The complexity of orientable graph manifolds

TL;DR

This paper establishes an upper bound for the Matveev complexity of all closed connected orientable prime graph manifolds, confirming its sharpness for a large catalog of such manifolds.

Contribution

It provides a universal upper bound for the complexity of orientable prime graph manifolds and verifies its sharpness across a comprehensive catalog.

Findings

Upper bound for Matveev complexity established

Bound is sharp for all cataloged manifolds

Enhances understanding of manifold complexity measures

Abstract

We give an upper bound for the Matveev complexity of the whole class of closed connected orientable prime graph manifolds that is sharp for all 14502 graph manifolds of the Recognizer catalogue (available at \texttt{http://matlas.math.csu.ru/?page=search}).

Figures

Equations

$$H=\left(\begin{array}[]{cc}0&1\\ 1&0\end{array}\right)\quad\textup{and}\quad U=\left(\begin{array}[]{cc}1&0\\ 1&1\end{array}\right),$$

$$A^{\prime}=U^{h}AU^{k}=\left(\begin{array}[]{cc}\alpha+k\beta&\beta\\ \gamma+h\alpha+k\delta+kh\beta&\delta+h\beta\end{array}\right)$$

$$\left(\begin{array}[]{ccc}a&c&e\\ b&d&f\end{array}\right),$$

$$c_A=\text{min}\left\{d(A{\Delta }_{-},{\Delta }_{-}),d(A{\Delta }_{-},{\Delta }_{+}),d(A{\Delta }_{+},{\Delta }_{-}),d(A{\Delta }_{+},{\Delta }_{+})\right\}.$$

$$\frac{a}{b}=a_1+\frac{1}{\ddots +\frac{1}{a_{k-1}+\frac{1}{a_k}}},$$

$$0<\frac{\alpha }{\gamma }<\frac{\beta +\alpha }{\gamma +\delta }<\frac{\beta }{\delta }<\frac{\beta -\alpha }{\delta -\gamma }.$$

$$\frac{\alpha }{\gamma }<\frac{\alpha }{\gamma }+\frac{1}{\delta \gamma }=\frac{\alpha \delta +1}{\delta \gamma }=\frac{\beta }{\delta }.$$

$$0<\frac{\alpha }{\gamma }=\frac{\frac{\alpha }{\gamma }+\frac{\alpha }{\delta }}{1+\frac{\gamma }{\delta }}<\frac{\frac{\beta }{\delta }+\frac{\alpha }{\delta }}{1+\frac{\gamma }{\delta }}=\frac{\beta +\alpha }{\delta +\gamma }=\frac{\frac{\beta }{\gamma }+\frac{\alpha }{\gamma }}{\frac{\delta }{\gamma }+1}<\frac{\frac{\beta }{\gamma }+\frac{\beta }{\delta }}{\frac{\delta }{\gamma }+1}=\frac{\beta }{\delta }=\frac{\frac{\beta }{\gamma }-\frac{\beta }{\delta }}{\frac{\delta }{\gamma }-1}<\frac{\frac{\beta }{\gamma }-\frac{\alpha }{\gamma }}{\frac{\delta }{\gamma }-1}=\frac{\beta -\alpha }{\delta -\gamma },$$

$$P^{\prime }=\bar{s}\left(B_0^{\prime }\setminus \left({\cup }_{l=1}^{|b|}\text{int}(p_0({\Phi }_l^{\prime }))\right)\right)\cup D^{\prime }\cup W^{\prime }\cup \partial {\Phi }_0^{\prime }{\cup }_{l=1}^{|b|}\left(\partial {\Phi }_l^{\prime }{\cup }_{{\psi }_l}\left(D^2\times \{\ast \}\right)\right)$$

$$f_{m,M}(b)=\left\{\begin{array}[]{ll}m-b&\textup{ if }b<m\\ 0&\textup{ if }m\leq b\leq M\\ b-M&\textup{ if }b>M\end{array}\right.,$$

$$c(M)$$

$$\sum _{j\in J_T}(S({\beta }_j/{\delta }_j)-1)+\sum _{i\in I}\left(3(d_i+r_i+2h_i-2)+\sum _{k=1}^{r_i}(S(p_k/q_k)-2)+f_{m_i,M_i}(b_i)\right)$$

$$c(M)$$

$${\Psi }_T=\{\psi :J_T^{\prime }\to \{+,-\}\},{\Psi }_T^{\prime }=\{{\psi }^{\prime }:J^{\prime }\setminus J_T^{\prime }\to \{++,+,+-,-+,-,--\}\}$$

$$c(M)$$

$$\sum _{j\in J_T^{\prime \prime }}S({\beta }_j/{\delta }_j)-1)+\sum _{i\in I}\left(3(d_i+r_i+2h_i-2)+\sum _{k=1}^{r_i}(S(p_k/q_k)-2)\right)+\sum _{i\in I}{f}_{m_i,M_i}(b_i)$$

Full text

MSC2010: Primary 57M27; Secondary 57N10, 57R22.

