Classifying compact 4-manifolds via generalized regular genus and G-degree

(d+1)-colored graphs, i.e. edge-colored graphs that are (d+1)-regular, have already been proved to be a useful representation tool for compact PL d-manifolds, thus extending the theory (known as crystallization theory) originally developed for the closed case. In this context, combinatorially defined PL invariants play a relevant role. The present paper focuses in particular on generalized regular genus and G-degree: the first one extending to higher dimension the classical notion of Heegaard genus for 3-manifolds, the second one arising, within theoretical physics, from the theory of random tensors as an approach to quantum gravity in dimension greater than two. We establish several general results concerning the two invariants, in relation with invariants of the boundary and with the rank of the fundamental group, as well as their behaviour with respect to connected sums. We also compute both generalized regular genus and G-degree for interesting classes of compact d-manifolds, such as handlebodies, products of closed manifolds by the interval and D^2-bundles over S^2. The main results of the paper concern dimension 4, where we obtain the classification of all compact PL manifolds with generalized regular genus at most one, and of all compact PL manifolds with G-degree at most 18; moreover, in case of empty or connected boundary, the classifications are extended to generalized regular genus two and to G-degree 24.


Introduction
In the PL d-dimensional setting (d ≥ 3), both the invariants generalized regular genus and G-degree have been recently introduced, making use of the possibility of representing all compact PL d-manifolds by means of regular (d + 1)-colored graphs (i.e. graphs whose vertices have degree d + 1, and so that the d + 1 edges adjacent to each vertex are injectively colored by the colors {0, 1, . . . , d}): see [16] and [15] respectively, or the following Section 2.
This kind of representation for compact PL manifolds and the study of the above invariants has been deeply motivated by the strong connections -accurately described in [15] -between random tensor models and the so called crystallization theory, which is a useful combinatorial tool for the topological and geometrical study of PL manifolds of arbitrary dimension (assumed to be closed, in the "classical" version of the theory) via edge-colored graphs.
Without going into details, we only recall that colored tensor models were introduced within theoretical physics as a random geometry approach to quantum gravity, following the successes of random matrix models ( [25]) and, in this context, the coefficients of the 1/N expansion of the correlation functions in dimension d are generating functions of regular bipartite (d + 1)-colored graphs; moreover, the quantity driving the 1/N expansion is the Gurau degree, whose definition involves the genera of the surfaces where the considered graphs regularly embed (see Definition 2 for details), exactly as the strictly related notions of generalized regular genus and G-degree ( [3], [4], [40]). Hence, any result obtained about generalized regular genus and/or G-degree is not only an achievement in the comprehension and possible classification of manifolds in the PL category, but may also bring insights in the interactions between geometry and physics. 1 Other types of tensor models whose Feynman graphs include or are a particular kind of colored graphs have also been introduced (for instance the O(N) 3 -invariant, the multi-orientable or the colored SYK models; see [5], [6], [38], [51] and references therein) and involve the definition of suitable "degrees" on colored graphs whose topological significance with respect to the dual colored triangulations can be investigated, as done for example in [29].
As far as the d-dimensional setting is concerned, the present paper proves in Section 4 several general properties of generalized regular genus and G-degree, relating them to the analogue invariants for the boundary manifold (Proposition 9), or for the summands of a connected sum decomposition (Proposition 10), and establishing an inequality between generalized regular genus and the rank of the fundamental group, in case of manifolds with empty or connected boundary (Proposition 6). Moreover, in Section 5, standard graphs representing some interesting classes of PL d-manifolds are obtained, yielding the computation of the associated invariants: see Proposition 11 concerning handlebodies, and Proposition 12 concerning the product between a closed d-manifold and the interval. A similar approach is then performed in Section 6, in the 4-dimensional setting, as regards the D 2 -bundles over S 2 .
