# Gem-induced trisections of compact PL $4$-manifolds

Maria Rita Casali  Department of Physics, Mathematics and Computer Science,
University of Modena and Reggio Emilia Via Campi 213 B, I-41125 Modena
(Italy<EMAIL_ADDRESS>Paola Cristofori  Department of Physics,
Mathematics and Computer Science, University of Modena and Reggio Emilia, Via
Campi 213 B, I-41125 Modena, Italy<EMAIL_ADDRESS>
###### Abstract

The idea of studying trisections of closed smooth $4$-manifolds via (singular)
triangulations, endowed with a suitable vertex-labelling by three colors, is
due to Bell, Hass, Rubinstein and Tillmann, and has been applied by Spreer and
Tillmann to colored triangulations associated to the so called simple
crystallizations of standard simply-connected $4$-manifolds. The present paper
performs a generalization of these ideas along two different directions:
first, we take in consideration also compact PL $4$-manifolds with connected
boundary, introducing a possible extension of trisections to the boundary
case; then, we analyze the trisections induced not only by simple
crystallizations, but by any 5-colored graph encoding a simply-connected
$4$-manifold. This extended notion is referred to as gem-induced trisection,
and gives rise to the G-trisection genus, generalizing the well-known
trisection genus. Both in the closed and boundary case, we give conditions on
a 5-colored graph which ensure one of its gem-induced trisections - if any -
to realize the G-trisection genus, and prove how to determine it directly from
the graph itself.
Moreover, the existence of gem-induced trisections and an estimation of the
G-trisection genus via surgery description is obtained, for each compact
simply-connected PL 4-manifold admitting a handle decomposition lacking in
1-handles and 3-handles. As a consequence, we prove that the G-trisection
genus equals $1$ for all $\mathbb{D}^{2}$-bundles of $\mathbb{S}^{2}$, and
hence it is not finite-to-one.

Keywords: trisections, 4-manifolds, triangulations, crystallizations,
tricolorings.

2000 Mathematics Subject Classification: 57Q15 - 57N13 - 57M15.

##  1 Introduction

The notion of trisection for any smooth, oriented closed $4$-manifold was
introduced in 2016 by Gay and Kirby ([31]), by generalizing the classical idea
of Heegaard splitting in dimension 3: via Morse functions, the $4$-manifold is
decomposed into three $4$-dimensional handlebodies, with disjoint interiors
and mutually intersecting in $3$-dimensional handlebodies, so that the
intersection of all three “pieces” is a closed orientable surface. The minimum
genus of the intersecting surface is called the trisection genus of the
$4$-manifold.

In the last years, the notion of trisection has been extensively studied, also
in its relationships with other manifold representation tools, such as handle-
decompositions, Lefschetz fibrations, framed links and Kirby diagrams: see,
for example, [34], [24], [6], [38], [36] and [25]. In particular, relying on
the coincidence between DIFF and PL categories in dimension $4$, Bell, Hass,
Rubinstein and Tillmann faced the study of unbalanced trisections via
(singular) triangulations, by making use of a vertex-labelling by three colors
(a tricoloring, which may be a c-tricoloring under suitable connectivity
assumptions, or better a ts-tricoloring in case of actually encoding a
trisection): see [6] for details. Obviously, this approach brings all the
advantages of a combinatorial description, and allows to algorithmically
construct trisections and estimate their complexity, starting from singular
triangulations.

In [38] the same idea has been applied to the special case of colored
triangulations associated to the so called simple crystallizations of simply-
connected PL $4$-manifolds (which turn out to be always c-tricolorings), thus
succeeding in computing the trisection genus of all standard simply-connected
$4$-manifolds (i.e. the connected sums of $\mathbb{CP}^{2}$ \- possibly with
reversed orientation -, $\mathbb{S}^{2}\times\mathbb{S}^{2}$,
$\mathbb{S}^{2}\tilde{\times}\mathbb{S}^{2}$ and the $K3$-surface).

The present paper performs a generalization of the ideas of [6] and [38],
along two different directions: first, we take in consideration also compact
PL $4$-manifolds with connected boundary, introducing a possible extension of
the notion of trisection to the boundary case (restricted to compact
$4$-manifolds whose associated singular manifold is simply-connected, and
different from the one studied in [23]); then, we prove that not only simple
crystallizations, but all edge-colored graphs encoding simply-connected
$4$-manifolds give rise to a c-tricoloring, which always satisfies “almost
all” conditions to be actually a ts-tricoloring, and hence to induce a
trisection.

Note that, in order to study exotic structures, it is of fundamental
importance to go beyond the class of simple crystallizations: in fact, as
pointed out in [16], there exist infinitely many simply-connected PL
$4$-manifolds, TOP-homeomorphic to the same standard simply-connected PL
$4$-manifold, which do not admit simple crystallizations.

Due to the features of generality of crystallization theory (i.e. the
representation tool for PL manifolds of arbitrary dimension, with or without
boundary, via edge-colored graphs) and related PL invariants (mainly the
regular genus, which extends to arbitrary dimension the classical genus of a
surface and the Heegaard genus of closed 3-manifolds), the approach performed
in the present paper appears to be open to further developments, including a
possible $n$-dimensional extension of the concept of trisection, based on a
suitable partition of the vertices of the colored triangulation associated to
a colored graph, for $n\geq 4$ (see [36] for a similar generalization to
higher dimension).

The basic result connecting 4-dimensional crystallization theory with
trisections is the following:

###### Theorem 1

Let $M^{4}$ be a compact PL $4$-manifold with empty (resp. connected)
boundary. For each 5-colored graph $(\Gamma,\gamma)$ representing $M^{4}$, a
triple $\mathcal{T}(\Gamma,\varepsilon)=(H_{0},H_{1},H_{2})$ of
$4$-dimensional submanifolds of $M^{4}$ is obtained,
$\varepsilon=(\varepsilon_{0},\dots,\varepsilon_{4})$ being a suitable
permutation of the color set, so that:

  * (i)

$M^{4}=H_{0}\cup H_{1}\cup H_{2}$ and the interiors of $H_{0},$ $H_{1},$
$H_{2}$ are pairwise disjoint;

  * (ii)

$H_{1},H_{2}$ are $4$-dimensional handlebodies; $H_{0}$ is a $4$-disk (resp.
is PL-homeomorphic to $\partial M^{4}\times[0,1]);$

  * (iii)

$H_{01}=H_{0}\cap H_{1}$ and $H_{02}=H_{0}\cap H_{2}$ are $3$-dimensional
handlebodies;

  * (iv)

$\Sigma(\mathcal{T}(\Gamma,\varepsilon))=H_{0}\cap H_{1}\cap H_{2}$ is a
closed connected surface (which is called _central surface_), whose genus
coincides with the regular genus, with respect to $\varepsilon,$ of the
subgraph $\Gamma_{\hat{\varepsilon_{4}}}$, obtained from $\Gamma$ by deleting
all $\varepsilon_{4}$-colored edges.

The triple $\mathcal{T}(\Gamma,\varepsilon)$ fits a weak notion of trisection,
that we call quasi-trisection: see Definition 7.

In case $H_{12}=H_{1}\cap H_{2}$ is a $3$-dimensional handlebody, too, then
$\mathcal{T}(\Gamma,\varepsilon)=(H_{0},H_{1},H_{2})$ is called a
B-trisection, or a gem-induced trisection of $M^{4}$ (see Definition 8 and
Definition 9 for details).

As shown in Section 5, it is not difficult to check that, if $M^{4}$ admits a
B-trisection, then the associated singular manifold $\widehat{M^{4}}$,
obtained by capping off the connected boundary of $M^{4}$ by a cone over it,
is simply-connected. Moreover, the letter “B” in B-trisections indicates the
fact that, if $M^{4}$ is a closed 4-manifold, a B-trisection is actually a
trisection of $M^{4}$ (according to the definition by Gay and Kirby), with the
property that one of the three involved 4-dimensional handlebodies is a
4-Ball, while in the boundary-connected case one of the 4-dimensional
handlebodies is replaced by a collar of the Boundary. Hence, the notion of
B-trisection may be considered as an extension of trisections also to compact
PL $4$-manifolds with connected boundary (whose associated singular manifold
is simply-connected), via an approach similar to the one suggested by
Rubinstein and Tillmann in [36, Subsection 5.2], taking inspiration from an
idea of Birman ([7]).

Through gem-induced trisections, an estimation of trisection genus may be
obviously performed in the closed simply-connected case. In fact, the
following extended notion arises quite naturally:

###### Definition 1

Let $M^{4}$ be a compact PL $4$-manifold with empty or connected boundary,
which admits at least a gem-induced trisection. The gem-induced trisection
genus \- or G-trisection genus for short - $g_{GT}(M^{4})$ of $M^{4}$ is the
minimum genus of the central surface of any gem-induced trisection
$\mathcal{T}(\Gamma,\varepsilon)$ of $M^{4}$:

$g_{GT}(M^{4})\ =\ \min\\{genus(\Sigma(\mathcal{T}(\Gamma,\varepsilon)))\ /\
\mathcal{T}(\Gamma,\varepsilon)\ \text{B-trisection of}\ M^{4}\\}.$

As pointed out, gem-induced trisections have already allowed the computation
of the trisection genus of all standard simply-connected PL 4-manifolds (see
[38]); in the present paper, we give conditions on a 5-colored graph
representing a closed simply-connected PL 4-manifold $M^{4}$ (resp. a compact
PL 4-manifold $M^{4}$ with connected boundary so that $\widehat{M^{4}}$ is
simply connected) which ensure one of its gem-induced trisections - if any -
to realize the trisection genus (resp. the G-trisection genus) of $M^{4}$, and
prove how to determine it directly from the given graph: see Proposition 18
and Corollary 19. Meanwhile, we detect a wide class of compact PL
$4$-manifolds (containing all standard simply-connected ones) for which the
G-trisection genus and the second Betti number coincide: see Corollary 19.

In particular, we obtain (Corollary 17) that the genus $h$ $4$-dimensional
handlebody ($\forall h\geq 1$) has G-trisection genus $h$, while all
$\mathbb{D}^{2}$-bundles over $\mathbb{S}^{2}$ have G-trisection $1$, thus
proving that the G-trisection genus is not finite-to-one in the boundary case.
On the other hand, it is an open problem whether $g_{GT}$ is finite-to-one or
not, if a fixed (possibly empty) boundary is assumed. Note that this problem
is related to Question 4.2 of [34], regarding the possible existence of
infinitely many closed 4-manifolds admitting minimal $(g;0)-$trisections, for
some $g\geq 3.$

Moreover, the existence of B-trisections and an estimation of the G-trisection
genus via surgery description is obtained, for each compact simply-connected
PL 4-manifold admitting a special handlebody decomposition, i.e. a handle
decomposition lacking in 1-handles and 3-handles (see [33, Section 3.3]).

As it is well-known, any such $4$-manifold may be obtained by adding to the
$4$-disk $2$-handles according to a framed link $(L,c)$
($c=(c_{1},c_{2},\dots,c_{l})$, $c_{i}\in\mathbb{Z}$ $\forall
i\in\\{1,\dots,l\\}$, $l$ being the number of components of the link $L$), and
by adding a $4$-handle, in case the resulting $4$-manifold has spherical
boundary (i.e.: in case the 3-manifold obtained by Dehn surgery on the same
framed link is $\mathbb{S}^{3}$). For this reason, we will denote it by
$M^{4}(L,c)$, both in the closed and boundary case.

###### Theorem 2

Given the framed link $(L,c)$, let $m_{\alpha}$ be the number of
$\alpha$-colored regions in a chess-board coloration (by colors $\alpha$ and
$\beta$) of a planar diagram $\mathcal{D}$ of $(L,c)$. Then, a colored graph
representing the associated PL $4$-manifold $M^{4}(L,c)$, with regular genus
$m_{\alpha}+l$, may be directly constructed from $\mathcal{D}$; moreover, it
gives rise to a gem-induced trisection of $M^{4}(L,c)$ whose central surface
has genus $m_{\alpha}$.

As a consequence, if $M^{4}$ admits a special handlebody decomposition, then:

$g_{GT}(M^{4})\leq\min\\{m_{\alpha}(L,c)\ /\ M^{4}(L,c)=M^{4}\\}.$

Note that the class of compact PL 4-manifolds involved in the above Theorem is
very large, and it is an open question whether it contains all closed simply-
connected ones or not. In fact, in the closed case, the existence of a special
handlebody decomposition is related to Kirby problem n. 50, and is of
particular interest with regard to exotic PL 4-manifolds: see, for example,
[1] and [2].

##  2 Colored graphs encoding manifolds

The present section is devoted to 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 [15], 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, connected
and orientable.

###### Definition 2

An $(n+1)$-colored graph ($n\geq 2$) is a pair $(\Gamma,\gamma)$, where
$\Gamma=(V(\Gamma),E(\Gamma))$ is a multigraph (i.e. it can contain multiple
edges, but no loops) which is regular of degree $n+1$, and $\gamma$ is an
edge-coloration, that is a map
$\gamma:E(\Gamma)\rightarrow\Delta_{n}=\\{0,\ldots,n\\}$ 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 $\Gamma$.