1 Introduction

Graph manifolds have been introduced and classified by Waldhausen in [15] and [16]. They are defined as compact 3-manifolds obtained by gluing Seifert fibre spaces along toric boundary components; so they can be described using labeled digraphs, as it will be explained in the next section.

Matveev in [12] (see also [13]) introduced the notion of complexity for compact 3-dimensional manifolds, as a way to measure how “complicated” a manifold is. Indeed, for closed irreducible and $\mathbb{P}^{2}$-irreducible manifolds the complexity coincides with the minimum number of tetrahedra needed to construct the manifold, with the only exceptions of $S^{3}$, $\mathbb{RP}^{3}$ and $L(3,1)$, all having complexity zero. Moreover, complexity is additive under connected sum and it is finite-to-one in the closed irreducible case. The last property has been used in order to construct a census of manifolds according to increasing complexity: for the orientable case, up to complexity 12, in the Recognizer catalogue111See http://matlas.math.csu.ru/?page=search. and for the non-orientable case, up to complexity 11, in the Regina catalogue222See https://regina-normal.github.io. .

Upper bounds for the complexity of infinite families of 3-manifolds are given in [11] for lens spaces, in [10] for closed orientable Seifert fibre spaces and for orientable torus bundles over the circle, in [6] for orientable Sefert fibre space with boundary and in [2] for non-orientable compact Seifert fibre spaces. All the previous upper bounds are sharp for manifolds contained in the above cited catalogues. Furthermore, in [7] and [8] it has been proved that the upper bound given in [11] is sharp for two infinite families of lens spaces. Very little is known for the complexity of graph manifolds: in [4] and [5] upper bounds are given only for the case of graph manifolds obtained by gluing along the boundary two or three Seifert fibre spaces with disk base space and at most two exceptional fibres.

The main goal of this paper is to furnish a potentially sharp upper bound for the complexity of all closed connected orientable prime graph manifolds different from Seifert fibre spaces and orientable torus bundles over the circle. It is worth noting that the upper bounds given in Theorems 1, 2 and 3 are sharp for all 14502 manifolds of this type included in the Recognizer catalogue.

The organization of the paper is the following. In Section 2 we recall the definition of graph manifolds and their representation via decomposition graphs, while in Section 3 we state our results.

We end this section by recalling the definition of complexity and skeletons.

A polyhedron $P$ is said to be almost simple if the link of each point $x\in P$ can be embedded into $K_{4}$, the complete graph with four vertices. In particular, the polyhedron is called simple if the link is homeomorphic to either a circle, or a circle with a diameter, or $K_{4}$. A true vertex of an (almost) simple polyhedron $P$ is a point $x\in P$ whose link is homeomorphic to $K_{4}$. A spine of a closed connected 3-manifold is a polyhedron $P$ embedded in $M$ such that $M\setminus P\cong B^{3}$, being $B^{3}$ an open 3-ball. The complexity $c(M)$ of $M$ is the minimum number of true vertices among all almost simple spines of $M$.

We will construct a spine for a given graph manifold by gluing skeletons of its Seifert pieces. Consider a compact connected 3-manifold $M$ whose boundary either is empty or consists of tori. Following [9] and [10], a skeleton of $M$ is a sub-polyhedron $P$ of $M$ such that (i) $P\cup\partial M$ is simple, (ii) $M\setminus(P\cup\partial M)\cong B^{3}$, (iii) for any component $T^{2}$ of $\partial M$ the intersection $T^{2}\cap P$ is a non-trivial theta graph333A non-trivial theta graph $\theta$ on a torus $T^{2}$ is a subset of $T^{2}$ homeomorphic to the theta graph (i.e., the graph with 2 vertices and 3 edges joining them), such that $T^{2}\setminus\theta$ is an open disk.. Note that if $M$ is closed then $P$ is a spine of $M$. Given two manifolds $M_{1}$ and $M_{2}$ as above with non-empty boundary, let $P_{i}$ be a skeleton of $M_{i}$, for $i=1,2$. Take two components $T_{1}\subseteq\partial M_{1}$ and $T_{2}\subseteq\partial M_{2}$ such that $P_{i}\cap T_{i}=\theta_{i}$ and consider an homeomorphism $\varphi:(T_{1},\theta_{1})\to(T_{2},\theta_{2})$. Then $P_{1}\cup_{\varphi}P_{2}$ is a skeleton for $M_{1}\cup_{\varphi}M_{2}$: we call this operation, as well as the manifold $M_{1}\cup_{\varphi}M_{2}$, an assembling of $M_{1}$ and $M_{2}$.

