Cataloguing PL 4-manifolds by gem-complexity

We describe an algorithm to subdivide automatically a given set of PL n-manifolds (via coloured triangulations or, equivalently, via crystallizations) into classes whose elements are PL-homeomorphic. The algorithm, implemented in the case n=4, succeeds to solve completely the PL-homeomorphism problem among the catalogue of all closed connected PL 4-manifolds up to gem-complexity 8 (i.e., which admit a coloured triangulation with at most 18 4-simplices). Possible interactions with the (not completely known) relationship among different classification in TOP and DIFF=PL categories are also investigated. As a first consequence of the above PL classification, the non-existence of exotic PL 4-manifolds up to gem-complexity 8 is proved. Further applications of the tool are described, related to possible PL-recognition of different triangulations of the K3-surface.


Introduction and main results
One of the most interesting features of piecewise-linear (PL) topology is the possibility of representing manifolds by combinatorial structures; the main developed theories concern the 3-dimensional case, where -thanks to recent advances in computing power -topologists succeeded in constructing exhaustive tables of "small" 3-manifolds based on different representation methods (see [43] and its bibliography for successive results about closed orientable irreducible 3-manifolds up to Matveev's complexity 11, and [3], [12] for analogous studies about closed non-orientable P 2 -irreducible 3-manifolds up to Matveev's complexity 10).
In dimension four, fewer combinatorial tools are available to represent PL-manifolds. On the other hand, classification results for topological (TOP) simply-connected 4manifolds are well-known, though more attention must be paid when considering equivalence of PL-structures.
Crystallization theory is a representation theory for PL-manifolds of arbitrary dimension by means of suitable edge-coloured graphs (called crystallizations), which are dual to coloured triangulations. Together with the Italian school that gave rise to the graphtheoretical tool (see [44], [30], [36], [31], [4] and their references), many authors around the world concurred to its development, with recent significant contributions, too: for example, [8], [9], [45], [28]. The totally combinatorial nature of the representing objects and the generality with respect to dimension are among the strong points of crystallization theory: topological and PL properties are reflected into combinatorial ones, and the problem of distinguishing manifolds (both in TOP and in PL category) can be simplified by combinatorial invariants computed on the graphs.
In particular, in dimension four and five the best achievements have been obtained as regards the attempts of classifying PL-manifolds via a suitable graph-defined invariant, called regular genus 1 : they concern both the case of "low" regular genus, and the case of "restricted gap" between the regular genus of the manifold and the regular genus of its boundary, and the case of "restricted gap" between the regular genus and the rank of the fundamental group of the manifold (see, for example, [24], [14] and [26]).
More recently, the interest focused on other combinatorial invariants internal to crystallization theory, i.e. GM-complexity and gem-complexity, which are related in dimension three to Matveev's complexity, too (see [16], [18], [23], [21]). In particular, gemcomplexity is the natural invariant used to create automatic catalogues of PL-manifolds via crystallizations: in fact, it is related to the minimum order of a crystallization of the manifold. On the other hand, suitable moves on edge-coloured graphs are defined, which preserve the represented manifold up to PL-homeomorphisms; even if they are not able to solve algorithmically the recognition problem for general PL-manifolds, they are a powerful tool to face the problem itself, for a given set of PL-manifolds. In dimension three, this approach already allowed the development of a classification algorithm, which succeeded to completely recognize PL-homeomorphism classes of all 3-manifolds up to gem-complexity 14 (i.e. representable by coloured triangulations with at most 30 tetrahedra): see [40], [18] and [19] for the orientable case and [15], [16] and [5] for the non-orientable one.
The present paper describes the n-dimensional extension of the above classifying al-gorithm, together with the results obtained by applying it to the crystallization catalogue representing all PL 4-manifolds up to gem-complexity 8 (i.e. whose associated coloured triangulations have at most 18 4-simplices).
The main classification results are collected into the following theorem, where k(M 4 ) denotes the gem-complexity of M 4 ; the last statement makes use also of a partial analysis (whose completion is in progress) of the crystallization catalogue representing all PL 4manifolds with gem-complexity 9, which has been generated, too.