For every $\\{c_{1},\dots,c_{h}\\}\subseteq\Delta_{n}$ let
$\Gamma_{\\{c_{1},\dots,c_{h}\\}}$ be the subgraph obtained from
$(\Gamma,\gamma)$ by deleting all the edges that are not colored by the
elements of $\\{c_{1},\dots,c_{h}\\}$. In this setting, the complementary set
of $\\{c\\}$ (resp. $\\{c_{1},\dots,c_{h}\\}$) in $\Delta_{n}$ will be denoted
by $\hat{c}$ (resp. $\hat{c}_{1}\cdots\hat{c}_{h}$). The connected components
of $\Gamma_{\\{c_{1},\dots,c_{h}\\}}$ are called
$\\{c_{1},\dots,c_{h}\\}$-residues or $h$-residues of $\Gamma$; their number
is denoted by $g_{\\{c_{1},\dots,c_{h}\\}}$ (or, for short, by
$g_{c_{1},c_{2}}$, $g_{c_{1},c_{2},c_{3}}$ and $g_{\hat{c}}$ if $h=2,$ $h=3$
and $h=n$ respectively).

An $n$-dimensional pseudocomplex $K(\Gamma)$ can be associated to an
$(n+1)$-colored graph $\Gamma$:

  * •

take an $n$-simplex for each vertex of $\Gamma$ and label its vertices by the
elements of $\Delta_{n}$;

  * •

if two vertices of $\Gamma$ are $c$-adjacent ($c\in\Delta_{n}$), glue the
corresponding $n$-simplices along their $(n-1)$-dimensional faces opposite to
the $c$-labeled vertices, so that equally labeled vertices are identified.

In general $|K(\Gamma)|$ is an $n$-pseudomanifold and $\Gamma$ is said to
represent it.

Note that, by construction, $K(\Gamma)$ is endowed with a vertex-labeling by
$\Delta_{n}$ that is injective on any simplex. Moreover, $\Gamma$ turns out to
be the 1-skeleton of the dual complex of $K(\Gamma)$. The duality establishes
a bijection between the $\\{c_{1},\dots,c_{h}\\}$-residues of $\Gamma$ and the
$(n-h)$-simplices of $K(\Gamma)$ whose vertices are labeled by
$\Delta_{n}-\\{c_{1},\dots,c_{h}\\}$.

Given a pseudocomplex $K$ and an $h$-simplex $\sigma^{h}$ of $K$, the disjoint
star of $\sigma^{h}$ in $K$ is the pseudocomplex obtained by taking all
$n$-simplices of $K$ having $\sigma^{h}$ as a face and identifying only their
faces that do not contain $\sigma^{h}.$ The disjoint link,
$lkd(\sigma^{h},K)$, of $\sigma^{h}$ in $K$ is the subcomplex of the disjoint
star formed by those simplices that do not intersect $\sigma^{h}.$

In particular, given an $(n+1)$-colored graph $\Gamma$, each connected
component of $\Gamma_{\hat{c}}$ ($c\in\Delta_{n}$) is an $n$-colored graph
representing the disjoint link of a $c$-labeled vertex of $K(\Gamma)$, that is
also (PL) homeomorphic to the link of this vertex in the first barycentric
subdivision of $K.$

Therefore:

$|K(\Gamma)|$ is a closed $n$-manifold iff, for each color $c\in\Delta_{n}$,

all $\hat{c}$-residues of $\Gamma$ represent the $(n-1)$-sphere,

while

$|K(\Gamma)|$ is a singular111A singular (PL) $n$-manifold is a closed
connected $n$-dimensional polyhedron admitting a simplicial triangulation
where the links of vertices are closed connected $(n-1)$-manifolds. The notion
extends also to polyhedra associated to colored graphs, provided that disjoint
links of vertices are considered, instead of links. $n$-manifold iff, for each
color $c\in\Delta_{n}$,

all $\hat{c}$-residues of $\Gamma$ represent closed connected
$(n-1)$-manifolds.

###### Remark 1

If $N$ is a singular $n$-manifold, then a compact $n$-manifold $\check{N}$ is
easily obtained by deleting small open neighbourhoods of its singular
vertices. Obviously $N=\check{N}$ iff $N$ is a closed manifold, otherwise
$\check{N}$ has non-empty boundary (without spherical components). Conversely,
given a compact $n$-manifold $M$, a singular $n$-manifold $\widehat{M}$ can be
constructed by capping off each component of $\partial M$ by a cone over it.
Note that, by restricting ourselves to the class of compact $n$-manifolds with
no spherical boundary components, the above correspondence is bijective and so
singular $n$-manifolds and compact $n$-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 $n$-manifolds are characterized by $M=\widehat{M}.$

In virtue of the bijection described in Remark 1, an $(n+1)$-colored graph
$\Gamma$ is said to represent a compact $n$-manifold $M$ with no spherical
boundary components (or, equivalently, to be a gem of $M$, where gem means
Graph Encoding Manifold) if and only if it represents the associated singular
manifold $\widehat{M}$.

The following theorem extends to the boundary case a well-known result -
originally stated in [35] \- founding the combinatorial representation theory
for closed manifolds of arbitrary dimension via colored graphs.

###### Theorem 3

([18]) Any compact orientable (resp. non orientable) $n$-manifold with no
spherical boundary components admits a bipartite (resp. non-bipartite)
$(n+1)$-colored graph representing it.

If $\Gamma$ is a gem of a compact $n$-manifold, an $n$-residue of $\Gamma$
will be called ordinary if it represents $\mathbb{S}^{n-1}$, singular
otherwise. Similarly, a color $c$ will be called singular if at least one of
the $\hat{c}$-residues of $\Gamma$ is singular.

The existence of a particular type of embedding of colored graphs into
surfaces, is the key result in order to define the important notion of regular
genus. Since this paper concerns only orientable manifolds, we restrict the
statement only to the bipartite case, although a similar result holds also for
non-bipartite graphs.

###### Proposition 4

([29]) Let $\Gamma$ be a bipartite $(n+1)$-colored graph of order $2p$. Then
for each cyclic permutation
$\varepsilon=(\varepsilon_{0},\ldots,\varepsilon_{n})$ of $\Delta_{n}$, up to
inverse, there exists a cellular embedding, called _regular_ , of $\Gamma$
into an orientable closed surface $F_{\varepsilon}(\Gamma)$ whose regions are
bounded by the images of the $\\{\varepsilon_{j},\varepsilon_{j+1}\\}$-colored
cycles, for each $j\in\mathbb{Z}_{n+1}$. Moreover, the genus
$\rho_{\varepsilon}(\Gamma)$ of $F_{\varepsilon}(\Gamma)$ satisfies

$2-2\rho_{\varepsilon}(\Gamma)=\sum_{j\in\mathbb{Z}_{n+1}}g_{\varepsilon_{j},\varepsilon_{j+1}}+(1-n)p.$

###### Definition 3

The regular genus of a $(d+1)$-colored graph $\Gamma$ is defined as

$\rho(\Gamma)=min\\{\rho_{\varepsilon}(\Gamma)\ |\ \varepsilon\ \text{cyclic
permutation of \ }\Delta_{n}\\};$

the (generalized) regular genus of a compact $n$-manifold $M$ is defined as

$\mathcal{G}(M)=min\\{\rho(\Gamma)\ |\ \Gamma\ \text{gem of \ }M\\}.$

###### Remark 2

Note that the (generalized) regular genus is a PL invariant, extending to
higher dimension the classical genus of a surface and the Heegaard genus of a
$3$-manifold. It succeeds in characterizing spheres in arbitrary dimension,
and a lot of classifying results via regular genus have been obtained,
especially in dimension $4$ and $5$ (see [19], [11], [15], [12] and their
references).

###### Definition 4

An $r$-dipole ($1\leq r\leq n$) of colors $c_{1},\ldots,c_{r}$ of a
$(n+1)$-colored graph $\Gamma$ is a subgraph of $\Gamma$ consisting in two
vertices joined by $r$ edges, colored by $c_{1},\ldots,c_{r}$, such that its
vertices belong to different connected components of
$\Gamma_{\hat{c}_{1}\ldots\hat{c}_{r}}$. An $r$-dipole can be eliminated from
$\Gamma$ by deleting the subgraph and welding the remaining hanging edges
according to their colors; in this way another $(n+1)$-colored graph
$\Gamma^{\prime}$ is obtained. The addition of the dipole to $\Gamma^{\prime}$
is the inverse operation.

The dipole is called proper if $|K(\Gamma)|$ and $|K(\Gamma^{\prime})|$ are
(PL) homeomorphic: this happens if and only if at least one of the two
connected components of $\Gamma_{\hat{c}_{1}\ldots\hat{c}_{r}}$ intersecting
the dipole represents the $(n-r)$-sphere ([30, Proposition 5.3]).222Note that,
if $\Gamma$ represents a compact $n$-manifold $M$, then all $r$-dipoles are
proper, for $1<r\leq n$; in particular, if $M$ has either empty or connected
boundary, then all 1-dipoles are proper, too.

###### Definition 5

An $(n+1)$-colored graph $\Gamma$ representing a compact $n$-manifold with
empty or connected boundary is said to be a crystallization of $M$ if, for
each color $c\in\Delta_{n}$, $\Gamma_{\hat{c}}$ is connected.

Note that any compact $n$-manifold $M$ with empty or connected boundary admits
a crystallization. In fact, it is sufficient to consider an arbitrary
$(n+1)$-colored graph representing $M$, and then to eliminate all possible
(proper) $1$-dipoles. Moreover, it is always possible to assume - up to
permutation of the color set - that any gem (and, in particular, any
crystallization) of such a manifold, has color $n$ as its (unique) possible
singular color.

We conclude the Section by recalling how to yield a presentation of the
fundamental group of $\widehat{M}$, directly from any gem $\Gamma$ of $M$.

###### Proposition 5

([12]) Let $\Gamma$ be a gem of the compact $n$-manifold $M.$ If
$i\in\Delta_{n}$ is such that each color in $\Delta_{n}-\\{i\\}$ is not
singular in $\Gamma$, then:

$\pi_{1}(\widehat{M})\ =\ <X_{i}\ /\ R_{i}\cup\bar{R}_{i}>,$

where $X_{i}$ is in bijection with the $i$-colored edges of $\Gamma$, $R_{i}$
is in bijection with the $\\{i,j\\}$-colored cycles of $\Gamma$ ($\,\forall
j\in\Delta_{n}-\\{i\\}$) and $\bar{R}_{i}$ is a subset of $X_{i}$
corresponding to a minimal set of $i$-colored edges of $\Gamma$ connecting
$\Gamma_{\hat{i}}.$

##  3 From framed links to $5$-colored graphs

In this section, we will recall a construction, first presented in [13], that
enables to obtain $5$-colored graphs representing a wide class of compact
(simply-connected) $4$-manifolds. The corresponding result (Proposition 7)
will be stated in a slightly different form with respect to the original one,
since further information about the obtained graph is of interest for the
purposes of the present paper.

First, it is useful to recall that a graph-based representation for compact
manifolds - different from the one considered in Section 2 \- was already
introduced by Gagliardi in the eighties (see [28]) by means of colored graphs
failing to be regular. More precisely any compact $n$-manifold can be
represented by a pair $(\Lambda,\lambda)$, where $\lambda$ is still an edge-
coloration on $E(\Lambda)$ by means of $\Delta_{n}$, but $\Lambda$ may miss
some (or even all) $n$-colored edges: such a $(\Lambda,\lambda)$ is said to be
an $(n+1)$-colored graph with boundary, regular with respect to color $n$, and
vertices missing the $n$-colored edge are called boundary vertices.

However, a connection between these different kinds of representation can be
established through an easy combinatorial procedure, called capping-off.

###### Proposition 6

([27]) Let $(\Lambda,\lambda)$ be an $(n+1)$-colored graph with boundary,
regular with respect to color $n$, representing the compact $n$-manifold $M$.
Chosen a color $c\in\Delta_{n-1}$, let $(\Gamma,\gamma)$ be the regular
$(n+1)$-colored graph obtained from $\Lambda$ by _capping-off with respect to
color $c$_, i.e. by joining two boundary vertices by an $n$-colored edge,
whenever they belong to the same $\\{c,n\\}$-colored path in $\Lambda.$ Then,
$(\Gamma,\gamma)$ represents the singular $n$-manifold $\widehat{M}$, and
hence $M$, too.

Given a framed link $(L,c)$ ($c=(c_{1},\dots,c_{l})$, with
$c_{i}\in\mathbb{Z}$ $\forall i\in\\{1,\dots,l\\}$, $l$ being the number of
components of $L$), let $M^{4}(L,c)$ denote the $4$-manifold with boundary
obtained from $\mathbb{D}^{4}$ by adding $l$ 2-handles according to the framed
link $(L,c)$. The boundary of $M^{4}(L,c)$ is the closed orientable 3-manifold
$M^{3}(L,c)$ obtained from $\mathbb{S}^{3}$ by Dehn surgery along $(L,c)$. In
case $M^{3}(L,c)\cong\mathbb{S}^{3}$, we will consider the closed $4$-manifold
- which will be still denoted by $M^{4}(L,c),$ \- obtained by adding a further
$4$-handle.

Moreover, given a planar diagram $\mathcal{D}$ of the link $L$, let us
consider a chess-board coloration of $\mathcal{D}$, by colors $\alpha$ and
$\beta$ say, with the convention that the infinite region is $\alpha$-colored.
We will denote by $m_{\alpha}$ the number of $\alpha$-colored regions.