2 Graph manifolds and their combinatorial description

Let us start by fixing some notations for Seifert fibre spaces. We will consider only orientable compact connected Seifert fibre spaces with non-empty boundary, described as $S=(g,d,(p_{1},q_{1}),\ldots,(p_{r},q_{r}),b)$ where: $g\in\mathbb{Z}$ coincides with the genus of the base space if it is orientable and with the opposite if it is non-orientable, $d>0$ is the number of boundary components of $S$, $(p_{j},q_{j})$ are lexicographically ordered pairs of coprime integers such that $0<q_{j}<p_{j}$, for $j=1,\ldots,r$, describing the type of the exceptional fibres of $S$ and $b\in\mathbb{Z}$ can be considered as a (non-exceptional) fibre of type $(1,b)$.

Up to fibre-preserving homeomorphism, we can assume (see [3]) that the Seifert pieces appearing in a graph manifold belong to the set $\mathcal{S}$ of the oriented compact connected Seifert fibre spaces with non-empty boundary that are different from fibred solid tori and from the fibred spaces $S^{1}\times S^{1}\times I$ and $N\tilde{\times}S^{1}$ (i.e., the orientable circle bundle over the Moebius strip $N$, which will be considered with the alternative Seifert fibre structure $(0,1,(2,1),(2,1),b)$).

A Seifert fibre space $S=(g,d,(p_{1},q_{1}),\ldots,(p_{r},q_{r}),b)\in\mathcal{S}$, with base space $B=p(S)$, is equipped with coordinate systems on the toric boundary components, as follows. Let $\widehat{S}$ be the closed Seifert fibre space $\widehat{S}=(g,0,(p_{1},q_{1}),\ldots,(p_{r},q_{r}),b)$, with base space $\widehat{B}=\widehat{p}(\widehat{S})$. Moreover, let $\widehat{s}:\widehat{B}^{\prime}\to\widehat{S}^{\prime}$ be the section naturally associated to $\widehat{S}$, up to isotopy, where $\widehat{B}^{\prime}=\widehat{p}(\widehat{S}^{\prime})$ and $\widehat{S}^{\prime}$ is obtained from $\widehat{S}$ by removing $r+1$ open fibred solid tori (with disjoint closures) which are regular neighborhoods of the exceptional fibres of $\widehat{S}$ and of the fibre of type $(1,b)$. The manifold $S$ is obtained from $\widehat{S}$ by removing $d$ open fibred solid tori which are regular neighborhoods of $d$ regular fibres of type $(1,0)$ of $\widehat{S}$. Therefore, $p=\widehat{p}_{|_{S}}$ and $\partial(B)\subset\widehat{B}^{\prime}$. The section $\widehat{s}$ induces a coordinate system (i.e., a positive basis of the first homology group) $(\mu_{h},\lambda_{h})$ on the $h$-th toric boundary component $T_{h}$ of $S$, for each $h=1,\ldots,d$, such that: $\mu_{h}$ is the image under $\widehat{s}$ of a boundary component $c^{\prime}_{h}$ of $B$ and $\lambda_{h}$ is a fibre of $S$. Moreover, $\mu_{h}$ and $\lambda_{h}$ are oriented in such a way that their intersection index is 1, with the orientation of $T_{h}$ induced by the one of $S$. When $B$ is orientable, we require that any $\mu_{h}$ is oriented according to the orientation of $c^{\prime}_{h}$ induced by a fixed orientation on $B$.