As far as the TOP category is concerned, the combinatorial properties of crystallizations, together with well-known results on TOP simply-connected 4-manifolds, yield the following interesting result related to the topological classification of simply-connected PL 4-manifolds with respect both to gem-complexity and to regular genus: Theorem 2 Let M 4 be a simply-connected PL 4-manifold M 4 . If either gem-complexity k(M 4 ) ≤ 65 or regular genus G(M 4 ) ≤ 43, then M 4 is TOP-homeomorphic to Theorem 2 summarizes Proposition 9 (for gem-complexity) and Proposition 12 (for regular genus) of Section 4.
As it is well-known, up to now there is no classification of smooth structures on any given smoothable topological 4-manifold; on the other hand, finding non-diffeomorphic smooth structures on the same closed simply connected topological manifold has long been an interesting problem.
We hope that further advances in the generation and classification of crystallization catalogues for PL 4-manifolds, according to gem-complexity, could produce examples of non-equivalent PL-structures on the same TOP 4-manifold. For example, if at least one among the infinitely many PL 4-manifolds TOP-homeomorphic but not PLhomeomorphic to CP 2 # 2 (−CP 2 ) (which are proved to exist in [1]) admits a so called simple crystallization (according to [9]), then it will appear in the catalogue of order 20 crystallizations.
Moreover, we point out that the program performing automatic recognition of PLhomeomorphic 4-manifolds may be an useful tool to approach open problems related to different triangulations of the same TOP 4-manifold, which are conjectured to represent the same PL 4-manifold, too. The first candidates are the two known 16-vertices and 17-vertices triangulations of the K3-surface: see [29] and [46], together with the attempts to settle the conjecture described in [10], [11] and [9].

Basic notions of crystallization theory
As already pointed out, crystallization theory allows to represent combinatorially PL manifolds of arbitrary dimension, without restrictions concerning orientability, connectedness or boundary properties, by means of suitable edge-coloured graphs or -equivalently -by means of coloured triangulations. A detailed account of the theory may be found in [33], [40] and [4], together with their references.
In the present paper, when not otherwise stated, we will restrict our attention to the case of closed, connected PL n-manifolds.
The elements of the set ∆ n = {0, 1, . . . , n} are called colours; moreover, for each i ∈ ∆ n , we denote by Γî the n-coloured graph obtained from (Γ, γ) by deleting all edges coloured by i. Definition 2. An (n+1)-coloured graph (Γ, γ) is said to be contracted if the subgraph Γî is connected, for each i ∈ ∆ n . Each (n + 1)-coloured graph uniquely determines an n-dimensional CW-complex K(Γ), which is said to be associated to Γ: • for every vertex v ∈ V (Γ), take an n-ball σ(v) abstractly isomorphic to an nsimplex, and label injectively its n + 1 vertices by the colours of ∆ n ; • for every i-coloured edge between v, w ∈ V (Γ), identify the (n − 1)-faces of σ(v) and σ(w) opposite to i-labelled vertices, so that equally labelled vertices coincide.
It is easy to check that the properties of (Γ, γ) imply K(Γ) to be actually a pseudocomplex: in particular, its balls may intersect in more than one face, but no selfidentification of faces are allowed. 3 Definition 3. An (n + 1)-coloured graph (Γ, γ) is said to represent a PL n-manifold M n (briefly, it is a gem of M n ) if M n is PL-homeomorphic to |K(Γ)|. If, in addition, (Γ, γ) is contracted, then it is called a crystallization of M n . In both cases, the pseudocomplex K(Γ), equipped with the vertex-labelling inherited from γ, is said to be a coloured triangulation of M n .
Remark 1 It is easy to prove that each PL n-manifold admits an (n+1)-coloured graph representing it: just take the barycentric subdivision H ′ of any (simplicial) triangulation H of M n , label any vertex of H ′ with the dimension of the open simplex containing it in H, and then consider the 1-skeleton of the dual cellular complex of H ′ , with each edge coloured by i iff it is dual to an (n − 1)-face whose vertices are labelled by ∆ n − {i}.
The following proposition collects some well-known facts in crystallization theory: there is a bijection between (n − h)-simplices of K(Γ) whose vertices are labelled by ∆ n −{B} and connected components of the h-coloured graph Γ B = (V (Γ), γ −1 (B)) (which are called h-residues involving colours B, or B-residues of Γ, and whose number will be denoted by g B ).