###### Proposition 7

[13] For any framed link $(L,c)$ with $l$ components, a regular bipartite
$5$-colored graph, representing $M^{4}(L,c)$ may be directly constructed from
any planar diagram $\mathcal{D}$ of $(L,c)$. Moreover, the obtained graph has
regular genus $m_{\alpha}+l$, while its $\hat{4}$-residue, representing
$M^{3}(L,c),$ has regular genus $m_{\alpha}.$

_Proof._ From any planar diagram $\mathcal{D}$ of a framed link $(L,c)$,
Theorem 2 (resp. Theorem 3) of [10] allows to construct a 4-colored graph
$\Lambda(\mathcal{D},c)$ (resp. a 5-colored graph with boundary
$\tilde{\Lambda}(\mathcal{D},c)$) representing $M^{3}(L,c)$ (resp.
$M^{4}(L,c)$).

$\Lambda(\mathcal{D},c)$ can be directly “drawn over“ a self-framed planar
diagram of $(L,c)$, i.e. a planar diagram obtained from $\mathcal{D}$ by
possibly adding a suitable number of “additional curls“, in order to make the
writhe of the $i$-th component of $L$ equal to $c_{i}$, for any
$i\in\\{1,\dots,l\\}$. See Figure 1 for an example, where $L$ is the trefoil
knot and $c=+1$.

Figure 1: The $4$-colored graph $\Lambda(\mathcal{D},c)$ representing
$M^{3}(L,c)$, for $c=+1$ and $L=$ trefoil

The main step of the construction from $\Lambda(\mathcal{D},c)$ to
$\tilde{\Lambda}(\mathcal{D},c)$ consists in the identification, for each
component of the link $L$, of a particular subgraph of order four - called
quadricolor 333Such a subgraph can always be detected, possibly by adding a
trivial pair of opposite additional curls. \- in $\Lambda(\mathcal{D},c)$,
which is substituted in $\tilde{\Lambda}(\mathcal{D},c)$ by a suitable order
ten 5-colored subgraph, as depicted in Figure 2.

Figure 2: main step from $\Lambda(\mathcal{D},c)$ to
$\tilde{\Lambda}(\mathcal{D},c)$

As proved in [10, Theorem 1], both $\Lambda(\mathcal{D},c)$ and
$\tilde{\Lambda}(\mathcal{D},c)$ admit a finite sequence of graph moves -
called “generalized dipole eliminations” - not affecting the represented
spaces, nor the quadricolor structures (see Figure 3), but reducing the
regular genus of both graphs.

Figure 3: A generalized dipole elimination

More precisely, each generalized dipole that is cancelled consists in a
$\\{0,3\\}$-colored cycle of length four, corresponding to a crossing of
$\mathcal{D}$, and one of its adjacent $\\{1,2\\}$-colored cycles, that does
not contain vertices of a quadricolor. It is possible to prove (see the above
cited Theorem) that the subsequent moves can be performed according to a
suitable choice of $\beta$-colored regions of $\mathcal{D}.$

As a consequence, a new 4-colored graph $\Omega(\mathcal{D},c)$ representing
$M^{3}(L,c)$ (resp. a new 5-colored graph with boundary
$\tilde{\Omega}(\mathcal{D},c)$ representing $M^{4}(L,c)$) is obtained, so
that, by taking the permutation $\varepsilon^{\prime}=(1,0,2,3)$ of
$\Delta_{3}$ (resp. $\varepsilon=(1,0,2,3,4)$ of $\Delta_{4}$), then
$\rho_{\varepsilon^{\prime}}(\Omega(\mathcal{D},c))=m_{\alpha}$ (resp.
$\rho_{\varepsilon}(\tilde{\Omega}(\mathcal{D},c))=m_{\alpha}+2l$ and
$\rho_{\varepsilon^{\prime}}(\tilde{\Omega}_{\hat{4}}(\mathcal{D},c))=m_{\alpha}+l$).
See Figure 4 for an example, $(L,c)$ being the trefoil knot with frame $c=+1$.

Figure 4: The $4$-colored graph $\Omega(\mathcal{D},c)$ representing
$M^{3}(L,c)$, for $c=+1$ and $L=$ trefoil

If the “capping off” procedure described in Proposition 6 is applied to
$\tilde{\Omega}(\mathcal{D},c)$ with respect to color $1$, the obtained
regular 5-colored graph (which represents the singular 4-manifold
$\widehat{M^{4}(L,c)}$) turns out to admit for each component of $L$ a
sequence of three (proper) 2-dipoles involving only vertices of a quadricolor
and never involving color $4$: in fact, for each component of $L$, they
consist of the pairs of vertices $\\{P_{3},R_{3}\\}$,
$\\{R_{3}^{\prime},R_{2}^{\prime}\\}$, $\\{P_{0},R_{1}^{\prime}\\}$. Since one
of these 2-dipoles has colors non-consecutive in the permutation
$\varepsilon$, it decreases by one the regular genus, while the other two
2-dipoles have colors consecutive in the same permutation and so their
elimination does not affect the regular genus. Finally, we obtain a regular
5-colored graph $\Gamma(\mathcal{D},c)$ (representing both
$\widehat{M^{4}(L,c)}$ and $M^{4}(L,c)$) with regular genus
$\rho_{\varepsilon}(\Gamma(\mathcal{D},c))=\rho_{\varepsilon}(\tilde{\Omega}(\mathcal{D},c))-l=m_{\alpha}+l.$
It is not difficult to prove that its unique singular residue (if any) is
$\Gamma_{\hat{4}}(\mathcal{D},c),$ which represents $M^{3}(L,c)$ and can be
obtained from $\tilde{\Omega}_{\hat{4}}(\mathcal{D},c)$ by cancelling
2-dipoles that are induced by the above ones. Therefore
$\rho_{\varepsilon^{\prime}}(\Gamma_{\hat{4}}(\mathcal{D},c))=\rho_{\varepsilon^{\prime}}(\tilde{\Omega}_{\hat{4}}(\mathcal{D},c))-l=m_{\alpha}.$
This proves the thesis.

See Figure 5 for the structure corresponding to each quadricolor in
$\Gamma(\mathcal{D},c)$, while all other $4$-colored edges are added between
pairs of $1$-adjacent vertices in $\Lambda(\mathcal{D},c)$. Figure 6 shows the
final graph $\Gamma(\mathcal{D},+1)$, in case $L$ being the trefoil knot.

$\Box$

Figure 5: main step yielding $\Gamma(\mathcal{D},c)$

Figure 6: The regular $5$-colored graph $\Gamma(\mathcal{D},c)$ representing
$M^{4}(L,c)$, for $c=+1$ and $L=$ trefoil

##  4 Weak semi-simple crystallizations

In [5] and [3] two particular types of crystallizations are introduced and
studied444Both semi-simple and weak semi-simple crystallizations generalize
the notion of simple crystallizations for closed simply-connected
$4$-manifolds: see [4] and [16].: they are proved to be “minimal” with respect
to regular genus, among all graphs representing the same closed $4$-manifold.

In the following we will extend the definitions to compact $4$-manifolds with
empty or connected boundary.

As already pointed out in Section 2, we may assume that all gems of such
manifolds have color $4$ as their (unique) possible singular color.

Furthermore, let us denote by $\mathcal{P}_{4}$ the set of all cyclic
permutations $\varepsilon=(\varepsilon_{0},\varepsilon_{1},\varepsilon_{2},$
$\varepsilon_{3},\varepsilon_{4})$ of $\Delta_{4}$ such that
$\varepsilon_{4}=4.$

###### Definition 6

Let $M^{4}$ be a compact $4$-manifold, with empty or connected boundary. A
$5$-colored graph $\Gamma$ representing $M^{4}$ is called semi-simple if
$g_{jkl}=1+m^{\prime}\ \ \forall\ j,k,l\in\Delta_{3}$ and $g_{jk4}=1+m\ \
\forall\ j,k\in\Delta_{3},$ where $rk(\pi_{1}(M^{4}))=m\geq 0$ and
$rk(\pi_{1}(\widehat{M}^{4}))=m^{\prime}\geq 0$ ($m^{\prime}\leq m$).
$\Gamma$ is called weak semi-simple with respect to a permutation
$\varepsilon\in\mathcal{P}_{4}$ if
$g_{\varepsilon_{i},\varepsilon_{i+2},\varepsilon_{i+4}}=1+m\ \ \forall\
i\in\\{0,2,4\\}$ and
$g_{\varepsilon_{i},\varepsilon_{i+2},\varepsilon_{i+4}}=1+m^{\prime}\ \
\forall\ i\in\\{1,3\\}$ (where the additions in subscripts are intended in
$\mathbb{Z}_{5}$).

Note that, as a consequence of the above definition, if $\Gamma$ is weak semi-
simple, then $g_{\hat{j}}=1,\ \forall\ j\in\Delta_{4}$, i.e. $\Gamma$ is a
crystallization of $M^{4}.$

In case $m=0$, semi-simple (resp. weak semi-simple) crystallizations are said
to be simple (resp. weak simple).

Generalizing [5], let us set

$g_{j,k,l}=(g_{\hat{r}}+g_{\hat{s}}-1)+m^{\prime}+t_{j,k,l},\ \ \text{with}\
t_{j,k,l}\geq 0,\ \\{r,s\\}=\Delta_{4}-\\{j,k,l\\},\ \ \forall
j,k,l\in\Delta_{3};$ $g_{j,k,4}=(g_{\hat{r}}+g_{\hat{s}}-1)+m+t_{j,k,4},\ \
\text{with}\ t_{j,k,4}\geq 0,\ \ \ \\{r,s\\}=\Delta_{3}-\\{j,k\\},\ \ \forall
j,k\in\Delta_{3}.$

Then $\Gamma$ is semi-simple (resp. weak semi-simple) iff $g_{\hat{j}}=1$ and
$t_{j,k,l}=0$ $\forall j,k,l\in\Delta_{4}$ (resp. $g_{\hat{i}}=1$ and
$t_{\varepsilon_{i},\varepsilon_{i+2},\varepsilon_{i+4}}=0$ $\forall
i\in\mathbb{Z}_{5}$)

In [12], for any gem of a compact $4$-manifold, the following relation is
proved to hold for each $i\in\Delta_{4}$:

$g_{\widehat{\varepsilon_{i-1}},\widehat{\varepsilon_{i+1}}}=g_{\varepsilon_{i},\varepsilon_{i+2},\varepsilon_{i+3}}\
=\
(g_{\widehat{\varepsilon_{i-1}}}+g_{\widehat{\varepsilon_{i+1}}}-1)+\rho_{\varepsilon}-\rho_{\varepsilon_{\widehat{i-1}}}-\rho_{\varepsilon_{\widehat{i+1}}}$
(1)

where
$\varepsilon_{\hat{i}}=(\varepsilon_{0},\ldots,\varepsilon_{i-1},\varepsilon_{i+1},\ldots,\varepsilon_{4}=4)$
and $\rho_{\varepsilon},$ $\rho_{\varepsilon_{\hat{i}}}$ respectively denote
$\rho_{\varepsilon}(\Gamma),$
$\rho_{\varepsilon_{\hat{i}}}(\Gamma_{\widehat{\varepsilon_{i}}}).$

Therefore, in the case of a crystallization of a compact $4$-manifold with
empty or connected boundary, we have:

$g_{\varepsilon_{i-1},\varepsilon_{i+1},\varepsilon_{i+3}}=1+\rho_{\varepsilon}-\rho_{\varepsilon_{\hat{i}}}-\rho_{\varepsilon_{\widehat{i+2}}}=1+m^{\prime}+t_{\varepsilon_{i-1},\varepsilon_{i+1},\varepsilon_{i+3}}\
\ \ \ \forall i\in\\{2,4\\}\ \ \text{and}$
$g_{\varepsilon_{i-1},\varepsilon_{i+1},\varepsilon_{i+3}}=1+\rho_{\varepsilon}-\rho_{\varepsilon_{\hat{i}}}-\rho_{\varepsilon_{\widehat{i+2}}}=1+m+t_{\varepsilon_{i-1},\varepsilon_{i+1},\varepsilon_{i+3}}\
\ \ \ \forall i\in\\{0,1,3\\},$

which trivially imply

$\displaystyle\rho_{\varepsilon}-\rho_{\varepsilon_{\hat{i}}}-\rho_{\varepsilon_{\widehat{i+2}}}-m^{\prime}=$
$\displaystyle\ t_{\varepsilon_{i-1},\varepsilon_{i+1},\varepsilon_{{i+3}}}\ \
\ \ \forall i\in\\{2,4\\}\ \ \text{and}$ (2)
$\displaystyle\rho_{\varepsilon}-\rho_{\varepsilon_{\hat{i}}}-\rho_{\varepsilon_{\widehat{i+2}}}-m=$
$\displaystyle\ t_{\varepsilon_{i-1},\varepsilon_{i+1},\varepsilon_{{i+3}}}\ \
\ \ \forall i\in\\{0,1,3\\}.$

By summing up over $i\in\Delta_{4}$, we obtain:

$5\rho_{\varepsilon}-2\sum_{i\in\Delta_{4}}\rho_{\varepsilon_{\hat{i}}}-5m+2(m-m^{\prime})=\sum_{i\in\Delta_{4}}t_{\varepsilon_{i},\varepsilon_{i+2},\varepsilon_{i+4}},$

i.e.