The focus of the paper is, indeed, on dimension d = 4, where crystallization theory already yielded classification results in the closed case both with respect to regular genus and with respect to gemcomplexity (see [12], [14] and their references). On the whole, experimental approaches to PL classification of 4-manifolds via combinatorial descriptions and associated PL-invariants are recent and still developing: as interesting examples we recall the use of Turaev's shadows and shadow complexity (see for instance [41], [44] for the closed case, and [48] for the case of compact acyclic 4-manifolds) and that of trisections and trisection genus (see [45], [46], [50]) 2 .
In the 4-dimensional setting, the present paper applies the general combinatorial properties of graphs representing compact d-manifold (obtained in Section 3), together with classical methods of crystallization theory and recent achievements about Dehn surgery, in order to yield classifying results for compact PL 4-manifolds M 4 with respect to both their generalized regular genusḠ (M 4 ) and their G-degree D G (M 4 ).
In particular, we prove in Section 7 the following statements (where S 1 × S 3 and S 1× S 3 denote the orientable and non-orientable sphere bundle over S 1 , Y 4 m andỸ 4 m denote the orientable and non-orientable 4-handlebody of genus m, ξ c denotes the D 2 -bundle over S 2 with Euler class c, while M 4 (K, d) denotes the compact PL 4-manifold obtained from the 4-disk by adding a 2-handle according to the framed knot (K, d), whose boundary is the 3-manifold M 3 (K, d) obtained from the 3-sphere by Dehn surgery on the same framed knot): 1 Note that, while the so-called colored tensor model is usually formulated in terms of complex tensors, a version involving real tensor fields, known as the Gurau-Witten model, has gained considerable interest in the study of quantum mechanical models with random interactions and their holographic properties ( [37], [52]). Its 1/N expansion involves both bipartite and non-bipartite graphs, and hence the investigation concerns both orientable and non-orientable manifolds.
2 Generalizing [50], a recent study also performs an approach to trisection genus via crystallizations, giving rise to the notion of gem-induced trisection genus, which applies also to manifolds with connected boundary: see [13].
b.Ḡ(M 4 ) = 1 if and only if whereM is a genus one closed 3-manifold; c. If M 4 has empty or connected boundary andḠ(M 4 ) = 2, then: Theorem 2 Let M 4 be a compact PL 4-manifold with no spherical boundary components. Then: No other compact PL 4-manifold (with no spherical boundary components) exists with D G (M 4 ) ≤ 23.
Moreover, if M 4 has empty or connected boundary, then: A direct consequence of Theorem 2 is the identification of all compact orientable PL 4-manifolds (resp. compact orientable PL 4-manifolds with empty or connected boundary), represented by regular graphs involved in the first four (resp. five) most significant non-null terms of the 1/N expansion of the correlation functions ( [3], [18]): see Remark 8. Further results of the present paper are also the characterization of 4-dimensional handlebodies as the only PL 4-manifolds with connected (non empty) boundary whose generalized regular genus equals that of their boundary (Theorem 3), while the equality between generalized regular genus and the rank of the fundamental group characterizes # ρ (S 1 × S 3 ) and # ρ (S 3 ×S 1 ) in the closed case, in the connected boundary case (Theorem 4). Note that, as a consequence of the above results, all D 2 -bundles over S 2 turn out to have generalized regular genus 2, thus proving that generalized regular genus is not finite-to-one in dimension four, and that, in the 4-dimensional case of non-empty boundary, the equality between generalized regular genus and the "classical" invariant regular genus does not hold, even if the boundary is assumed to be connected (Corollary 18).

Preliminaries
In the present section we will briefly review some basic notions of the so called crystallization theory, which is a representation tool for general piecewise linear (PL) compact manifolds, without assumptions about dimension, connectedness, orientability or boundary properties (see the "classical" survey paper [28], or the more recent one [14], concerning the 4-dimensional case).
From now on, unless otherwise stated, all spaces and maps will be considered in the PL category, and all manifolds will be assumed to be compact and connected.
) is a multigraph (i.e. multiple edges are allowed, but loops are forbidden) which is regular of degree d + 1, and γ is an edge-coloration, that is a map γ : E(Γ) → ∆ d = {0, . . . , d} which is injective on adjacent edges.