Consider a finite connected non-trivial digraph $G=(V,E,\iota)$, where $V=\{v_{i}\mid i\in I\}$ is the set of vertices and $E=\{e_{j}\mid j\in J\}$ is the set of oriented edges of $G$, with incidence structure $\iota:E\to V\times V$ given by $\iota(e_{j})=(v_{i^{\prime}_{j}},v_{i^{\prime\prime}_{j}})$. Let

[TABLE]

then associate:

• •

to each vertex $v_{i}\in V$ having degree $d_{i}$ a Seifert fibre space $S_{i}=(g_{i},d_{i},(p_{1},q_{1}),\ldots,(p_{r_{i}},q_{r_{i}}),b_{i})\in\mathcal{S}$ (i.e., the degree of $v_{i}$ is equal to the number of components of $\partial S_{i}$);

• •

to each edge $e_{j}\in E$ a matrix A_{j}=\left(\begin{array}[]{cc}\alpha_{j}&\beta_{j}\\ \gamma_{j}&\delta_{j}\end{array}\right)\in\textup{GL}^{-}_{2}(\mathbb{Z}) such that $\beta_{j}\neq 0$ and $0\leq\epsilon_{j}\alpha_{j},\epsilon_{j}\delta_{j}<|\beta_{j}|$, where $\epsilon_{j}=\beta_{j}/|\beta_{j}|$. We call normalized a matrix of $\textup{GL}_{2}^{-}(\mathbb{Z})$ satisfying these conditions. Moreover:

• (i)

$A_{j}\neq\pm H$ when either $S_{i^{\prime}_{j}}$ or $S_{i^{\prime\prime}_{j}}$ is the space $(0,1,(2,1),(2,1),-1)$;

• (ii)

when $|V|=2$, $|E|=1$ and $S_{1}=(0,1,(2,1),(2,1),b_{1})$, $S_{2}=(0,1,(2,1),(2,1),b_{2})$,

• (a)

if $A=\pm H$ then $(b_{1},b_{2})\neq(0,0),(-2,-2)$;

• (b)

if A=\pm\left(\begin{array}[]{cc}1&\beta\\ 1&\beta-1\end{array}\right), with $\beta>1$, then $(b_{1},b_{2})\neq(-1,-2)$;

• (c)

if A=\pm\left(\begin{array}[]{cc}\beta-1&\beta\\ 1&1\end{array}\right), with $\beta>1$, then $(b_{1},b_{2})\neq(0,-1)$.

The graph manifold $M$ associated to the above data is obtained by gluing, for each edge $e_{j}\in E$ with $\iota(e_{j})=(v_{i^{\prime}_{j}},v_{i^{\prime\prime}_{j}})$, a toric boundary component of $S_{i^{\prime}_{j}}$ with one of $S_{i^{\prime\prime}_{j}}$, using the homeomorphism represented by $A_{j}$ with respect to the fixed coordinate systems on the tori.444If $\beta_{j}=0$ for some $A_{j}$, then the gluing map sends a fiber of $S_{i^{\prime}_{j}}$ into a fiber of $S_{i^{\prime\prime}_{j}}$. This implies that the decomposition of $M$ in Seifert pieces is not minimal with respect to the number of cutting tori. For the same reason conditions (i) and (ii) hold (see [3, p. 279]). Observe that we can assume $\beta_{j}>0$, for any $j$ such that $e_{j}$ belongs to a fixed spanning tree of $G$. Clearly, $M$ is a closed, orientable and connected graph manifold. On the other hand, each closed connected orientable prime graph manifold different from a Seifert fibre space and an orientable torus bundle over the circle can be obtained in this way (see [3, §11]). We will call $G$ a decomposition graph of $M$.

If $G^{\prime}=(V,E^{\prime})$ is a spanning subgraph of a decomposition graph $G$ we denote by $M_{G^{\prime}}$ the graph manifold (with boundary if $G^{\prime}\neq G$) obtained by performing only the attachments corresponding to the elements of $E^{\prime}$.

Remark 1**.**

There is no restriction in assuming that all matrices associated to the edges of a decomposition graph are normalized: this is because of the following two operations that do not change the resulting graph manifold (see [3, §11] and [14]):

replacement of the matrix $A_{j}$ with $A_{j}U^{k}$ and of the parameter $b_{i^{\prime}_{j}}$ of the Seifert space $S_{i^{\prime}_{j}}$ with the parameter $b_{i^{\prime}_{j}}+k$;

• 2)

replacement of the matrix $A_{j}$ with $U^{k}A_{j}$ and of the parameter $b_{i^{\prime\prime}_{j}}$ of the Seifert space $S_{i^{\prime\prime}_{j}}$ with $b_{i^{\prime\prime}_{j}}-k$.