$\sum_{i\in\Delta_{4}}\rho_{\varepsilon_{\hat{i}}}\ =\
\frac{5}{2}(\rho_{\varepsilon}-m)+(m-m^{\prime})-\frac{1}{2}\sum_{i\in\Delta_{4}}t_{\varepsilon_{i},\varepsilon_{i+2},\varepsilon_{i+4}}.$
(3)

###### Corollary 8

Let $\Gamma$ be a crystallization of a compact $4$-manifold $M^{4}$ with empty
or connected boundary, with $rk(\pi_{1}(M^{4}))=m\geq 0$,
$rk(\pi_{1}(\widehat{M}^{4}))=m^{\prime}\geq 0$ ($m^{\prime}\leq m$). Then
$\Gamma$ is weak semi-simple with respect to a cyclic permutation
$\varepsilon\in\mathcal{P}_{4}$ if and only if

$\rho_{\varepsilon_{\hat{i}}}=\frac{1}{2}(\rho_{\varepsilon}-m)\ \ \forall
i\in\Delta_{3}\ \ \ \ \text{and}\ \ \ \
\rho_{\varepsilon_{\hat{4}}}=\frac{1}{2}(\rho_{\varepsilon}-m)+(m-m^{\prime}).$

_Proof._ Let us suppose that $\Gamma$ is a weak semi-simple crystallization
with respect to $\varepsilon$, then
$t_{\varepsilon_{i},\varepsilon_{i+2},\varepsilon_{i+4}}=0$ $\forall
i\in\Delta_{4}.$ Hence, from formulas (3) and (2),
$\sum_{i\in\Delta_{4}}\rho_{\varepsilon_{\hat{i}}}\ =\
\frac{5}{2}(\rho_{\varepsilon}-m)+(m-m^{\prime})$ holds, together with
$\rho_{\varepsilon}-\rho_{\varepsilon_{\hat{i}}}-\rho_{\varepsilon_{\widehat{i+2}}}=m^{\prime}$
$\forall i\in\\{2,4\\}$ and
$\rho_{\varepsilon}-\rho_{\varepsilon_{\hat{i}}}-\rho_{\varepsilon_{\widehat{i+2}}}=m$
$\forall i\in\\{0,1,3\\}.$

Let us now consider the five integers $h_{i}$, where
$h_{i}=\rho_{\varepsilon_{\hat{i}}}-\frac{1}{2}(\rho_{\varepsilon}-m)$
$\forall i\in\Delta_{3}$ and
$h_{4}=\rho_{\varepsilon_{\hat{4}}}-\frac{1}{2}(\rho_{\varepsilon}-m)$
$-(m-m^{\prime}).$ It is easy to check that $h_{i}+h_{i+2}=0$ $\forall
i\in\Delta_{4}$; therefore, if an integer $j\in\Delta_{4}$ existed, so that
$h_{j}<0$, then both $h_{j+2}>0$ and $h_{j-2}>0$ would follow from the
equations
$\rho_{\varepsilon}-\rho_{\varepsilon_{\hat{i}}}-\rho_{\varepsilon_{\widehat{i+2}}}=m^{\prime}$
$\forall i\in\\{2,4\\}$ and
$\rho_{\varepsilon}-\rho_{\varepsilon_{\hat{i}}}-\rho_{\varepsilon_{\widehat{i+2}}}=m$
$\forall i\in\\{0,1,3\\}$. Now, $h_{j+2}>0$ and $h_{j-2}>0$ would imply both
$h_{j+4}<0$ and $h_{j-4}<0$, from the same equations. Since
$\varepsilon_{j-4}=\varepsilon_{j+1}$ and
$\varepsilon_{j+4}=\varepsilon_{j-1}$ are non-consecutive in $\varepsilon$,
this gives a contradiction. As a consequence, $h_{i}=0$ follows, $\forall
i\in\Delta_{4}.$

On the other hand, if
$\rho_{\varepsilon_{\hat{i}}}=\frac{1}{2}(\rho_{\varepsilon}-m)$ $\forall
i\in\Delta_{3}$ and
$\rho_{\varepsilon_{\hat{4}}}=\frac{1}{2}(\rho_{\varepsilon}-m)+(m-m^{\prime})$
hold, then formula (2) trivially implies
$t_{\varepsilon_{i-1},\varepsilon_{i+1},\varepsilon_{i+3}}=0$ $\forall
i\in\Delta_{4},$ thus proving $\Gamma$ to be weak semi-simple with respect to
$\varepsilon$.

$\Box$

Example:  The crystallization of $\xi_{c}$, the $\mathbb{D}^{2}$-bundle over
$\mathbb{S}^{2}$ with Euler class $c$ (resp. the crystallization of
$\mathbb{Y}^{4}_{h}$, the genus $h$ $4$-dimensional handlebody), that is
constructed in [12], turns out to be weak simple (resp. to be semi-simple). A
direct check can be performed by using the definition or, equivalently,
through Corollary 8. Both graphs will be considered again in Section 6.

The following result extends to compact 4-manifolds properties already proved
in [20, Proposition 8] for the closed case.

###### Proposition 9

Let $\Gamma$ be an order $2p$ crystallization of a compact $4$-manifold
$M^{4}$ with empty or connected boundary, such that $rk(\pi_{1}(M^{4}))=m\geq
0$, $rk(\pi_{1}(\widehat{M}^{4}))=m^{\prime}\geq 0$ ($m^{\prime}\leq m$).
Then, for each pair ($\varepsilon,\varepsilon^{\prime})$ of associated
555Given $\varepsilon$, the associated permutation $\varepsilon^{\prime}$ is
defined as
$\varepsilon^{\prime}=(\varepsilon_{0},\varepsilon_{2},\varepsilon_{4},\varepsilon_{1},\varepsilon_{3})$
cyclic permutations of $\Delta_{4}$, with
$\rho_{\varepsilon}(\Gamma)\leq\rho_{\varepsilon^{\prime}}(\Gamma)$:

$\rho_{\varepsilon^{\prime}}(\Gamma)-\rho_{\varepsilon}(\Gamma)=\sum_{i\in\mathbb{Z}_{5}}t_{\varepsilon_{i},\varepsilon_{i+1},\varepsilon_{i+2}}-\sum_{i\in\mathbb{Z}_{5}}t_{\varepsilon_{i},\varepsilon_{i+2},\varepsilon_{i+4}}\
\leq\
q=\sum_{i,j,k\in\Delta_{4}}t_{\varepsilon_{i},\varepsilon_{j},\varepsilon_{k}}.$
(4)

Moreover, the following statements are equivalent:

  * (a)

a cyclic permutation $\varepsilon$ of $\Delta_{4}$ exists, such that
$\rho_{\varepsilon^{\prime}}(\Gamma)-\rho_{\varepsilon}(\Gamma)=q;$

  * (b)

$\Gamma$ is weak semi-simple with respect to $\varepsilon$;

  * (c)

$\rho(\Gamma)=\rho_{\varepsilon}(\Gamma)=2\chi(\widehat{M}^{4})+5m-2(m-m^{\prime})-4$.

_Proof._ In virtue of [20, Proposition 6(b)],

$\rho_{\varepsilon^{\prime}}(\Gamma)-\rho_{\varepsilon}(\Gamma)\ =\
\sum_{j\in\mathbb{Z}_{5}}g_{\varepsilon_{j},\varepsilon_{j+1},\varepsilon_{j+2}}-\sum_{j\in\mathbb{Z}_{5}}g_{\varepsilon_{j},\varepsilon_{j+2},\varepsilon_{j+4}}$

holds for any $5$-colored graph representing a singular $4$-manifold $N^{4}$;
hence, an easy computation yields:

$\displaystyle\rho_{\varepsilon^{\prime}}(\Gamma)-\rho_{\varepsilon}(\Gamma)$
$\displaystyle=\
[5(m+1)-2(m-m^{\prime})+\sum_{i\in\mathbb{Z}_{5}}t_{\varepsilon_{i},\varepsilon_{i+1},\varepsilon_{i+2}}]+$
$\displaystyle\ \ \ \
-[5(m+1)-2(m-m^{\prime})+\sum_{i\in\mathbb{Z}_{5}}t_{\varepsilon_{i},\varepsilon_{i+2},\varepsilon_{i+4}}]\
=$ $\displaystyle=\
\sum_{i\in\mathbb{Z}_{5}}t_{\varepsilon_{i},\varepsilon_{i+1},\varepsilon_{i+2}}-\sum_{i\in\mathbb{Z}_{5}}t_{\varepsilon_{i},\varepsilon_{i+2},\varepsilon_{i+4}}\
=$ $\displaystyle=\
\sum_{j,k,l\in\Delta_{4}}t_{jkl}-2\sum_{i\in\mathbb{Z}_{5}}t_{\varepsilon_{i},\varepsilon_{i+2},\varepsilon_{i+4}}\
=$ $\displaystyle=\
q-2\sum_{i\in\mathbb{Z}_{5}}t_{\varepsilon_{i},\varepsilon_{i+2},\varepsilon_{i+4}}\
\leq\ q.$

Now, if $\Gamma$ is a weak semi-simple crystallization with respect to a
cyclic permutation $\varepsilon$ of $\Delta_{4}$, then
$t_{\varepsilon_{i},\varepsilon_{i+2},\varepsilon_{i+4}}=0$ for each
$i\in\mathbb{Z}_{5}.$ Hence,
$\rho_{\varepsilon^{\prime}}(\Gamma)-\rho_{\varepsilon}(\Gamma)=q$ easily
follows from relation (4).

On the other hand, if a cyclic permutation $\varepsilon$ of $\Delta_{4}$
exists, so that
$\rho_{\varepsilon^{\prime}}(\Gamma)-\rho_{\varepsilon}(\Gamma)=q,$ relation
(4) yields $t_{\varepsilon_{i},\varepsilon_{i+2},\varepsilon_{i+4}}=0,$ which,
together with $g_{\hat{j}}=1\ \forall\ j\in\Delta_{4}$, is equivalent to the
definition of weak semi-simple crystallization.

Hence, (a) and (b) are proved to be equivalent.

Moreover, [20, Proposition 7] gives:

$\chi(N^{4})\ =\
\Big{(}\rho_{\varepsilon}(\Gamma)+\rho_{\varepsilon^{\prime}}(\Gamma)\Big{)}-p+3$

for any order $2p$ crystallization of a singular 4-manifold $N^{4}$ with one
singular vertex at most. Then, by making use of relation (4):

$\chi(\widehat{M}^{4})\ =\
2\rho_{\varepsilon}(\Gamma)+3-p+q-2\sum_{i\in\mathbb{Z}_{5}}t_{\varepsilon_{i},\varepsilon_{i+2},\varepsilon_{i+4}}.$
(5)

On the other hand, an easy computation (making use of [17, Lemma 21]) yields:

$\chi(\widehat{M}^{4})=5-\frac{1}{3}\sum_{i,j,k\in\Delta_{4}}g_{ijk}+\frac{1}{3}p.$

Hence, by comparison with (5):

$\chi(\widehat{M}^{4})=2\rho_{\varepsilon}(\Gamma)+3-3\chi(\widehat{M}^{4})+5-10m+4(m-m^{\prime})-q+q-2\sum_{i\in\mathbb{Z}_{5}}t_{\varepsilon_{i},\varepsilon_{i+2},\varepsilon_{i+4}},$

from which

$\rho_{\varepsilon}(\Gamma)=2\chi(\widehat{M}^{4})+5m-2(m-m^{\prime})-4+\sum_{i\in\mathbb{Z}_{5}}t_{\varepsilon_{i},\varepsilon_{i+2},\varepsilon_{i+4}}$
(6)

easily follows, as well as

$\rho_{\varepsilon^{\prime}}(\Gamma)=2\chi(\widehat{M}^{4})+5m-2(m-m^{\prime})-4+\sum_{i\in\mathbb{Z}_{5}}t_{\varepsilon_{i},\varepsilon_{i+1},\varepsilon_{i+2}}.$
(7)

Since $g_{\hat{j}}=1\ \forall\ j\in\Delta_{4}$ is assumed, relation (6)
directly yields the co-implication between statements (b) and (c).

$\Box$

As a consequence, we prove that weak semi-simple crystallizations realize the
minimum (generalized) regular genus, also in the extended setting of compact
4-manifolds with empty or connected boundary, thus generalizing the analogous
result obtained in [3] for the closed case.

###### Corollary 10

Let $\Gamma$ be a crystallization of a compact $4$-manifold $M^{4}$ with empty
or connected boundary, with $rk(\pi_{1}(M^{4}))=m\geq 0$ and
$rk(\pi_{1}(\widehat{M}^{4}))=m^{\prime}\geq 0$ ($m^{\prime}\leq m$). Then:

$\mathcal{G}(M^{4})\geq 2\chi(\widehat{M}^{4})+5m-2(m-m^{\prime})-4,$

and
$\mathcal{G}(M^{4})=\rho(\Gamma)=\rho_{\varepsilon}(\Gamma)=2\chi(\widehat{M}^{4})+5m-2(m-m^{\prime})-4$
if and only if $\Gamma$ is a weak semi-simple crystallization with respect to
the permutation $\varepsilon\in\mathcal{P}_{4}.$

In this case, we have

$\rho_{\varepsilon_{\hat{i}}}=\chi(\widehat{M}^{4})+m+m^{\prime}-2\ \ \forall
i\in\Delta_{3}\ \ \ \ \text{and}\ \ \ \
\rho_{\varepsilon_{\hat{4}}}=\chi(\widehat{M}^{4})+2m-2.$

_Proof._ It is a trivial consequence of relation (6).