It is not difficult to understand that, generally, many crystallizations of the same PL n-manifold exist; hence, it is a basic problem how to recognize crystallizations (or, more generally, gems) of the same PL n-manifold.
The easiest case is that of two colour-isomorphic gems, i.e. if there exists an isomorphism between the graphs, which preserves colours up to a permutation of ∆ n . It is quite trivial to check that two colour-isomorphic gems produce the same polyhedron.
The following result assures that colour-isomorphic graphs can be effectively detected by means of a suitably defined numerical code, which can be directly computed on each of them (see [25]). Proposition 5 Two (n + 1)-coloured graphs are colour-isomorphic iff their codes coincide.
The problem of recognizing non-colour-isomorphic gems representing the same manifold is also solved, but not algorithmically: a finite set of moves -the so called dipole moves -is proved to exist, with the property that two gems represent the same manifold iff they can be related by a finite sequence of such moves. A dipole move consists in the insertion or elimination of particular configurations involving h parallel edges, called h-dipoles (1 ≤ h ≤ n): see [31] for details, or Figure 1 for an example in dimension n = 4, with h = 2.

Figure 1: dipole move
In this paper, however, we will also make use of another set of moves, which appears to be more suitable for algorithmic procedures (see the notion of blob and flip in section 5). Even if they still do not solve algorithmically the problem for general PL n-manifolds (nor for general PL 4-manifolds), nevertheless we will prove that a fixed sequence of blobs and flips is sufficient to classify -via PL-homeomorphism -all PL 4-manifolds admitting a coloured triangulation with at most 18 4-simplices (see Section 5).
In order to define the class of gems involved in our catalogues, further preliminary notions are required.

Definition 4.
A pair (e, f ) of distinct i-coloured edges in an (n + 1)-coloured graph (Γ, γ) is said to form a ρ h -pair iff e and f both belong to exactly h common bicoloured cycles of Γ. Figure 2 shows the combinatorial move called ρ-pair switching.  The effect of ρ-pair switching on crystallizations is explained by the following result, where H denotes an n-dimensional handle, i.e. either the orientable or non-orientable S n−1 -bundle over S 1 (respectively denoted by S n−1 × S 1 and S n−1 ×S 1 ), according to the orientability of M n : Proposition 6 ([7]) Let (Γ, γ) be a crystallization of a PL n-manifold M n , n 3 and let (Γ ′ , γ ′ ) be obtained by switching a ρ h -pair in Γ. Then: Definition 5. An (n + 1)-coloured graph is said to be rigid (resp. rigid dipole-free) if it has no ρ h -pairs, with h ∈ {n − 1, n} (resp. if it is rigid and has no dipoles).
Catalogues of PL n-manifolds are obviously constructed with respect to increasing "complexity" of the representing combinatorial objects. Within crystallization theory, the following quite natural invariant is considered: Definition 6. Given a PL n-manifold M n , its gem-complexity is the non-negative integer k(M n ) = p − 1, where 2p is the minimum order of a crystallization of M n .
In order to generate an exhaustive catalogue of (n + 1)-coloured graphs representing all closed connected PL n-manifold up to a fixed gem-complexity, the restriction to the class of rigid dipole-free crystallizations yields no loss of generality, as Proposition 7 below proves.
In the following, # h M denotes the connected sum of h copies of a given n-manifold M .
Definition 7. A PL n-manifold M n is said to be handle-free if it admits no handles as connected summands.
Proposition 7 Let M n be a PL n-manifold (n ≥ 3). Then: (a) If M n is handle-free, then a rigid dipole-free order 2p crystallization (Γ, γ) of M n exists, so that k(M n ) = p − 1.
(b) Otherwise, a rigid dipole-free order 2p crystallization of a PL n-manifolds N n exists, so that Proof. Statement (a) directly follows from [7,Theorem 5.3]. Let now M n ∼ = P L N n # h H be any connected sum decomposition of M n , with h ≥ 0. 4 Since a standard order 2(n + 1) crystallization of S n−1 × S 1 (resp. S n−1 ×S 1 ) is wellknown (see [39]), and since the so called graph connected sum (see [33]) yields an order 2(p 1 + p 2 − 1) gem of N 1 #N 2 from any order 2p 1 (resp. 2p 2 ) gem of N 1 (resp. N 2 ), inequality k(M n ) ≤ p − 1 + n · h easily follows, 2p being the order of any crystallization of N n .