In the following, for sake of concision, when the coloration is clearly understood, we will denote colored graphs simply by Γ.
A d-dimensional pseudocomplex K(Γ) can be associated to a (d + 1)-colored graph Γ: • take a d-simplex for each vertex of Γ and label its vertices by the elements of ∆ d ; • if two vertices of Γ are c-adjacent (c ∈ ∆ d ), glue the corresponding d-simplices along their (d − 1)dimensional faces opposite to the c-labeled vertices, so that equally labeled vertices are identified.
In general |K(Γ)| is a d-pseudomanifold and Γ is said to represent it.
Note that, by construction, K(Γ) is endowed with a vertex-labeling by ∆ d that is injective on any simplex. Moreover, Γ turns out to be the 1-skeleton of the dual complex of K(Γ). The duality establishes a bijection between the {c 1 , . . . , c h }-residues of Γ and the (d − h)-simplices of K(Γ) whose vertices are labeled by ∆ d − {c 1 , . . . , c h }.
Given a pseudocomplex K and an h-simplex σ h of K, the disjoint star of σ h in K is the pseudocomplex obtained by taking all d-simplices of K having σ h as a face and identifying only their faces that do not contain σ h . The disjoint link, lkd(σ h , K), of σ h in K is the subcomplex of the disjoint star formed by those simplices that do not intersect σ h .
In particular, given a (d + 1)-colored graph Γ, each connected component of Γĉ (c ∈ ∆ d ) is a d-colored graph representing the disjoint link of a c-labeled vertex of K(Γ), that is also (PL) homeomorphic to the link of this vertex in the first barycentric subdivision of K(Γ). Note that, in case of polyhedra arising from colored graphs, the condition about links of vertices obviously implies the one about links of h-simplices, with h > 0. Therefore: • |K(Γ)| is a singular d-manifold iff, for each color c ∈ ∆ d , allĉ-residues of Γ represent closed connected (d − 1)-manifolds.
Remark 1 If N is a singular d-manifold, then a compact d-manifoldŇ is easily obtained by deleting small open neighbourhoods of its singular vertices. Obviously N =Ň iff N is a closed manifold; otherwise,Ň has non-empty boundary (without spherical components). Conversely, given a compact d-manifold M , a singular d-manifold M can be constructed by capping off each component of ∂M by a cone over it. Note that, by restricting ourselves to the class of compact d-manifolds with no spherical boundary components, the above correspondence is bijective and so singular d-manifolds and compact d-manifolds of this class can be associated to each other in a well-defined way.
For this reason, throughout the present work, we will restrict our attention to compact manifolds without spherical boundary components. Obviously, in this wider context, closed d-manifolds are characterized by M = M .
In virtue of the bijection described in Remark 1, a (d+1)-colored graph Γ is said to represent a compact d-manifold M with no spherical boundary components if and only if it represents the associated singular manifold M .
The following theorem extends to the boundary case a well-known result -originally stated in [49] founding the combinatorial representation theory for closed manifolds of arbitrary dimension via colored graphs.
If Γ represents a compact d-manifold, a d-residue of Γ will be called ordinary if it represents S d−1 , singular otherwise. Similarly, a color c will be called singular if at least one of theĉ-residues of Γ is singular.
The Gurau degree (often called degree in the tensor models literature, see [36]) and the regular genus of a colored graph are defined in terms of the embeddings of Proposition 2.
As a consequence of the definition of regular genus of a colored graph and of Proposition 1, two PL invariants for compact d-manifolds can be defined: and the Gurau degree (or G-degree) of M is defined as For any (d + 1)-colored graph Γ, the following inequality obviously holds: Hence, for any compact d-manifold M : Remark 2 Note that, in case M is a closed PL d-manifold, the generalized regular genus coincides by definition with the PL invariant regular genus (see Section 4), extending to higher dimension the Heegaard genus of a 3-manifold ( [23]). Regular genus zero succeeds in characterizing spheres in arbitrary dimension, 4 and many classifying results via regular genus have been obtained, especially in dimension 4 and 5 (see [2], [10], [14], [17] and their references). Also G-degree zero characterizes spheres in arbitrary dimension, and some classifying results via this invariant have recently been obtained for compact 3-manifolds and for closed PL 4-manifolds: see [15] and [16].