$\Box$

###### Remark 3

Note that, if $M^{4}$ is a closed simply-connected $4$-manifold, the
inequality of Corollary 10 becomes $\mathcal{G}(M^{4})\geq 2\beta_{2}$,
$\beta_{2}$ being the second Betti number of $M^{4}$, and $\Gamma$ is a weak
simple crystallization of $M^{4}$ if and only if equality holds. Moreover, in
this last case all $\hat{\varepsilon}_{i}$-residues ($i\in\mathbb{Z}_{5}$) of
$\Gamma$ have regular genus $\beta_{2}$ with respect to the permutations
induced by $\varepsilon.$

##  5 B-trisections induced by colored graphs

As already pointed out in Section 1, the fundamental result on which our
approach to trisections is based on consists in a weak notion of trisection
(quasi-trisection), which arises from colored graphs representing compact
$4$-manifolds. Then, under suitable conditions, the notion of B-trisection,
actually extending the classical one of trisection, is introduced.

We are now able to formalize both the above notions.

###### Definition 7

Let $M^{4}$ be a compact $4$-manifold with empty (resp. connected) boundary. A
quasi-trisection of $M^{4}$ is a triple $\mathcal{T}=(H_{0},H_{1},H_{2})$ of
$4$-dimensional submanifolds of $M^{4}$, whose interiors are pairwise
disjoint, such that $M^{4}=H_{0}\cup H_{1}\cup H_{2}$ and

  * (i)

$H_{1},H_{2}$ are $4$-dimensional handlebodies; $H_{0}$ is a $4$-disk (resp.
is (PL) homeomorphic to $\partial M^{4}\times[0,1]$);

  * (ii)

$H_{01}=H_{0}\cap H_{1}$ and $H_{02}=H_{0}\cap H_{2}$ are $3$-dimensional
handlebodies;

  * (iii)

$\Sigma(\mathcal{T})=H_{0}\cap H_{1}\cap H_{2}$ is a closed connected surface
(which is called central surface).

###### Definition 8

Let $M^{4}$ be a compact $4$-manifold with empty or connected boundary. A
B-trisection of $M^{4}$ is a quasi-trisection
$\mathcal{T}=(H_{0},H_{1},H_{2})$ of $M^{4}$ such that $H_{12}=H_{1}\cap
H_{2}$ is a $3$-dimensional handlebody, too.

###### Remark 4

Note that the central surface $\Sigma(\mathcal{T})$ of a B-trisection
$\mathcal{T}=(H_{0},H_{1},H_{2})$ of $M^{4}$ is an Heegaard surface for the
3-manifold $\partial H_{i}=\\#_{k_{i}}(\mathbb{S}^{1}\times\mathbb{S}^{2})$
($k_{i}\geq 0$), for each $i\in\\{1,2\\}$, splitting it into the 3-dimensional
handlebodies $H_{ij}$ and $H_{ik}$, with $\\{j,k\\}=\\{0,1,2\\}-\\{i\\}$.
Moreover, in the closed (resp. boundary) case,
$(H_{01},H_{02},\Sigma(\mathcal{T}))$ is an Heegaard splitting of $\partial
H_{0}=\mathbb{S}^{3}$ (resp. of $\partial M^{4}$, and more precisely of the
boundary component of $\partial H_{0}$ intersecting $H_{1}\cup H_{2}$).
Hence, obviously, we have $k_{i}\leq genus(\Sigma(\mathcal{T}))$ for each
$i\in\\{1,2\\}$, and $\mathcal{H}(\partial M^{4})\leq
genus(\Sigma(\mathcal{T}))$ in the boundary case (where $\mathcal{H}(M^{3})$
denotes the classical Heegaard genus of the 3-manifold $M^{3}$).
Moreover, it is not difficult to check that, if $M^{4}$ admits a B-trisection,
the associated singular manifold $\widehat{M^{4}}$ is simply-connected
666Here, we obtain a generalization of the well-known fact that, if a closed
$4$-manifold $M^{4}$ admits a trisection in which one of the “pieces” is a
$4$-disk, then $M^{4}$ is simply-connected.: in fact, the existence of the
above Heegaard splittings of $\partial H_{1}$ and $\partial H_{2}$, implies
that there are surjections from $\pi_{1}(H_{12})$ to $\pi_{1}(\partial
H_{1})\cong\pi_{1}(H_{1})$ and $\pi_{1}(\partial H_{2})\cong\pi_{1}(H_{2})$
respectively. Therefore, by applying Van-Kampen’s theorem to the pair
$(H_{1},H_{2})$, surjections are obtained
$\pi_{1}(H_{i})\twoheadrightarrow\pi_{1}(H_{1}\cup H_{2})$ for each
$i\in\\{1,2\\}.$
If $H^{\prime}_{0}$ denotes the cone over the boundary component of $H_{0}$
not intersecting $H_{1}\cup H_{2}$, then we have $(H_{1}\cup H_{2})\cap
H^{\prime}_{0}=H_{01}\cup H_{02}=\partial H^{\prime}_{0};$ since
$\pi_{1}(\Sigma)\twoheadrightarrow\pi_{1}(H_{0i})\twoheadrightarrow\pi_{1}(\partial
H_{i})\cong\pi_{1}(H_{i}),\ \forall i\in\\{1,2\\}$, then there is a surjection
$\pi_{1}(H_{01}\cup H_{02})\twoheadrightarrow\pi_{1}(H_{1}\cup H_{2}).$
Obviously, we have $H^{\prime}_{0}\cup H_{1}\cup H_{2}=\widehat{M}^{4}$ and
Van Kampen’s theorem applied to the pair $(H_{1}\cup H_{2},H^{\prime}_{0})$
implies $\pi_{1}(H^{\prime}_{0})\twoheadrightarrow\pi_{1}(\widehat{M}^{4}).$
Since $H^{\prime}_{0}$ is contractible, $\widehat{M}^{4}$ is simply-connected.

###### Remark 5

Note that, given a B-trisection $\mathcal{T}=(H_{0},H_{1},H_{2})$ of a
4-manifold $M^{4}$ with non-empty boundary, if both $H_{1}$ and $H_{2}$ are
4-balls, the Heegaard splitting of $\partial M^{4}$ induced by $\mathcal{T}$
defines a special Heegaard sewing of this 3-manifold according to the
definition in [7]. More precisely, in that paper Birman introduced a class of
simply-connected 4-manifolds constructed by means of the partial
identification of the boundaries of two 4-balls, in a way that, in our
terminology, gives rise to a B-trisection of the 4-manifold itself;
consequently its boundary, which is a closed oriented 3-manifold, is said to
be represented by a special Heegaard sewing. Moreover, Birman proved that each
closed orientable 3-manifold admits such a representation. Rubinstein and
Tillmann in [36] pointed out this construction as a possible direction of
generalization of trisections from closed to compact 4-manifolds with non-
empty boundary. As we will see in the next Proposition 11, this approach turns
out to be quite natural starting from colored graphs.

In order to prove Theorem 1, a previous analysis on the singular $4$-manifolds
turns out to be useful. We will perform it by making use of the colored
triangulations dual to colored graphs, and then by yielding from them a
tricoloring, as in [6] and [38].

Let us denote by $G_{s}^{(4)}$ the set of $5$-colored graphs having only one
$\hat{4}$-residue and such that all $\hat{i}$-residues, with $i\in\Delta_{3}$,
are ordinary. Obviously, if $\Gamma\in G_{s}^{(4)},$ $|K(\Gamma)|$ is a
singular $4$-manifold with only one $4$-labelled, possibly singular, vertex,
while all other vertices are not singular.

Observe that the class $G_{s}^{(4)}$ properly contains (up to permutation of
the color set) all crystallizations of compact $4$-manifolds with empty or
connected boundary. Hence, any singular $4$-manifold $N^{4}$ with at most one
singular vertex can be represented by an element of $G_{s}^{(4)}$.

Furthermore, as in the previous Section, let $\mathcal{P}_{4}$ denote the set
of all cyclic permutations
$\varepsilon=(\varepsilon_{0},\varepsilon_{1},\varepsilon_{2},\varepsilon_{3},\varepsilon_{4})$
of $\Delta_{4}$ such that $\varepsilon_{4}=4.$

###### Proposition 11

Given a singular $4$-manifold $N^{4}$ having at most one singular vertex, a
$5$-colored graph $\Gamma\in G_{s}^{(4)}$ representing $N^{4}$ and a cyclic
permutation $\varepsilon$ of $\mathcal{P}_{4}$, there exists a triple
$(H_{b},H_{r},H_{g})$ such that

  * (i)

$H_{r},H_{g}$ are $4$-dimensional handlebodies; $H_{b}$ is either a $4$-ball,
if $N^{4}$ is a closed $4$-manifold, or it is the cone over the disjoint link
of the unique singular vertex of $N^{4}.$

  * (ii)

$N^{4}=H_{b}\cup H_{r}\cup H_{g}$ and the interiors of $H_{b},H_{r},H_{g}$ are
pairwise disjoint.

  * (iii)

$H_{rb}=H_{r}\cap H_{b}$ and $H_{gb}=H_{g}\cap H_{b}$ are $3$-dimensional
handlebodies.

  * (iv)

$H_{rg}=H_{r}\cap H_{g}$ collapses to a $2$-dimensional complex
$\overline{H}(\Gamma,\varepsilon).$

  * (v)

$H_{b}\cap H_{r}\cap H_{g}$ is a closed connected surface of genus
$\rho_{\hat{4}}(\Gamma_{\hat{4}}).$

_Proof._ Let us denote by $\sigma$ the standard $2$-simplex, by
$v_{b},v_{r},v_{b}$ its vertices and by $\sigma^{\prime}$ its first
barycentric subdivision.

Following [6] and [38], let us consider the tricoloring of $K(\Gamma)$ induced
by the following partition of $\Delta_{4}:$ $b=\\{4\\},\
g=\\{\varepsilon_{1},\varepsilon_{3}\\},\
r=\\{\varepsilon_{0},\varepsilon_{2}\\}.$

In the following, for sake of simplicity, we suppose
$\varepsilon=(0,1,2,3,4)$.

Let $\mu\ :\ K(\Gamma)\to\sigma$ be the the simplicial map sending all equally
colored vertices of $K(\Gamma)$ to one vertex of the standard $2$-simplex;
then $H_{b}$ (resp. $H_{r}$) (resp. $H_{g}$) is the preimage by $\mu$ of the
star of $v_{b}$ (resp. $v_{r}$) (resp. $v_{g}$) in $\sigma^{\prime}.$
Therefore it is easy to see that $H_{b}$ is a regular neighbourhood of the
(unique) $4$-colored vertex of $K(\Gamma)$ and precisely it is the cone over
its disjoint link. On the other hand $H_{r}$ (resp. $H_{g}$) is a regular
neighbourhood of the $1$-dimensional subcomplex $K_{02}(\Gamma)$ (resp.
$K_{13}(\Gamma)$) of $K(\Gamma)$ generated by its $i$-colored vertices, with
$i\in\\{0,2\\}$ (resp. $i\in\\{1,3\\}$). Since $K(\Gamma)$ is a
pseudomanifold, $K_{02}(\Gamma)$ (resp. $K_{13}(\Gamma)$) is connected;
therefore $H_{r}$ (resp. $H_{g}$) is a $4$-dimensional handlebody of genus
$g_{134}-g_{\hat{0}}-g_{\hat{2}}+1$ (resp.
$g_{024}-g_{\hat{1}}-g_{\hat{3}}+1$).

This proves property (i). Property (ii) is straightforward.

Figure 7: The 2-simplex $\sigma$

$H_{rb}$ is the preimage under $\mu$ of the edge of $\sigma^{\prime}$ with
endpoints the barycenter of $\sigma$ and the barycenter of the edge opposite
to $v_{g}$ (the “green” edge in Figure 7); it can be obtained by taking one
triangular prism for each $4$-simplex of $K(\Gamma)$ \- hence for each vertex
of $\Gamma$ \- as in Figure 8, where the vertices of the prism are barycenters
of $1$\- and $2$-simplexes of $K(\Gamma)$. We denoted a barycenter with the
colors of the vertices spanning it.

Figure 8: $H_{rb}$

The prisms are glued to each other in the following way:

for each $i\in\\{1,3\\}$ and for each $i$-colored edge $e$ of $\Gamma$, we
identify the quadrangular (resp. triangular) faces of the two prisms
associated to the endpoints of $e$ whose vertices do not contain the label
$i$.

Note that the quadrangular face of each prism that is opposite to the “green”
edge remains free. As a consequence, $H_{rb}$ collapses to the $1$-dimensional
complex made by all the “green” edges, i.e. it collapses to a graph with
$g_{123}+g_{013}$ vertices (i.e. the barycenters of the $\\{2,4\\}-$ and
$\\{0,4\\}-$colored edges of $K(\Gamma)$) and $g_{13}$ edges (each one is an
arc inside a $\\{0,2,4\\}-$colored triangle of $K(\Gamma)$ joining the
barycenter of the triangle with the barycenters of the the $\\{2,4\\}-$ and
$\\{0,4\\}-$colored edges; see Figure 9).