Hence, if we set
where the minimum is taken over all decompositions M n ∼ = P L N n # h H (with h ≥ 0) and over all rigid dipole-free crystallizations of N n .
In order to prove the reversed inequality (and hence statement (b)), let us consider an arbitrary crystallization (Γ,γ) of M n , with order 2p. If (Γ,γ) is a rigid dipole-free crystallization, thenp − 1 ∈ S (with respect to the trivial decomposition of M n ), and sop − 1 ≥ min S trivially holds. If (Γ,γ) contains ρ n−1 -pairs and/or or dipoles, a order 2p ′ rigid dipole-free crystallization (Γ ′ ,γ ′ ) of M n is easily obtained via a suitable number of ρ n−1 -switching, each one followed by a 1-dipole elimination, and/or dipole eliminations; hence,p ′ − 1 ∈ S (with respect to the trivial decomposition of M n ), and sop − 1 >p ′ − 1 ≥ min S trivially holds.
Finally, let us assume (Γ,γ) to admit no ρ n−1 -pairs, no dipoles and h ≥ 1 ρ n -pairs. After each ρ n -pair switching, n 1-dipoles appear, one for each colour not involved in the ρ-pair. Let (Γ ′′ ,γ ′′ ) be the order 2p ′′ graph obtained by all ρ n -switchings and elimination of the resulting n · h 1-dipoles; according to Proposition 6(b), Elementary notions of crystallization theory -and in particular the graph-connected sum quoted in the above proof -allow to easily extend to any dimension both the sphererecognition property and the finiteness property and the sub-additivity (with respect to connected sum) of gem-complexity, already stated in [15,Proposition 10] for the 3dimensional case.
Moreover, in analogy to the 3-dimensional case, it seems reasonable to conjecture the additivity of gem-complexity with respect to connected sum (which is also confirmed, in dimension 4, by the classifying results of the present paper): for any pair of PL n-manifolds.

4-dimensional generation algorithm
By Proposition 3(d), the generation of catalogues of crystallizations of n-manifolds with a fixed number of vertices 2p is essentially inductive on dimension and requires the prior generation and recognition of all gems (with 2p vertices) representing the (n − 1)-sphere. It is so easy to understand why the first results have been obtained in dimension three, since 2-spheres recognition can be performed easily by computing the Euler characteristic and null Euler characteristic characterizes closed 3-manifolds.
However, the generating algorithm, even in low dimension, becomes quickly very intensive as the number of vertices increases and requires large computing resources. A way to face this problem is to find combinatorial configurations in the graphs, which can be eliminated without changing the manifold. Examples of such configurations are dipoles and ρ-pairs. Proposition 7 assures that restricting the catalogues to rigid dipole-free crystallizations does not affect their completeness.
Let now fix our attention to the 4-dimensional case. For each p ≥ 1, we will denote by C (2p) (resp.C (2p) ) the catalogue of all not colour-isomorphic rigid dipole-free bipartite (resp. non-bipartite) crystallizations of 4-manifolds with 2p vertices.
S (2p) will be the starting set of the procedure generating C (2p) andC (2p) , which consists essentially in adding 4-coloured edges to all elements of S (2p) , so as to obtain crystallizations of 4-manifolds.
The set S (2p) is constructed by a suitable adaptation of the 3-dimensional generation algorithm; recognition of the 3-sphere is performed by cancelling dipoles and switching ρ-pairs in order to obtain a rigid crystallization and by comparing the resulting graph with the list of rigid crystallizations of S 3 , which appear in the 3-dimensional catalogue.
As a matter of fact, the recognition is very easy for p < 12, since the only rigid crystallization of S 3 up to this order is the standard one with two vertices.
So, the generating algorithm in dimension four runs as follows: 2) For each i = 1, 2, . . . , n p : 4-coloured edges in all possible ways so to produce 4-coloured graphs; -for each produced graph Γ, check absence of ρ-pairs and 2-dipoles; -for each c ∈ ∆ 3 , check that Γĉ represents S 3 .