Finally, we recall that, within crystallization theory, a finite set of combinatorial moves have been defined, which translate the homeomorphism problem of the represented polyhedra.
The elimination of an r-dipole in Γ can be performed by deleting the subgraph and welding the remaining hanging edges according to their colors; in this way another (d + 1)-colored graph (Γ , γ ) is obtained. The inverse operation is called the addition of the dipole to Γ .
The dipole is called proper if |K(Γ)| and |K(Γ )| are PL homeomorphic. It is known that this happens when at least one of the two connected components of Γĉ 1 ...ĉr intersecting the dipole represents a (d − r)sphere ([32, Proposition 5.3]). 5

Remark 3
Neither the G-degree nor the regular genus of a (d+1)-colored graph are affected by elimination of 1-dipoles. Therefore, from any (d + 1)-colored graph Γ representing a compact PL d-manifold M with empty or connected boundary, by eliminating (proper) 1-dipoles, a (d + 1)-colored graph can be obtained, still representing M , with the same G-degree and regular genus as Γ and having only oneî-residue for each i ∈ ∆ d . Such a (d + 1)-colored graph is said to be a crystallization of M .

Combinatorial properties of graphs representing singular dmanifolds
In [7], [9] and [17], interesting combinatorial formulae have been obtained, regarding both regular edgecolored graphs representing closed d-manifolds and edge-colored graphs with boundary (see [28], or the next Section 4) representing d-manifolds with non-empty boundary. Here, we will generalize them to regular edge-colored graphs representing (via singular d-manifolds) all compact (PL) d-manifolds.
In the following, let (Γ, γ) be a (possibly disconnected) (d + 1)-colored graph representing a (possibly where by ρ ε (H i ) we denote the regular genus of H i with respect to the permutation induced by ε on the subset B of ∆ d ).
• if #B = m and m ≤ d − 1, m even: • if B = ∆ d − {i}, with i non-singular color and d odd: • if B = ∆ d − {i}, with i non-singular color and d even: Moreover: Proof. By definition of generalized regular genus with respect to the permutation ε: By applying the same relation to the (possibly disconnected) subgraph Γ ε i (i ∈ ∆ d ), we have: In order to prove relations (1) and (2), recall that each connected component of Γ B represents the disjoint link of a (d − m)-simplex in the singular d-manifold |K(Γ)| = N d , which -under the hypothesis m ≤ d − 1 -is homeomorphic to the (m − 1)-sphere. Hence, its Euler characteristic equals 2 if m is odd and 0 if m is even. The quoted formulae simply perform the computation of the Euler characteristic from the combinatorial features of the representing graph.
Proof. As a consequence of relations (6) and (13), we have: The first statement now easily follows. As regards the second one, it is sufficient to note that, in case B = {i − 1, i, i + 1, r}, Γ B−{i} represents a 2-dimensional sphere, and hence its regular genus is zero. 2

General properties of generalized regular genus
Within crystallization theory, two standard methods are known, in order to obtain a presentation of the fundamental group of a closed manifold directly from a graph representing it. The following extensions to compact manifolds and singular manifolds hold: Proposition 5 Let (Γ, γ) be a (d + 1)-colored graph representing the singular d-manifold N and the associated compact d-manifoldŇ .
• For each i, j ∈ ∆ d , let X ij (resp. R ij ) be a set in bijection with the connected components of Γî (resp. with the {i, j}-colored cycles of Γ), and letR ij be a subset of X ij corresponding to the a maximal tree of the subcomplex K ij of K(Γ) (consisting only of vertices labelled i and j, and edges connecting them). Then: • For each i ∈ ∆ d , let X i (resp. R i ) be a set in bijection with the i-colored edges of Γ (resp. with the {i, j}-colored cycles of Γ, for any j ∈ ∆ d − {i}) and letR i be a subset of X i corresponding to a minimal set of i-colored edges of Γ connecting Γî. Then: Proof. It is a direct consequence of some general results concerning the fundamental groups of pseudocomplexes associated to colored graphs: see [20]. 2 The following statement yields an interesting inequality between the generalized regular genus and the rank of the fundamental group, for any compact manifold with connected boundary. The analogous inequality for closed manifolds is well known: see [21,Proposition B].