Figure 9:

Therefore $H_{rb}$ is a $3$-dimensional handlebody.

The proof that also $H_{gb}$ is a handlebody is analogous to the previous one
with the exchange of colors $1,3$ with $0,2$ and of the vertex $v_{g}$ with
$v_{r}$ (the “red” one in Figure 7) of $\sigma$ (see Figure 10).

Figure 10: $H_{gb}$

With regard to $H_{rg}$, this complex is the preimage under $\mu$ of the edge
of $\sigma^{\prime}$ with endpoints the barycenter of $\sigma$ and the
barycenter of the edge opposite to $v_{b}$ (the “blue” edge in Figure 7); it
can be obtained by taking a cube for each vertex of $\Gamma$ as in Figure 11,
where, as above, the vertices are barycenters of $1$\- and $2$-simplexes of
$K(\Gamma)$. The gluings of the cubes is performed according to the
$i$-adjancencies of the vertices of $\Gamma$, for each $i\in\Delta_{4}$, i.e.
two cubes corresponding to the endpoints of an $i$-colored edge of $\Gamma$
are glued along their faces whose vertices do not contain the color $i.$

Figure 11: $H_{rg}$

As a consequence, the face of each cube that misses no color (the top face in
Figure 11) is a free face and $H_{rg}$ collapses to a $2$-dimensional complex
$\overline{H}(\Gamma,\varepsilon)$ consisting of one square for each
$4$-colored edge of $\Gamma$. It is not difficult to see that the edges of
each square are dual to $\\{i,4\\}$-colored cycles of $\Gamma$ for each
$i\in\hat{4}$ (see Figure 12), therefore two squares are glued along an edge
iff their corresponding $4$-colored edges of $\Gamma$ share a common
$\\{i,4\\}$-colored cycle of $\Gamma.$

Figure 12: $\overline{H}(\Gamma,\varepsilon)$

With regard to statement (v), note that $\Sigma=H_{b}\cap H_{r}\cap
H_{g}=H_{rb}\cap H_{gb}\cap H_{rg}$ is a closed connected surface formed by
the free faces of the cubes of $H_{rg}$, therefore by $2p$ squares, $4p$ edges
and $g_{01}+g_{12}+g_{23}+g_{03}$ vertices, where $2p=\\#V(\Gamma)$.
Therefore, we have (via Proposition 4):

$\chi(\Sigma)=g_{01}+g_{12}+g_{23}+g_{03}-2p=2-2\rho_{\hat{4}}(\Gamma_{\hat{4}}).$

$\Box$

By means of Proposition 11, the existence of a quasi-trisection for each
compact $4$-manifold (Theorem 1) easily follows.

Proof of Theorem 1. Let $M^{4}$ be a compact $4$-manifold with empty or
connected boundary and $\Gamma$ a $5$-colored graph representing it.

If $\partial M^{4}\neq\emptyset$, up to a permutation of colors, we can always
suppose color $4$ to be the only singular color of $\Gamma$. Furthermore, note
that, if $\Gamma_{\hat{4}}$ is not connected, a suitable sequence of
eliminations of (proper) 1-dipoles of color $4$ yields a new graph
$\Gamma^{\prime}$, still representing $M^{4}$, having only one
$\hat{4}$-residue. In this case it is not difficult to prove that, for each
cyclic permutation $\varepsilon\in\mathcal{P}_{4}$,
$\rho_{\varepsilon_{\hat{4}}}(\Gamma^{\prime}_{\hat{4}})=\rho_{\varepsilon_{\hat{4}}}(\Gamma_{\hat{4}})$,
where $\rho_{\varepsilon_{\hat{4}}}(\Gamma_{\hat{4}})$ is intended as the sum
of the regular genera of the connected components of $\Gamma_{\hat{4}}$ with
respect to the permutation induced by $\varepsilon$ on
$\Delta_{3}=\Delta_{4}-\\{\varepsilon_{4}\\}.$

Therefore, without loss of generality, we can assume $\Gamma\in G_{s}^{(4)}$
and fix a permutation $\varepsilon\in\mathcal{P}_{4}.$

If $(H_{b},H_{r},H_{g})$ is the triple constructed as in Proposition 11
starting from $\Gamma$ and $\varepsilon$, it is easy to check - via the
bijection between $M^{4}$ and $\widehat{M^{4}}$ \- that $(H_{b},H_{r},H_{g})$
defines a quasi-trisection
$\mathcal{T}(\Gamma,\varepsilon)=(H_{0},H_{1},H_{2})$ of $M^{4}$ where
$H_{1}=H_{r}$ and $H_{2}=H_{g}$, while $H_{0}=H_{b}$, if $\partial
M^{4}=\emptyset$, or $H_{0}$ is a collar of $\partial M^{4}$ obtained by
removing from $H_{b}$ a suitable neighbourhood of the singular vertex, if
$\partial M^{4}\neq\emptyset$. Moreover, in both cases, $H_{b}\cap H_{r}\cap
H_{g}=H_{0}\cap H_{1}\cap H_{2}.$

$\Box$

The notion of gem-induced trisection (from which the G-trisection genus is
defined: see Definition 1) is used to identify B-trisections arising from
colored graphs.

###### Definition 9

Let $M^{4}$ be a compact $4$-manifold $M^{4}$ with empty or connected
boundary, and let $\mathcal{T}(\Gamma,\varepsilon)$ be the quasi-trisection of
$M^{4}$ associated to the 5-colored graph $\Gamma\in G_{s}^{(4)}$ and to the
permutation $\varepsilon\in\mathcal{P}_{4}.$ In case
$\mathcal{T}(\Gamma,\varepsilon)$ is a B-trisection, it is called a gem-
induced trisection of $M^{4}.$

Obviously, gem-induced trisections give upper bounds for the value of the
G-trisection genus of a compact $4$-manifold. In the particular case of closed
simply-connected $4$-manifolds, where gem-induced trisections are actually
trisections, the following statement is a direct consequence of Theorem 1 and
Proposition 11:

###### Corollary 12

If $\Gamma$ is a crystallization of a closed simply-connected $4$-manifold
$M^{4}$ admitting a gem-induced trisection $\mathcal{T}(\Gamma,\varepsilon)$
($\varepsilon\in\mathcal{P}_{4}$), then the trisection genus of $M^{4}$ is
less or equal to $\rho_{\varepsilon_{\hat{4}}}(\Gamma_{\hat{4}}).$

In order to analyze when a gem-induced trisection attains the (minimal)
G-trisection genus of a compact $4$-manifold, the following lemma (similar to
what already pointed out in the closed case by [25, Section 1]) is very
useful.

###### Lemma 13

Let $N^{4},\Gamma$ and $\varepsilon$ be as in Proposition 11 and let
$\mathcal{T}(\Gamma,\varepsilon)=(H_{0},H_{1},H_{2})$ be the induced quasi-
trisection of $\check{N}^{4}.$ Then:

$\chi(N^{4})=2+genus(\Sigma(\mathcal{T}(\Gamma,\varepsilon))-genus(H_{1})-genus(H_{2}).$

_Proof._ In [12], the following formula for the computation of the Euler
characteristic of a singular $4$-manifold is obtained:

$\chi(N^{4})=2-2\rho_{\varepsilon}+\sum_{i\in\Delta_{4}}\rho_{\varepsilon_{\hat{i}}}$
(8)

where $\rho_{\varepsilon}=\rho_{\varepsilon}(\Gamma)$ and
$\rho_{\varepsilon_{\hat{i}}}=\rho_{\varepsilon_{\hat{i}}}(\Gamma_{\hat{i}}).$

On the other hand, formula (1) yields

$g_{\varepsilon_{1}\varepsilon_{3}\varepsilon_{4}}=g_{\hat{\varepsilon}_{0}}+g_{\hat{\varepsilon}_{2}}-1+\rho_{\varepsilon}-\rho_{\varepsilon_{\hat{0}}}-\rho_{\varepsilon_{\hat{2}}}$

$g_{\varepsilon_{0}\varepsilon_{2}\varepsilon_{4}}=g_{\hat{\varepsilon}_{1}}+g_{\hat{\varepsilon}_{3}}-1+\rho_{\varepsilon}-\rho_{\varepsilon_{\hat{1}}}-\rho_{\varepsilon_{\hat{3}}}$

Hence,

$g_{\varepsilon_{1}\varepsilon_{3}\varepsilon_{4}}+g_{\varepsilon_{0}\varepsilon_{2}\varepsilon_{4}}=\sum_{i=0}^{3}g_{\hat{\varepsilon}_{i}}-2+2\rho_{\varepsilon}-\sum_{i=0}^{3}\rho_{\varepsilon_{\hat{i}}}.$

By substituting in formula (8), we have

$\displaystyle\chi(N^{4})$
$\displaystyle=\rho_{\varepsilon_{\hat{4}}}+\sum_{i=0}^{3}g_{\hat{\varepsilon}_{i}}-g_{\varepsilon_{1}\varepsilon_{3}\varepsilon_{4}}-g_{\varepsilon_{0}\varepsilon_{2}\varepsilon_{4}}=$
$\displaystyle=\rho_{\varepsilon_{\hat{4}}}-(g_{\varepsilon_{1}\varepsilon_{3}\varepsilon_{4}}-g_{\hat{\varepsilon}_{0}}-g_{\hat{\varepsilon}_{2}}+1)-(g_{\varepsilon_{0}\varepsilon_{2}\varepsilon_{4}}-g_{\hat{\varepsilon}_{1}}-g_{\hat{\varepsilon}_{3}}+1)+2=$
$\displaystyle=2+genus(\Sigma(\mathcal{T}(\Gamma,\varepsilon))-genus(H_{1})-genus(H_{2}).$

$\Box$

As a consequence, a large class of gems, including weak simple
crystallizations, is proved to realize the minimum genus of the central
surface among all quasi-trisections associated to $5$-colored graphs of
$G_{s}^{(4)}.$

###### Proposition 14

Let $\Gamma\in G_{s}^{(4)}$ be a gem of a compact $4$-manifold $M^{4}$ with
empty or connected boundary and let $\mathcal{T}(\Gamma,\varepsilon)$ be the
quasi-trisection of $M^{4}$ associated to $\Gamma$ and a cyclic permutation
$\varepsilon\in\mathcal{P}_{4}.$ Then,

  * (i)

if
$g_{\varepsilon_{1}\varepsilon_{3}\varepsilon_{4}}=m+(g_{\hat{\varepsilon}_{0}}+g_{\hat{\varepsilon}_{2}}-1)$
and
$g_{\varepsilon_{0}\varepsilon_{2}\varepsilon_{4}}=m+(g_{\hat{\varepsilon}_{1}}+g_{\hat{\varepsilon}_{3}}-1),$
with $\ m=rk(\pi_{1}(M^{4}))$, then
$genus(\Sigma(\mathcal{T}(\Gamma,\varepsilon)))\leq
genus(\Sigma(\mathcal{T}(\Gamma^{\prime},\varepsilon^{\prime})))$ for all
$\Gamma^{\prime}\in G_{s}^{(4)}$ such that
$|K(\Gamma)|\cong|K(\Gamma^{\prime})|$ and for all
$\varepsilon^{\prime}\in\mathcal{P}_{4};$

  * (ii)

if $\Gamma$ is weak semi-simple with respect to $\varepsilon$, then

$genus(\Sigma(\mathcal{T}(\Gamma,\varepsilon)))=\rho_{\varepsilon_{\hat{4}}}(\Gamma)=\frac{1}{2}(\rho_{\varepsilon}(\Gamma)-m)+(m-m^{\prime}),$

where $m^{\prime}=rk(\pi_{1}(\widehat{M}^{4}))$, and
$genus(\Sigma(\mathcal{T}(\Gamma,\varepsilon)))\leq
genus(\Sigma(\mathcal{T}(\Gamma^{\prime},\varepsilon^{\prime})))$ for all
$\Gamma^{\prime}\in G_{s}^{(4)}$ such that
$|K(\Gamma)|\cong|K(\Gamma^{\prime})|$ and for all
$\varepsilon^{\prime}\in\mathcal{P}_{4}.$

In particular, if $\Gamma$ is weak simple with respect to $\varepsilon$, then
the genus of $\Sigma(\mathcal{T}(\Gamma,\varepsilon)))$ is
$\frac{1}{2}\rho_{\varepsilon}(\Gamma)$ and it is minimal among all quasi-
trisections of the (simply-connected) $4$-manifold $M^{4}$ associated to
$5$-colored graphs of $G_{s}^{(4)}.$

_Proof._ If
$g_{\varepsilon_{1}\varepsilon_{3}\varepsilon_{4}}=m+(g_{\hat{\varepsilon}_{0}}+g_{\hat{\varepsilon}_{2}}-1)$
and
$g_{\varepsilon_{0}\varepsilon_{2}\varepsilon_{4}}=m+(g_{\hat{\varepsilon}_{1}}+g_{\hat{\varepsilon}_{3}}-1),$
then $genus(H_{1})=genus(H_{r})=m$ and $genus(H_{2})=genus(H_{g})=m$; hence,
Lemma 13 gives $\
genus(\Sigma(\mathcal{T}(\Gamma,\varepsilon))=\chi(N^{4})-2+2m.$

Now, if $(H^{\prime}_{0},H^{\prime}_{1},H^{\prime}_{2})$ is another quasi-
trisection of $M^{4}$ associated to an element of $G_{s}^{(4)}$, whose central
surface has genus $g^{\prime}$, we have
$g^{\prime}=\chi(N^{4})-2+genus(H^{\prime}_{1})+genus(H^{\prime}_{2})\geq
genus(\Sigma(\mathcal{T}(\Gamma,\varepsilon)),$ since obviously
$genus(H^{\prime}_{1})\geq m$ and $genus(H^{\prime}_{2})\geq m.$ This proves
statement (i).