3) Compute and compare the codes in order to exclude colour-isomorphic duplicates. 5 It is easy to see that in dimension three, rigidity and contractedness imply absence of dipoles. However, the above algorithm is practically useless due to the great computational time it requires; therefore it needs some modifications in order to be effective. More precisely, a branch and bound technique is used to prune the tree of possible attachments of edges on each element of S (2p) .
LetΓ be an 5-coloured graph obtained from an element of S (2p) by addition of r < p 4-coloured edges, thenΓ will be kept for further additions if and only if: (i) it contains no three edges with the same endpoints (otherwise there will be ρ-pairs in the final regular graph); (ii) for each i ∈ ∆ 3 ,Γî represents a 3-sphere with holes.
Unfortunately condition (ii) is very heavy to check, since it implies recognition of 3-spheres with holes. Instead, we use a weaker condition, which is equivalent to requirē Γî to be a 3-manifold (with boundary), i.e.
(ii ′ ) for each pair of colours i, j ∈ ∆ 3 , each 3-residue ofΓ not involving colours i, j must represent a 2-sphere, possibly with holes.
Condition (ii ′ ) can be checked by direct computation onΓ in the following way. Note thatΓ is a so-called 5-coloured graph with boundary, i.e.Γ4 is a 4-coloured graph. Then K(Γ), which is obtained exactly in the same way as in the closed case, is a pseudocomplex with non-empty boundary. The 3-simplices which triangulate ∂K(Γ) correspond bijectively to the vertices ofΓ missing the 4-coloured edge (boundary vertices).
Then, a suitable extension of the Euler characteristic computation of Proposition 3(c) implies that condition (ii') is equivalent to require the following equality to hold: where g kt is the number of {k, t}-coloured cycles ofΓ,p is the number of boundary vertices ofΓ and gîĵ (resp.ḡîĵ) is the number of (∆ 4 − {i, j})-residues ofΓ (resp. ∂Γ).
The above described restrictions succeed in reducing considerably both the computation time and the size of the resulting catalogues. Moreover, a parallelization strategy, which has been adopted in the implementation, has allowed to reduce further the computation time: see [42] for details.
As a consequence we could produce catalogues C (2p) andC (2p) for each p ≤ 10 (see Table 1 below).   Table 1 Remark 3 We point out that the unique rigid dipole-free crystallization of C (2) (resp. of C (8) ) is the standard crystallization of S 4 (resp. CP 2 : see [37]), while the unique nonbipartite rigid dipole-free crystallization appearing up to 20 vertices is the standard one of RP 4 with 16 vertices ( [38]).

Remark 4 A further restriction on the catalogues could be imposed
is a graph connected sum (see [33], or the Proof of Proposition 7) of two graphs Γ 1 and Γ 2 , then |K(Γ)| ∼ = P L |K(Γ 1 )|#|K(Γ 2 )| and we call Γ splittable. The problem of recognizing the manifold |K(Γ)| is thus traced back to the (easier) problem of recognizing |K(Γ 1 )| and |K(Γ 2 )|. However, the low number of splittable crystallizations (about 0.6% of the whole for the catalogues up to 18 vertices) makes of little advantage this restriction of the catalogues if compared with the possibility of recognizing any crystallization which is related to a splittable one by moves keeping the PL-homeomorphism of |K(Γ)| (see Section 5).

TOP classification via combinatorial invariants
Before describing the classification performed in our crystallization catalogues via suitable sequences of moves which realize the PL-homeomorphism of the represented PL 4-manifolds, we devote the present section to the much weaker problem of classifying the involved PL 4-manifolds within the TOP category.
The starting point is the direct computation of the Betti numbers and of the rank of the fundamental group for each PL 4-manifold represented by a crystallization of our catalogues. all the remaining ones represent (simply-connected) PL 4-manifolds M 4 with β 2 = 3.