Proposition 6 Let M be a compact d-manifold with empty or connected boundary. Then: Proof. Let (Γ, γ) be a (d + 1)-colored graph realizing the generalized regular genus of M , with respect to the permutation ε of ∆ d , i.e. ρ ε (Γ) =Ḡ (M ). Let i and j be two not singular colors that are not consecutive in the permutation ε: they certainly exist since M has empty or connected boundary and so Γ has at most one singular color. It is now sufficient to consider the presentation of the fundamental group of M given by Proposition 5(a), with respect to colors i and j and to recall that, in virtue of formulae (6) and (15), Let us now recall that another graph-based representation theory for compact (PL) manifolds exists, making use of colored graphs which fail to be regular. More precisely, any compact d-manifold can be represented by a pair (Λ, λ), where λ is still an edge-coloration on E(Λ) by means of ∆ d , but Λ may miss some (or even all) d-colored edges: such a pair is said to be a (d + 1)-colored graph with boundary, regular with respect to color d, and vertices missing the d-colored edge are called boundary vertices (see [28]).
An easy combinatorial procedure, called capping off, enables to connect this representation to the one -involving only regular colored graphs -considered in Section 2.

Proposition 7 ([27])
Let (Λ, λ) be a (d + 1)-colored graph with boundary, regular with respect to color d, representing the compact d-manifold M . Chosen a color c ∈ ∆ d−1 , let (Γ, γ) be the regular (d + 1)-colored graph obtained from Λ by capping off with respect to color c, i.e. by joining two boundary vertices by a d-colored edge, whenever they belong to the same {c, d}-colored path in Λ. Then, (Γ, γ) represents the singular d-manifold M , and hence M , too.
By means of (non-regular) edge-colored graphs with boundary, together with a suitable extension of Proposition 2, Gagliardi introduced within crystallization theory a "classical" notion of regular genus for compact d-manifolds, too (see [31] and [33]). The following result establishes a comparison between regular genus and generalized regular genus (as defined in Section 2: see Definitions 3 and 4) for any compact d-manifold. Proof. The general inequality is a consequence of the "capping off" procedure, recalled in Proposition 7.
In fact, let us assume the regular genus of M to be realized by the (not regular) graph with boundary Λ with respect to the cyclic permutation ε = (ε 0 , ε 1 , . . . , ε d−1 , ε d = d) of ∆ d . Then, it is not difficult to prove that, if c ∈ {ε 0 , ε d−1 } is chosen, and Γ is the (regular) (d + 1)-colored graph obtained from Λ by capping off with respect to color c, the generalized regular genus of Γ with respect to ε equals the regular genus of Λ with respect to the same permutation: ρ ε (Γ) = ρ ε (Λ) = G(M ). Equality (a) is trivial by definition (as already pointed out in Remark 2). Regarding statement (b), first note that, obviously, G(S d ) = G(S d ) = 0; moreover, the main theorem of [26] ensures that, if Γ represents a closed d-manifold M , ρ(Γ) = 0 implies M to be a PL d-sphere. In order to complete the proof of both co-implications, let us consider a (regular) (d + 1)-colored graph Γ such that there exists a cyclic permutation ε of ∆ d with ρ ε (Γ) = 0; we want to prove that |K(Γ)| is a closed d-manifold. If d = 2 then |K(Γ)| ∼ = S 2 , since ρ ε (Γ) trivially coincides with the genus of the surface |K(Γ)|. Suppose now our claim to be true in each dimension < d; given i ∈ Z d+1 , let Ξ be a connected component of Γ ε i , which is a d-colored graph. Since ρ ε (Ξ) ≤ ρ ε (Γ) (see inequality (15)) then, by induction, Ξ represents a PL (d − 1)-sphere and, therefore, |K(Γ)| is a closed PL d-manifold, and statement (b) is proved.