Statement (ii) comes from statement (i) and Corollary 8.

Finally, if $\Gamma$ is weak simple with respect to $\varepsilon$, both
statements can be applied with $m=m^{\prime}=0.$

$\Box$

Trisections of closed standard simply-connected $4$-manifolds induced by
simple crystallizations are obtained in [38].

Now, we are able to prove Theorem 2, which ensures the existence of gem-
induced trisections for all $4$-manifolds with empty or connected boundary
associated to a (non-dotted) framed link, i.e. admitting a handle
decomposition consisting only of 0-, 2- (and possibly 4-) handles.

Proof of Theorem 2. Let $\Gamma(\mathcal{D},c)$ be the $5$-colored graph of
Proposition 7, which is constructed starting from $\mathcal{D}$ and represents
the singular manifold $\widehat{M^{4}(L,c)}.$ Note that
$\Gamma_{\hat{4}}(\mathcal{D},c)$ is connected and $4$ is only possible
singular color of $\Gamma(\mathcal{D},c)$; hence $\Gamma(\mathcal{D},c)\in
G_{s}^{(4)}.$

Let $(H_{b},H_{r},H_{g})$ be the decomposition of $\widehat{M^{4}(L,c)}$
constructed starting from $\Gamma(\mathcal{D},c)$, as in Proposition 11, where
$\varepsilon=(1,0,2,3,4)$.

Because of the structure of $\Gamma(\mathcal{D},c)$, pointed out in the proof
of Proposition 7, the edges of color $4$ of $\Gamma(\mathcal{D},c)$ double
those of color $1$ except “inside” the quadricolors, where they double one
$i$-colored edge for each $i\in\\{0,2,3\\}.$ Hence each square of the
quadrangulation of $\overline{H}(\Gamma,\varepsilon)$ has a free edge
(corresponding to a length two $\\{1,4\\}$-colored or $\\{i,4\\}$-colored
cycle of $\Gamma$). This ensures the collapse of
$\overline{H}(\Gamma,\varepsilon)$ to a graph.

Therefore, $\mathcal{T}(\Gamma,\varepsilon)$ is a gem-induced trisection of
$M^{4}(L,c)$, whose genus is
$\rho_{\varepsilon_{\hat{4}}}(\Gamma_{\hat{4}}(\mathcal{D},c))=m_{\alpha}.$

$\Box$

In the general case of 5-colored graphs representing compact 4-manifolds but
not arising from the construction described in Section 3, the existence of
gem-induced trisections is not always ensured. However, it is possible to
establish an easy combinatorial condition on the presentation of the
fundamental group of $|K(\Gamma)|$ with respect to the color
$\varepsilon_{4}=4$ (obtained from Proposition 5, with $\bar{R}_{4}=\emptyset$
since $g_{\hat{4}}=1$ holds for each $\Gamma\in G_{s}^{(4)}$), which implies
that the quasi-trisection $\mathcal{T}(\Gamma,\varepsilon)$ associated to such
a graph $\Gamma$ is actually a B-trisection, for each permutation
$\varepsilon\in\mathcal{P}_{4}.$

###### Proposition 15

Let $\Gamma\in G_{s}^{(4)}$ be a gem of a compact $4$-manifold $M^{4}$ with
empty or connected boundary, and let $<X,R>$ be the presentation of
$\pi(\widehat{M}^{4})$ with $X=\\{x_{1},\dots,x_{p}\\}$ in bijection with
4-colored edges of $\Gamma$ and $R=\\{r_{1},\dots,r_{q}\\}$ in bijection with
$\\{4,i\\}$-cycles of $\Gamma$, for each $i\in\Delta_{3}$.

With the notations of Proposition 11, the $2$-dimensional spine
$\overline{H}(\Gamma,\varepsilon)$ of $H_{rg}$ collapses to a graph if and
only if the presentation $<X,R>$ collapses to the trivial one, through a
finite sequence of moves of type:

$\displaystyle\text{if}\ \ r_{j}=x_{s}\ (j\in\mathbb{N}_{q},\
s\in\mathbb{N}_{p}),$ $\displaystyle\text{then delete}\ x_{s}\ \text{from the
generator set,}$ (9) and from each relation containing it, too.

Hence: if the presentation $<X,R>$ collapses to the trivial one through a
finite sequence of moves of type (9), then $\Gamma$ admits a gem-induced
trisection $\mathcal{T}(\Gamma,\varepsilon)$, for each
$\varepsilon\in\mathcal{P}_{4}$.

_Proof._ It is not difficult to check that each elementary collapse of a
square in $\overline{H}(\Gamma,\varepsilon)$ from one of its free edges
exactly corresponds to the “collapse” of a generator appearing in a length one
relator (i.e. a generator corresponding to a $4$-colored edge which is
involved in a length two $\\{i,4\\}$-colored cycle, for some
$i\in\Delta_{3}$): see Figure 12.

Hence, if $<X,R>$ collapses to the trivial one by a suitable sequence of the
above moves, $\overline{H}(\Gamma,\varepsilon)$ collapses to a $1$-dimensional
subcomplex, thus proving that $H_{rg}$ is a 3-dimensional handlebody, as
required to have $\mathcal{T}(\Gamma,\varepsilon)$ a gem-induced trisection,
for each $\varepsilon\in\mathcal{P}_{4}.$

$\Box$

###### Remark 6

It is easy to check that the condition of Proposition 15 holds, for example,
for all graphs arising from the construction of Section 3 (see the above proof
of Theorem 2), as well as for simple crystallizations of all standard simply-
connected $4$-manifolds (see [38, proof of Theorem 2]).
Note that in the closed case quasi trisections can be constructed as in
Proposition 11 for all permutations of $\Delta_{4}$ and that the group
presentation of Proposition 15 depends only on the choice of one color, which
is associated to three distinct tricolorings of the dual triangulation.
The implementation of the above condition into a C++ program allows us to
state that all (rigid) crystallizations of closed $4$-manifolds up to 18
vertices777Simple crystallizations of a closed simply-connected $4$-manifold
$M^{4}$ are characterized by having exactly $6\beta_{2}(M^{4})+2$ vertices
(see [16]). induce trisections for all tricolorings associated to at least
three colors. The same program was also applied to two simple crystallizations
obtained from the 16- and 17-vertices (possibly non PL-homeomorphic)
triangulations of the $K3$ surface constructed in [22] and [37] respectively;
both crystallizations turned out to induce trisections for all permutations of
$\Delta_{4}.$

##  6 Results on the G-trisection genus

The results of Section 5 may be applied to yield information about the
G-trisection genus (and hence, in the closed case, about the trisection
genus).

First of all, the G-trisection genus is proved to satisfy the following
properties:

###### Proposition 16

  * (i)

$g_{GT}(M^{4})=0\ \ \Leftrightarrow\ \ M^{4}\cong\mathbb{S}^{4}$.

  * (ii)

Let $M^{4}_{1},M^{4}_{2}$ be two compact $4$-manifolds with empty or connected
boundary, then:

$g_{GT}(M^{4}_{1}\\#M^{4}_{2})\leq g_{GT}(M^{4}_{1})+g_{GT}(M^{4}_{2})$ in
case at least one between $M^{4}_{1}$ and $M^{4}_{2}$ is closed;

$g_{GT}(M^{4}_{1}\,{}^{\partial}\\#M^{4}_{2})\leq
g_{GT}(M^{4}_{1})+g_{GT}(M^{4}_{2})$ in case both $M^{4}_{1}$ and $M^{4}_{2}$
have non-empty boundary.

  * (iii)

If $M^{4}$ has connected non-empty boundary,
$g_{GT}(M^{4})\geq\mathcal{H}(\partial M^{4}).$

_Proof._ If $g_{GT}(M^{4})=0$, there exists a $5$-colored graph $\Gamma\in
G_{s}^{(4)}$ representing $M^{4}$ and a permutation
$\varepsilon\in\mathcal{P}_{4}$ so that
$\mathcal{T}(\Gamma,\varepsilon)=(H_{0},H_{1},H_{2})$ is gem-induced
trisection, whose central surface
$\Sigma=\Sigma(\mathcal{T}(\Gamma,\varepsilon))$ is a 2-sphere. Since $\Sigma$
is a (genus zero) Heegaard surface for the 3-manifold $\partial H_{i}$, for
each $i\in\\{0,1,2\\}$, and since $\partial H_{0}$ is either $\mathbb{S}^{3}$
(in the closed case) or $\partial M^{4}$ (in the connected boundary case, with
the assumption that no spherical boundary is allowed), $M^{4}$ turns out to be
a closed $4-$manifold, admitting a trisection by three 4-disks.
$M^{4}\cong\mathbb{S}^{4}$ directly follows, proving statement (i).

As concerns statement (ii), let us consider two $5$-colored graphs
$\Gamma_{1},\Gamma_{2}\in G_{s}^{(4)}$ so that, for each $i\in\\{1,2\\}$,
$\Gamma_{i}$ represents $M^{4}_{i}$ and
$\mathcal{T}(\Gamma_{i},\varepsilon_{i})$, with
$\varepsilon_{i}\in\mathcal{P}_{4},$ is a gem-induced trisection of
$M^{4}_{i}$ such that
$g_{GT}(M^{4}_{i})=genus(\Sigma(\mathcal{T}(\Gamma_{i},\varepsilon_{i}))).$ If
necessary, perform a permutation on the color set of one of the two graphs, so
to have $\varepsilon_{1}=\varepsilon_{2}=\bar{\varepsilon}.$

Note that, given a sequence of collapsing moves by which the submanifold
$H_{rg}^{(i)}$ (made by the cubes in Figure 11) of $|K(\Gamma_{i})|$
($i\in\\{1,2\\}$) collapses to a graph, it is always possible to arrange it in
order to perform first the collapses of the 3-dimensional cubes. Let us choose
now a vertex $v_{1}\in V(\Gamma_{1})$ (resp. $v_{2}\in V(\Gamma_{2})$)
corresponding to a cube containing the 2-dimensional face of $H_{rg}^{(1)}$
that collapses last (resp. of $H_{rg}^{(2)}$ that collapses first) and let us
perform the graph connected sum between $\Gamma_{1}$ and $\Gamma_{2}$ with
respect to vertices $v_{1}$ and $v_{2}$ (i.e. the graph
$\Gamma_{1}\\#_{v_{1},v_{2}}\Gamma_{2}$ obtained by deleting $v_{i}$ in
$\Gamma_{i}$, $\forall i\in\\{1,2\\}$, and by welding the hanging edges of the
same color: see [32, Section 7] for details about the construction).
$\Gamma_{1}\\#_{v_{1},v_{2}}\Gamma_{2}$ is known to represent
$M^{4}_{1}\\#M^{4}_{2}$ in case at least one between $M^{4}_{1}$ and
$M^{4}_{2}$ is closed, and $M^{4}_{1}\,{}^{\partial}\\#M^{4}_{2}$ in case both
$M^{4}_{1}$ and $M^{4}_{2}$ have connected non-empty boundary.

It is not difficult to see that the quasi-trisection induced by the graph
$\Gamma_{1}\\#_{v_{1},v_{2}}\Gamma_{2}$ and permutation $\bar{\varepsilon}$ is
indeed a B-trisection, since the collapsing sequences of $H_{rg}^{(1)}$ and
$H_{rg}^{(2)}$ can be used to get a collapsing sequence of
$H_{rg}^{(1)}\\#H_{rg}^{(2)}$. Moreover, the central surface of this gem-
induced trisection is the connected sum of
$\Sigma(\mathcal{T}(\Gamma_{1},\bar{\varepsilon}))$ and
$\Sigma(\mathcal{T}(\Gamma_{2},\bar{\varepsilon})).$

In particular, note that, in case the collapsing sequence of $H_{rg}^{(i)}$ to
a graph realizes first the collapse of $H_{rg}^{(i)}$ to
$\overline{H}(\Gamma_{i},\bar{\varepsilon})$ and then the collapse of
$\overline{H}(\Gamma_{i},\bar{\varepsilon})$ to a graph, for each
$i\in\\{1,2\\}$, then the vertex $v_{1}$ (resp. $v_{2}$) is an end-point of
the $4$-colored edge of $\Gamma_{1}$ (resp. $\Gamma_{2}$) corresponding to the
last (resp. first) deleted generator in the sequence of moves on the
presentation of $\pi_{1}(M^{4}_{1})$ (resp. $\pi_{1}(M^{4}_{2})$) described in
Proposition 15.

