8], we have On the other hand, g ij p/2 for every distinct colors i, j ∈ ∆ 5 , so we get i<j g ij (15p)/2.Thus we obtain the inequalities in (a).If µ − (M ) = 2, then there is a contracted triangulation K of M such that 15p = 2 i<j g ij , hence g ij = p/2 for every i, j ∈ ∆ 5 .By Lemma 2.2 [8] we get g ijk = p/2 for every distinct colors i, j, k ∈ ∆ 5 .We see also that g hkrs = p/2 for every 4-tuple of distinct colors.So the last formula of Lemma 2.2 [8] gives p = 2. Then K consists of two 5-simplexes with identified boundary, i.

A question on average order
The work by Luo and Stong [13] and Tamura [17,18] implies the following result, stated in [20]: Theorem 11.For every closed connected topological 3-manifold M and every rational number 4.5 < r < 6 there is a simplicial triangulation T of M for which the average edge-order µ(T ) is r.
The fact that we can find such a triangulation T independent of the topology of M is significant.This is a point of difference with the statement of Theorem 12 (below).
The following question has been posed by Trout in [20]: Question: Does anyone know of results similar to Theorem 11 but for n 4?
We provide an answer to question above for n = 4.
Theorem 12.For any rational number This completes the proof.
Connections between the average order and the edge diameter of a manifold can be derived from the work by Trout [19].
• Spherical manifolds with finite noncyclic fundamental groups (which are Seifert fibered manifolds): The relevant crystallizations of minimal order 26 and 28 for closed connected prime 3-manifolds can be found in [12], §5.

Theorem 3 .∼S 3 .
(a) There are no closed connected 4-manifolds M of reduced complexity 0 < c(M ) < 6.The unique closed connected 4-manifold of reduced complexity 6 is the complex projective plane CP 2 .(b) M 4 be a closed connected 4-manifold.If c(M ) = 8, then M is PL homeomorphic to either S 1 × S 3 or S 1 × There are no closed connected 4-manifolds of reduced complexity 10.(c)The unique closed connected prime 4-manifold of reduced complexity 12 is the topological product S 2 × S 2 .

Theorem 4 .∼ S 4 .
(a) The only reduced complexity zero 5-manifold is S 5 , and there are no closed connected 5-manifolds M of reduced complexity 0 < c(M ) < 10.The only closed connected 5-manifolds of reduced complexity 10 are S 1 × S 4 and S 1 × (b) There are no closed connected 5-manifolds M of reduced complexity 10 < c(M ) < 20.