The same paper also presents examples of the strict inequality (d): if F is a closed surface of genus g, then G(F × I) = g < G(F × I) = 2g. 2 Remark 5 In Section 7 we will prove that the equality between the two invariants does not hold for 4-manifolds with boundary, even if the boundary is assumed to be connected: see Corollary 18(b).
As regards the invariant regular genus, a well-known relation (i.e. G(M ) ≥ G(∂M )) compares the regular genus of any compact manifold with the regular genus of its boundary; in the case of connected boundary, the following extensions hold, concerning both the generalized regular genus and the G-degree: Proposition 9 Let M be a compact d-manifold with (non-empty) connected boundary. Then: Proof. The first inequality is an easy consequence of (15), applied to a regular graph Γ representing M , so thatḠ(M ) = ρ ε (Γ) (ε being a cyclic permutation of ∆ d ) and having color i as its (only) singular color. The second inequality may be obtained in a similar way, by making use of the relation ω G (Γ) ≥ d · ω G (Γî), proved in [39, Lemma 4.6] for each (d + 1)-colored graph and for each color i ∈ ∆ d . 2 A d-dimensional extension of the construction described in [24, Proposition 5(i)] (resp. in [24, Proposition 5(ii)]), performed in [35,Section 7] in a general setting including graphs representing singular d-manifolds, allows to easily obtain graphs representing connected sums (resp. boundary connected sums) of compact PL d-manifolds directly from the graphs representing the summands.
Proposition 10 Let M 1 and M 2 be compact d-manifolds. Then: Proof. It is an easy consequence of the above described constructions: see the quoted papers, together with [15, Proposition 10]. 2

Proposition 11
For any d ≥ 4, a bipartite (resp. non-bipartite) (d + 1)-colored graph exists, with order 2d and regular genus one with respect to any permutation of ∆ d , representing the genus one d-dimensional handlebody Y d 1 (resp.Ỹ d 1 ). Hence, for each d ≥ 4 : Moreover, for each d ≥ 4 and for each m ≥ 1 : Proof. For any d ≥ 3, an order 2(d + 1) (d + 1)-colored graph with boundary (H, h) (resp. (H , h )) is well-known, which represents the genus one d-dimensional handlebody Y d 1 (resp.Ỹ d 1 ) : see [33]. By applying to (H, h) (resp. (H , h )) the "capping off" procedure described in Proposition 7, a (regular) order 2(d + 1) (d + 1)-colored graph representing Y d 1 (resp.Ỹ d 1 ) is obtained. It is easy to check that it admits a (proper) 2-dipole, whose elimination yields a (minimal) order 2d regular (d + 1)-colored graph (Ĥ,ĥ) (resp. (Ĥ ,ĥ )) representing Y d 1 (resp.Ỹ d 1 ): see Figure 1 for the orientable 4-dimensional case. A direct computation gives ρ ε (Ĥ) = 1 (resp. ρ ε (Ĥ ) = 1) for each permutation ε of ∆ d . Hence, the classification of compact PL d-manifolds with generalized regular genus zero (and with G-degree zero) easily allows to prove (16): see Proposition 8 (b). Now, it is not difficult to check that, for each m ≥ 1, the graph connected sum construction hinted to in the previous Section, with suitable choices of the vertices, enables to obtain a bipartite (resp. non-bipartite) (d + 1)-colored graph representing the genus m d-dimensional  (see Proposition 9), both equalities of (17) easily follow. 2 Remark 6 Note that, by a suitable application of the procedure of graph connected sum, it is easy to obtain also a (bipartite or non-bipartite) (d + 1)-colored graph representing the connected sum of m (m ≥ 2) (orientable or non-orientable) d-dimensional handlebodies. IfỸ d r denotes either the orientable or non-orientable genus r d-dimensional handlebody, then the graph representingỸ d r 1 # · · · #Ỹ d rm has order 2d(r 1 + · · · + r m ) (since the procedure has to be preceded by the insertion of m d-dipoles, in order to obtain m ordinary d-residues) and its regular genus is r 1 + · · · + r m with respect to any permutation of ∆ d . See Figure 3 for an example, in case d = 4, m = 2 and r 1 = r 2 = 1.

