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.