Proof. In virtue of Proposition 3(e), an estimation of rk(π 1 (M 4 )), for each PL 4-manifold M 4 represented by our catalogues, may be obtained by computing the number g rst of 3-residues involving colours {r, s, t}, with 0 ≤ r < s < t ≤ 4, for each element of C (2p) , 1 ≤ p ≤ 10. The calculation has been done by means of a suitable procedure of the program DUKE III 6 and the program output ensures that each crystallization of C (2p) , 1 ≤ p ≤ 10, has g rst = 1 for at least a choice of distinct r, s, t ∈ ∆ 4 . Hence, the simply-connectedness of all involved orientable PL 4-manifolds is proved. In order to calculate the Betti numbers of the same PL 4-manifolds, it is necessary to apply Proposition 3(c), yielding the Euler characteristic of M 4 by a direct computation on each order 2p crystallization of M 4 : 7 Now, the simply-connectedness implies χ(M 4 ) = 2 + β 2 (M 4 ); hence, β 2 (M 4 ) follows by a direct computation of both the number of 3-residues and 2-residues, for each element of C (2p) , 1 ≤ p ≤ 10. The statement is proved by making use of suitable procedures of the program DUKE III. ✷ The fact that all orientable PL 4-manifolds represented by crystallizations of our catalogues are simply-connected has important consequences as regards their classification in TOP category.
In fact, the following result proves that, up to a significantly high gem-complexity, the classification of PL 4-manifolds up to TOP-homeomorphism is quite easy, at least in the simply-connected case (which -as a matter of fact -turns out to be the most frequent case): 8 6 "DUKE III: A program to handle edge-coloured graphs representing PL n-dimensional manifolds" is available on the Web: http://cdm.unimo.it/home/matematica/casali.mariarita/DUKEIII.htm 7 Recall that contractedness implies Γ to have exactly one 4-residue involving ∆ 4 − {i}, for each i ∈ ∆ 4 . 8 The statement of Proposition 9 was announced in [17] and in [20].

Proposition 9 Any simply-connected PL 4-manifold
is the second Betti number of M 4 .
Proof. Let Γ be an order 2p crystallization of M 4 . As already recalled, formula (2) yields the direct computation of the Euler characteristic of M 4 .
On the other hand, the planarity of each 3-residue of Γ yields (via Proposition 3(c), too) 2g ijk = g ij + g ik + g jk − p for each triple (i, j, k) ∈ ∆ 4 , from which the following relation is obtained: Hence, the Euler characteristic computation gives Now, if M 4 is assumed to be simply-connected, 6 + 3β 2 (M 4 ) = 15 + p − i<j<k g ijk follows; since g ijk ≥ 1 trivially holds, we have 3β may be stated, for each simply-connected PL 4-manifold M 4 . Now, the classical theorems of Freedman and Donaldson ( [34]) about the TOP classification of simply-connected closed 4-manifolds, together with more recent results by Furuta ([35]), ensure that intersection forms of type As a consequence of the above result, we can already deduce the complete TOP classification of all PL 4-manifolds represented in our cystallization catalogues, i.e. up to gem-complexity 9.
Proof. In virtue of Proposition 7, the gem-complexity of a closed connected nonorientable PL 4-manifold M 4 is congruent mod 4 to p − 1, 2p being the order of a rigid dipole-free crystallization (i.e. an element of C (2p) ∪C (2p) ). Moreover, the generation algorithm output (Table 1) and Proposition 8 ensure that all elements of 1≤p≤10 C (2p) ∪C (2p) , except the standard order 16 crystallization of RP 4 , represent simply-connected PL 4-manifolds; hence, no PL 4-manifold with handles appears. Now, the arguments are exactly the same as in the proof of Proposition 10, when taking into account also the results of the generation algorithm for the non-bipartite case. ✷ Another interesting consequence of the quoted results by Freedman, Donaldson and Furuta ([34], [35]) is related to the topological classification of simply-connected PL 4-manifolds with respect to the invariant regular genus.