Moreover
: Proof. Let ε = (ε 0 , ε 1 , . . . , ε d ) be the cyclic permutation of ∆ d so that ρ(Λ) = ρ ε (Λ). If (Λ,λ) is obtained from Λ by adding a (d + 1)-colored edge between any pair of ε 0 -adjacent vertices, then it is easy to check thatΛ represents M × I and ρ ε (Λ) = ρ(Λ), where ε = (ε 0 , ε 1 , . . . , ε d , d + 1) : see [27], and d + d 2 (d − 1)p − r,s∈∆ d g rs (proved in [15] for any (d + 1)-colored graph) to (Λ,λ) yields (in virtue of the combinatorial structure ofΛ): Moreover, by making use of the similar computation for ω G (Λ), yielding r,s∈∆ d g εrεs = d + d , we obtain: which the last formula of the statement follows from.  graphs have regular genus equal to three. This allows to prove -by means also of some theoretical results about the "gap" between regular genus and the rank of the fundamental group of a PL 4-manifold with boundary -that G(ξ c ) = G(S 2 × D 2 ) = 3. The regular 5-colored graphs obtained from the above graphs by means of the "capping off" procedure described in Proposition 7 (which represent the singular 4-manifoldsξ c and S 2 × D 2 , and hence the compact 4-manifolds ξ c and S 2 × D 2 , too) have regular genus three by construction, have the same order as the starting graphs with boundary (i.e. 4c + 6 for ξ c , ∀c ∈ Z + − {1}, and 14 for S 2 × D 2 ), but admit a (proper) 2-dipole involving colors non-consecutive in the permutation ε realizing the minimum generalized regular genus, together with two 2-dipoles involving colors consecutive in the permutation ε. Now, it is easy to check, via Proposition 2 and Definition 5, that the elimination of a 2-dipole involving colors non-consecutive (resp. consecutive) in the permutation ε decreases by one (resp. does not affect) the regular genus with respect to ε. Hence, the elimination of the three 2-dipoles yields a regular 5-colored graph (Λ c , λ c ), ∀c ∈ Z + − {1} (resp. (Λ 0 , λ 0 )) representing ξ c (resp. S 2 × D 2 ) with the same order 4c (resp. 8) as the standard crystallization of L(c, 1) (resp. of S 1 × S 2 ): see Figure 5 (resp. Figure 6).
As a consequence, we have: Actually, in the following Corollary 17, we will prove that all compact 4-manifolds of this infinite class turn out to have generalized regular genus equal to two.
Hence:  7 Classifying results in dimension 4

Classifying with respect to generalized regular genus
In the 4-dimensional setting, formula (10) enables to prove the following useful results.
Both point (a) and point (b) of the statement are nothing but particular cases of the general statement; indeed, as regards point (b), the second part of Proposition 4 yields, for d = 4:

Remark 7
Note that point (a) of the above proposition could be independently proved simply by noting that relation (22) yields which ensures -via Lemma 5 of [7] -that K(i − 2, i − 1, i + 1) collapses to a graph, i.e. N (i − 2, i − 1, i + 1) is a handlebody. Since also N (i, i + 2) is obviously a handlebody, statement (a) follows via a well-known theorem by Montesinos and Laudenbach-Poenaru (see [47] and [43]). Also point (b) could be independently proved by noting that, by formula (21) We are now able to classify all compact 4-manifolds with generalized regular genus one.
Proof. Three cases occur: • M 4 is a closed 4-manifold; • M 4 is a compact 4-manifold with (non-empty) connected boundary; • M 4 is a compact 4-manifold with disconnected boundary.