With regard to statement (iii), Theorem 1 immediately implies, in the boundary
case, that, if $\Sigma(\mathcal{T}(\Gamma,\varepsilon))$ is the central
surface of a gem-induced trisection of the compact $4$-manifold $M^{4}$ (with
connected non-empty boundary), then

$genus(\Sigma(\mathcal{T}(\Gamma,\varepsilon)))=\rho_{\hat{4}}(\Gamma_{\hat{4}})\geq\mathcal{G}(\partial
M^{4})=\mathcal{H}(\partial M^{4}).$

$\Box$

As a consequence of the construction described in Section 3 and of Theorem 2,
we are able to compute the G-trisection genus of all $\mathbb{D}^{2}$-bundles
over $\mathbb{S}^{2}$ and of of all $4$-dimensional handlebodies, and to
estimate the G-trisection genus of any $4$-manifold associated to the framed
Hopf link.

###### Corollary 17

  * (a)

Let $\xi_{c}$ be the $\mathbb{D}^{2}$-bundle over $\mathbb{S}^{2}$ with Euler
class $c$; then $g_{GT}(\xi_{c})=1$, for each $c\in\mathbb{Z}.$

  * (b)

Let $L_{H}$ be the Hopf link; then $g_{GT}(M^{4}(L_{H},c))\leq 2$, for each
frame $c=(c_{1},c_{2})\in\mathbb{Z}\times\mathbb{Z}.$

  * (c)

Let $\mathbb{Y}^{4}_{h}$ be the genus $h$ $4$-dimensional handlebody; then
$g_{GT}(\mathbb{Y}^{4}_{h})=h$, for each $h\geq 1.$

_Proof._ Theorem 2 directly yields statement (b), as well as inequality
$g_{GT}(\xi_{c})\leq 1$, since $\xi_{c}$ is the $4$-manifold associated to the
$c$-framed unknot, for each $c\in\mathbb{Z}$.

On the other hand, $g_{GT}(\xi_{c})\geq 1$ follows from Proposition 16 in case
$c=\pm 1$ (since $\xi_{\pm 1}=\pm\mathbb{CP}^{2}$), and from Proposition
16(iii) for each $c\in\mathbb{Z}-\\{\pm 1\\},$ since $\partial\xi_{c}=L(c,1)$
has Heegaard genus 1 (where, as usual, $L(0,1)$ denotes
$\mathbb{S}^{1}\times\mathbb{S}^{2}$).

With regard to statement (c), let us consider the $5$-colored graph $\Gamma$
in Figure 13, which is known to represent the genus one handlebody ([12]), and
let us consider $\varepsilon=(0,1,2,3,4).$ It is not difficult to see that the
quasi-trisection $\mathcal{T}(\Gamma,\varepsilon)$ of $\mathbb{Y}^{4}_{1}$ is
indeed a gem-induced trisection of genus one, since for each
$i,j\in\\{0,1,2\\},H_{ij}=H_{i}\cap H_{j}$ is a genus one $3$-dimensional
handlebody. On the other hand, by performing a graph connected sum on $h$
copies of the above graph $\Gamma$, it is easy to obtain a gem of the genus
$h$ handlebody (see Figure 14 for the case $h=2$, and [12] for details about
the described gem of $\mathbb{Y}^{4}_{h}$), which gives rise to a gem-induced
trisection of genus $h$ of $\mathbb{Y}^{4}_{h}$, according to Proposition
16(ii). Hence, $g_{GT}(\mathbb{Y}^{4}_{h})\leq h$ trivially follows.

Moreover, the minimality of this value for the genus of a central surface of a
gem-induced trisection of $\mathbb{Y}^{4}_{h}$ is guaranteed by Proposition
16(iii), since $\mathcal{H}(\partial\mathbb{Y}^{4}_{h})=h$.

$\Box$

###### Remark 7

Corollary 17(a) proves that gem-induced trisection genus is not finite-to-one
in the boundary case, exactly as it happens for regular genus (see [12]).
Moreover, the two invariants share also the open problem about their
finiteness-to-one, if a fixed (possibly empty) boundary is assumed. The
problem is obviously relevant in order to search for invariants able to
distinguish different PL structures on the same TOP $4$-manifold. And, in this
context, simple crystallizations are not useful, since both their G-trisection
genus and their regular genus are determined by the TOP-structure.

Figure 13: $5$-colored graph representing $\mathbb{Y}^{4}_{1}$

Figure 14: $5$-colored graph representing $\mathbb{Y}^{4}_{2}$

###### Remark 8

It is pointed out in [25] that, if $(H_{0},H_{1},H_{2})$ is a trisection of a
closed $4$-manifold $M^{4}$ with central surface $\Sigma$, then
$g(\Sigma)\geq\beta_{1}(M^{4})+\beta_{2}(M^{4}).$ Hence, as a particular case
of Corollary 17(a) and (b), we obtain that the trisection genus of
$\mathbb{CP}^{2}$ is realized by a gem-induced trisection and equals $1$, as
well as the trisection genus of both $\mathbb{S}^{2}\times\mathbb{S}^{2}$ and
$\mathbb{S}^{2}\tilde{\times}\mathbb{S}^{2}$ is realized by a gem-induced
trisection and equals $2$; moreover, they all coincide with the second Betti
number of the involved $4$-manifolds. This coincidence - which Proposition
16(ii) enables to extend to any connected sum of $\mathbb{CP}^{2}$ (possibly
with reversed orientation) and/or $\mathbb{S}^{2}\times\mathbb{S}^{2}$ \- has
been already proved in [38] for all standard simply-connected closed
$4$-manifolds, via gem-induced trisections of the $K3-$surface.

A more general formula extends the above coincidence to a wider class of
compact $4$-manifolds with empty or connected boundary.

###### Proposition 18

Let $M^{4}$ be a compact $4$-manifold with empty or connected boundary which
admits a gem-induced trisection. Then,

$g_{GT}(M^{4})\
\geq\beta_{2}(M^{4})+\beta_{1}(M^{4})+2\left(m-\beta_{1}(M^{4})\right),\ \
\text{with}\ \ m=rk(\pi_{1}(M^{4})),$

and the equality holds if and only if a $5$-colored graph $\Gamma\in
G_{s}^{(4)}$ representing $M^{4}$ and a cyclic permutation
$\varepsilon\in\mathcal{P}_{4}$ exist, so that
$g_{\varepsilon_{1}\varepsilon_{3}\varepsilon_{4}}=m+(g_{\hat{\varepsilon}_{0}}+g_{\hat{\varepsilon}_{2}}-1)$,
$g_{\varepsilon_{0}\varepsilon_{2}\varepsilon_{4}}=m+(g_{\hat{\varepsilon}_{1}}+g_{\hat{\varepsilon}_{3}}-1)$
and $\mathcal{T}(\Gamma,\varepsilon)$ is a gem-induced trisection.

In particular, $g_{GT}(M^{4})\geq\beta_{2}(M^{4})$ for each simply-connected
such manifold, and the equality holds if and only if a a $5$-colored graph
$\Gamma\in G_{s}^{(4)}$ representing $M^{4}$ and a cyclic permutation
$\varepsilon\in\mathcal{P}_{4}$ exist, so that
$g_{\varepsilon_{1}\varepsilon_{3}\varepsilon_{4}}=g_{\hat{\varepsilon}_{0}}+g_{\hat{\varepsilon}_{2}}-1$,
$g_{\varepsilon_{0}\varepsilon_{2}\varepsilon_{4}}=g_{\hat{\varepsilon}_{1}}+g_{\hat{\varepsilon}_{3}}-1$
and $\mathcal{T}(\Gamma,\varepsilon)$ is a gem-induced trisection.

_Proof._ Let $\Gamma\in G_{s}^{(4)}$ be a gem of $M^{4}$ and
$\mathcal{T}(\Gamma,\varepsilon)=(H_{0},H_{1},H_{2})$ an associated gem-
induced trisection, with $\varepsilon\in\mathcal{P}_{4}$. By Lemma 13, setting
$N^{4}=\widehat{M^{4}}$, we have

$genus(\Sigma(\mathcal{T}(\Gamma,\varepsilon))=\chi(N^{4})-2+genus(H_{1})+genus(H_{2})\geq\chi(N^{4})-2+2m$

(recall that $genus(H_{i})\geq rk(\pi_{1}(M^{4}))=m$ trivially holds for each
$i\in\\{1,2\\}$), and the equality
$genus(\Sigma(\mathcal{T}(\Gamma,\varepsilon))=\chi(N^{4})-2+2m$ holds if and
only if $genus(H_{i})=m$ for each $i\in\\{1,2\\}.$

Now, since $\mathcal{T}(\Gamma,\varepsilon)$ is a gem-induced trisection,
$N^{4}$ turns out to be simply-connected (recall Remark 4); hence,
$genus(\Sigma(\mathcal{T}(\Gamma,\varepsilon))\geq\
\beta_{2}(N^{4})-\beta_{3}(N^{4})+2m,$ and the equality holds if and only if
$genus(H_{i})=m$ for each $i\in\\{1,2\\}.$

On the other hand, the exact sequence of relative homology of the pair
$(N,C)$, where $C$ is the cone over $\partial M^{4}$, together with the fact
that the inclusion $(M,\partial M)\hookrightarrow(N,C)$ induces an isomorphism
in homology, imply

$H_{k}(N^{4})\cong H_{k}(N^{4},C)\cong H_{k}(M^{4},\partial M^{4})\cong
H^{4-k}(M^{4})\qquad\forall k\in\\{2,3\\}$

where the last isomorphism comes from Poincaré-Lefschetz duality.

Hence $\beta_{2}(N^{4})=\beta_{2}(M^{4}),\ \beta_{3}(N^{4})=\beta_{1}(M^{4})$
and, therefore,

$genus(\Sigma(\mathcal{T}(\Gamma,\varepsilon))\geq\beta_{2}(M^{4})-\beta_{1}(M^{4})+2m=\beta_{2}(M^{4})+\beta_{1}(M^{4})+2\left(m-\beta_{1}(M^{4})\right);$

moreover, in virtue of formula (1), we obtain that the equality holds if and
only if
$genus(H_{1})=m=g_{\varepsilon_{1}\varepsilon_{3}\varepsilon_{4}}-(g_{\hat{\varepsilon}_{0}}+g_{\hat{\varepsilon}_{2}}-1)$
and
$genus(H_{2})=m=g_{\varepsilon_{0}\varepsilon_{2}\varepsilon_{4}}-(g_{\hat{\varepsilon}_{1}}+g_{\hat{\varepsilon}_{3}}-1).$

$\Box$

###### Corollary 19

Let $M^{4}$ be a compact $4$-manifold with empty or connected boundary which
admits a weak semi-simple crystallization $\Gamma$ giving rise to a gem-
induced trisection $\mathcal{T}(\Gamma,\varepsilon)$. Then,

$g_{GT}(M^{4})\ =\
\beta_{2}(M^{4})+\beta_{1}(M^{4})+2\left(m-\beta_{1}(M^{4})\right)=\frac{1}{2}(\rho_{\varepsilon}(\Gamma)+m),$

where $m=rk(\pi_{1}(M^{4})).$

In particular,

$g_{GT}(M^{4})=\beta_{2}(M^{4})=\frac{1}{2}\rho_{\varepsilon}(\Gamma)$

for each compact (simply-connected) $4$-manifold $M^{4}$, with empty or
connected boundary, which admits a weak simple crystallization $\Gamma$ giving
rise to a gem-induced trisection $\mathcal{T}(\Gamma,\varepsilon)$.

_Proof._ It is sufficient to note that
$g_{\varepsilon_{1}\varepsilon_{3}\varepsilon_{4}}=g_{\varepsilon_{0}\varepsilon_{2}\varepsilon_{4}}=m+1$
(resp.
$g_{\varepsilon_{1}\varepsilon_{3}\varepsilon_{4}}=g_{\varepsilon_{0}\varepsilon_{2}\varepsilon_{4}}=1$)
is always satisfied by weak semi-simple (resp. weak simple) crystallizations
with respect to $\varepsilon$; hence, a weak semi-simple (resp. weak simple)
crystallization $\Gamma$ such that $\mathcal{T}(\Gamma,\varepsilon)$ is a gem-
induced trisection always realizes
$g_{GT}(M^{4})=\beta_{2}(M^{4})+\beta_{1}(M^{4})+2\left(m-\beta_{1}(M^{4})\right)$
(resp. $g_{GT}(M^{4})=\beta_{2}(M^{4})$) in virtue of Proposition 18. Finally,
Proposition 11, together with Corollary 8, enables to compute $g_{GT}(M^{4})$
directly from the above weak semi-simple (resp. weak simple) crystallization
$\Gamma$:

$g_{GT}(M^{4})=genus(\Sigma(\mathcal{T}(\Gamma,\varepsilon))=\rho_{\varepsilon_{\hat{4}}}(\Gamma)=\frac{1}{2}(\rho_{\varepsilon}(\Gamma)+m)$
$\text{(resp.:}\ \ \
g_{GT}(M^{4})=genus(\Sigma(\mathcal{T}(\Gamma,\varepsilon))=\rho_{\varepsilon_{\hat{4}}}(\Gamma)=\frac{1}{2}\rho_{\varepsilon}(\Gamma)).$

$\Box$

Acknowledgements:  This work was supported by GNSAGA of INDAM and by the
University of Modena and Reggio Emilia, project: “Discrete Methods in
Combinatorial Geometry and Geometric Topology”

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