First, recall that the genus of a bipartite 9 (n + 1)-coloured graph (Γ, γ) with respect to a cyclic permutation ε = (ε o , ε 1 , . . . , ε n−1 , ε n = n) of ∆ n is the genus ρ ε (Γ) of the surface F ε into which Γ regularly embeds (see [36] for details); moreover, ρ ε (Γ) may be directly computed by the following formula: Then, the regular genus of Γ is defined as ρ(Γ) = min ε {ρ ε (Γ)}, while the regular genus of an orientable PL n-manifold M n is defined as: With the above notations, the following statement holds: Proof. Let Γ be a crystallization of a PL 4-manifold M 4 , and let ρ ε (resp. ρε i ) be the regular genus of Γ (resp. Γε i ) with respect to a given cyclic permutation ε of ∆ 4 . By applying formula (4) both to Γ and to each 4-residue of Γ, and by making use of formula (2), too, the following relations easily follow (see, for example, formulae (3)-(13) in the proofs of Lemma 1 and Lemma 2 in [13], where they are given in the more general setting of crystallizations of bounded PL 4-manifolds): Since g ε i−1 ,ε i ,ε i+1 ≥ 1 trivially holds, inequality 2 i∈Z 4 ρε i ≤ 5ρ ε directly follows from (b). By substituting it into (c), with the additional hypothesis π 1 (M 4 ) = 0, we have means the integer part of x), from which β 2 (M 4 ) ≤ [ ρε 2 ] follows. As a consequence, relation (already obtained in [27,Proposition 2], too) is proved to hold. Now, the thesis directly follows from the fact that G(M 4 ) ≤ 43 implies β 2 (M 4 ) ≤ 21, and from the already quoted well-known results about the classification of simplyconnected closed 4-manifolds (exactly as in the proof of Proposition 9). ✷ As already pointed out, Proposition 9 and Proposition 12, together, prove Theorem 2 settled in Section 1, related to TOP classification of the PL 4-manifolds represented by crystallizations.
The (more interesting!) PL classification will be independently achieved in Section 6, by making use of an implementation of the classifying algorithm described in Section 5.

Classification algorithm and possible applications
In order to complete the PL classification of the manifolds appearing in our catalogues of crystallizations, we exploit in the n-dimensional setting an idea already applied in dimension three.
First of all, let us call admissible a sequence of combinatorial moves which transforms a rigid dipole-free crystallization of a PL n-manifold M , into a rigid dipole-free crystallization of a PL n-manifold M ′ such that Let now X be a list of rigid dipole-free crystallizations; for any given set S of admissible sequences, it is possible to subdivide X into equivalence classes with regard to S.
Note that no theoretical proof exists ensuring that |K(Γ)| ∼ = P L |K(Γ ′ )| implies cl S (Γ) = cl S (Γ ′ ) (as well as its generalization: Nevertheless, in dimension three, it has been proved the existence of a set of admissible moves which are sufficient to perform the topological (=PL) classification of all 3-manifolds admitting a coloured triangulation with at most 30 tetrahedra ( [19], [5]).
As we will see in the next section, the same turns out to be true, in the 4-dimensional setting, for all elements of 1≤p≤9 C (2p) , though with respect to a different set of moves.
In fact, the 3-dimensional classification algorithm employs dipole moves and ρ-pairs switchings (which are available in any dimension), together with generalized dipole moves ( [31]), which are defined only for n = 3.
Instead, in the n-dimensional setting we make use of a further set of moves, introduced by Lins and Mulazzani in [41].
Definition 8. Let Γ be a gem of a PL n-manifold M n . Then: • A blob is the insertion or cancellation of an n-dipole.  Flips and blobs on a gem do not change the represented manifold: [41,Proposition 3]. Actually, even if two crystallizations are known to represent the same manifold, there is no algorithmic procedure to determine a sequence of blobs and flips connecting them, nor an upper bound to the number of moves to be performed. However, in the next section we will show that such an algorithm exists for the crystallizations appearing in our catalogues and that only one blob is sufficient.
In order to define the set of admissible movesS which have been chosen to work on the catalogue 1≤p≤9 C (2p) , let us introduce some definitions and notations.
Given an order 2p (n + 1)−coloured graph Γ there is a natural ordering of its vertices induced by the rooted numbering algorithm generating its code (see [25]); so we can write If Γ is a rigid dipole-free crystallization of a PL n-manifold, given i ∈ N 2p = {1, . . . , 2p}, c ∈ ∆ n , an n-tuple x = (x 1 , . . . , x n ) with x i ∈ N 2p and a permutation τ ofĉ = ∆ n − {c}, we denote by θ i,c,x,τ (Γ) the rigid dipole-free crystallization obtained from Γ in the following way: -cancel dipoles and switch ρ-pairs in the resulting graph.