Thus M 4 ∼ =M × I, whereM is a genus one closed 3-manifold (homeomorphic to both lkd(v c ) and lkd(v c+1 )), now easily follows. 2 Corollary 17 Let ξ c be the D 2 -bundle over S 2 with Euler class c, ∀c ∈ Z + − {1}. Then, On the other hand, in Section 6 (resp. in Section 5), we have obtained 5-colored graphs with generalized regular genus two representing S 2 × D 2 and ξ c , ∀c ∈ Z + − {1} (resp. representing The results about non-finiteness-to-one of generalized regular genus (already pointed out in Section 1 and in Remark 5) now easily follow: Hence, the only remaining cases concern simply-connected 4-manifolds M 4 (K, d) having lens spaces L(α, β), with α ≥ 3, as boundary. 2 We are now able to prove the theorem, already stated in Section 1, that summarizes the obtained classification results for compact 4-manifolds according to their generalized regular genus.
Proof. (Theorem 1) Statement (a) is nothing but the case d = 4 of Proposition 8(b). Statement (b) is a direct consequence of Propositions 11, 12 and 16, together with the well-known existence of 5-colored graphs of regular genus one representing the two S 3 -bundles over S 1 .
With regard to statement (c), the result comes directly from Proposition 19, since for each c ∈ Z, ξ c , the D 2 -bundle over S 2 with Euler class c, is exactly M 4 (K 0 , c), (K 0 , c) being the c-framed trivial knot. Proof. Let c be the singular color of Γ and let ε be the cyclic permutation of ∆ 4 such that ρ(Γ) = ρ ε (Γ) = 2; further, let us assume -without loss of generality -ε = (0, 1, 2, 3, 4) and gî = 1 for each i ∈ ∆ 4 − {c} (see Remark 3). It is easy to check that, if K(Γ) has more than one singular vertex, Γ may be assumed to have exactly twoĉ-residues, both with regular genus one with respect to the permutation induced by ε. Hence, ρĉ = 2. Arguments similar to those used in the proof of Proposition 15(b) ensure that K(c, c + 2) consists of two edges, with a common end-point (i.e. the (c + 2)-labelled vertex) and with the other end-points consisting in the two singular c-labelled vertices of K(Γ), v c and v c , say. This easily implies that N (c, c+2) is homeomorphic to the boundary connected sum of a 4-disk, the cone v c * lkd(v c ) and the cone v c * lkd(v c ); hence, the boundary of N (c, c + 2) is lkd(v c )#lkd(v c ).
Since the boundaries of N (c, c + 2) and N (c − 2, c − 1, c + 1) have to be identified, then m = 2 and both lkd(v c ) and lkd(v c ) must be homeomorphic to an orientable or non-orientable sphere bundle over S 1 .
In particular, if Γ representsM × I, the hypothesis gî = 1 (∀i ∈ ∆ 4 ) implies the existence of two colors c 1 , c 2 ∈ ∆ 4 so that both the 4-residues Γ c 1 and Γ c 2 of Γ representM . Then, for each i ∈ {1, 2},  No other 5-colored graph representing a compact 4-manifold exists with ω G (Γ) ≤ 23. Moreover, if (Γ, γ) has one singular color at most: d. if ω G (Γ) = 24, then above formula are the ones corresponding to even (integer) powers of 1/N. Hence, for d = 4, Proposition 23 yields the identification of all compact orientable PL 4-manifolds (resp. compact orientable PL 4-manifolds with empty or connected boundary), represented by regular graphs involved in the first four (resp. five) most significant non-null terms of the 1/N expansion of formula (28).
Proposition 23 allows us to prove the second main result of the present paper (already stated in Section 1).
Then it is easy to check that the 5-colored graphs obtained from the minimal (order eight) crystallizations of L(2, 1), S 1 × S 2 and S 1× S 2 by applying the procedure described in Proposition 12 have G-degree equal to 18 and represent L(2, 1) × I, (S 1 × S 2 ) × I and (S 1× S 2 ) × I respectively.