We denote byS the set of all sequences θ i,c,x,τ , where i ∈ N 2p , c ∈ ∆ n , x is an n-tuple of elements of N 2p and τ is a permutation ofĉ.
Remark 5 Note that the above definition ofS, as well as the classification algorithm itself, are independent from dimension. As a consequence, the partition into equivalence classes with respect toS can be performed on any list of crystallizations of n-manifolds, in order to prove their PL-equivalence. Furthermore, note that the PL manifold represented by an equivalence class is completely identified once at least one of the crystallizations of the class is "known"; hence the algorithm can be also effective for the PL recognition of the manifolds involved in the list.

Remark 6
It is not difficult to prove that ρ n -pairs cannot appear when the classification algorithm with respect toS is applied to a set of bipartite crystallizations representing simply-connected n-manifolds (see Proposition 6(b)). Hence, in this case, cl S (Γ) = cl S (Γ ′ ) surely implies |K(Γ)| ∼ = P L |K(Γ ′ )|.
In order to obtain PL classification results, the classification algorithm, with respect toS and for n = 4, has been implemented in a C++ program -called "Γ4-class".
As already pointed out in Section 1, program Γ4-class can be applied to attempt the proof of PL-equivalence between different (pseudo)-triangulations of the same topological 4-manifold; in fact, it is very easy to produce automatically a rigid dipole-free crystallization starting from any (pseudo)-triangulation (see Remark 1 and Proposition 7).
In particular, an application of Γ4-class to the case of the 16-vertices (resp. 17vertices) triangulation of the K3-surface (obtained in [29] and [46] respectively) is in progress. The idea is similar to the one described in [9], [10] and [11], but the elementary moves involved in the automatic procedures are different (blob and flips, together with dipole eliminations and ρ-pair switching, instead of edge-contraction and bistellar moves). Hence, it is possible that one sequence succeeds when the others fail, or viceversa, with equal computational time employed.

Classification results in PL=DIFF category
The application of the program Γ4-class to the catalogue 1≤p≤9 C (2p) yields the complete PL classification of the involved crystallizations (or, equivalently, of the dual contracted triangulations) as shown in the following proposition. 10 Proposition 14 There is a bijective correspondence between the partition obtained by the program Γ4-class and the set of PL 4-manifolds represented by 1≤p≤9 C (2p) ∪C (2p) . Moreover, the PL classification coincides with the TOP classification.
Proof. Since there is only one non-bipartite crystallization in all our catalogues, the statement is trivial for 1≤p≤9C (2p) .
Program Γ4-class applied to the set 1≤p≤9 C (2p) produces a partition into five classes, which coincides with the partition induced by the second Betti number. More precisely, all crystallizations with β 2 = 0 (resp. β 2 = 1) belong to the same class as the standard crystallization of S 4 (resp. CP 2 ), while the crystallizations with β 2 = 2 are subdivided into three classes, containing the standard crystallization of CP 2 #CP 2 , CP 2 #(−CP 2 ) and S 2 × S 2 respectively. ✷ The following proposition -whose statement already appeared in a partial and preliminary version in [17] and in [20] -summarizes the complete PL classification of orientable (resp. non-orientable) PL 4-manifolds having gem-complexity up to 8 (resp. up to 9).
Proof. The statements concerning PL 4-manifolds up to gem-complexity 8 are consequences of the previous Proposition 14, together with Proposition 7. The last statement, concerning gem-complexity 9, directly follows from Proposition 8. ✷ Note that the above Proposition 15 implies Theorem 1 (stated in Section 1), when the attention is restricted to the handle-free PL 4-manifolds.
As a consequence of the (partial) analysis of the 4-dimensional crystallization catalogue 1≤p≤10 C (2p) , together with a suitable application of the classification program Γ4-class, the following result may also be stated: Proposition 16 A rigid crystallization of S 4 exists, with 20 vertices (see Figure 5). Apart from the standard order-two crystallization, it is the only rigid dipole-free crystallization of S 4 up to 20 vertices.
As a consequence of our catalogues, we can state:
More generally, we hope that further developments in the generation and classification of 4-dimensional crystallization catalogues, for increasing gem-complexity, could be useful to face open problems concerning different PL-structures on the same TOP 4-manifold.