Indeed, given a matrix A=\left(\begin{array}[]{cc}\alpha&\beta\\ \gamma&\delta\end{array}\right)\in\textup{GL}^{-}_{2}(\mathbb{Z}), let $k=-\lfloor\frac{\alpha}{\beta}\rfloor$ and $h=-\lfloor\frac{\delta}{\beta}\rfloor$ where $\lfloor x\rfloor$ denotes the floor of $x$. Then the matrix

[TABLE]

is normalized. Note that for a normalized matrix A=\left(\begin{array}[]{cc}\alpha&\beta\\ \gamma&\delta\end{array}\right) the following properties hold:

• •

$\beta\gamma>0$;

• •

if $\beta=\pm 1$ then $A=\beta H$;

• •

if $A\neq\pm H$ then $\beta/\delta>0$.

Moreover $A\in\textup{GL}^{-}_{2}(\mathbb{Z})$ is normalized if and only if $-A$ is normalized.

2.1 Theta graphs and Farey triangulation

Consider the upper half-plane model of the hyperbolic plane $\mathbb{H}^{2}$ and let $\mathbb{F}$ be the ideal Farey triangulation (see [1]). The vertices of $\mathbb{F}$ coincide with the points of $\mathbb{Q}\cup\{\infty\}\subset\mathbb{R}\cup\{\infty\}=\partial\mathbb{H}^{2}$ and the edges of $\mathbb{F}$ are geodesics in $\mathbb{H}^{2}$ with endpoints the pairs $a/b$, $c/d$ such that $ad-bc=\pm 1$, with $\pm 1/0=\infty$. Let $\Delta_{\frac{a}{b},\frac{c}{d},\frac{e}{f}}$ be the triangle of the Farey triangulation with vertices $a/b,c/d,e/f\in\mathbb{Q}\cup\{\infty\}$ and set $\Delta_{+}=\Delta_{\infty,0,1}$, $\Delta_{-}=\Delta_{\infty,0,-1}$.

Let $T^{2}$ be a torus, it is a well-known fact that the vertex set of $\mathbb{F}$ is in bijection with the set of slopes (i.e., isotopy classes of non-contractible simple closed curves) on $T^{2}$ via $a/b\leftrightarrow a\mu+b\lambda$, where $(\mu,\lambda)$ is a fixed basis of $H_{1}(T^{2})$. This bijection induces a bijection between the set of triangles of the Farey triangulation and the set $\Theta(T^{2})$ of non-trivial theta graphs on $T^{2}$, considered up to isotopy. Indeed, given $\theta\in\Theta(T^{2})$, consider the three slopes $l_{1},l_{2},l_{3}$ on $T^{2}$ formed by the pairs of edges of $\theta$. The triangle associated to $\theta$ is $\Delta_{l_{1},l_{2},l_{3}}$. Note that this bijection is well defined since the intersection index of $l_{i}$ and $l_{j}$, with $i\neq j$, is always $\pm 1$.

The graph $\mathbb{F}^{*}$ dual to $\mathbb{F}$ is an infinite tree. Given two triangles $\Delta$ and $\Delta^{\prime}$ in $\mathbb{F}$ the distance $d(\Delta,\Delta^{\prime})$ between them is the number of edges of the unique simple path joining the vertices $v_{\Delta}$ and $v_{\Delta^{\prime}}$ corresponding to $\Delta$ and $\Delta^{\prime}$ in $\mathbb{F}^{*}$, respectively. Given two theta graphs $\theta,\theta^{\prime}\in\Theta(T^{2})$ it is possible to pass from one to the other by a sequence of flip moves (see Figure 1): the distance on the set of triangles of the Farey triangulation induces a distance on $\Theta(T^{2})$ such that $d(\theta,\theta^{\prime})$ turns out to be the minimal number of flips necessary to pass from $\theta$ to $\theta^{\prime}$ (see [10]).

The group $\textup{GL}_{2}(\mathbb{Z})$ acts on $\mathbb{H}^{2}$ as isometries and $\mathbb{F}$ is invariant under this action: given a triangle $\Delta_{\frac{a}{b},\frac{c}{d},\frac{e}{f}}\in\mathbb{F}$ if we associate to it the matrix

[TABLE]

then the group $\textup{GL}_{2}(\mathbb{Z})$ acts on the set of triangles of the Farey triangulation by left multiplication.

The complexity $c_{A}$ of a matrix $A\in\textup{GL}_{2}(\mathbb{Z})$ is defined as

[TABLE]

Now we state a result about the complexity of normalized matrices. Let $S:\mathbb{Q}^{+}\to\mathbb{N}$ defined by $S(a/b)=a_{1}+\cdots+a_{k}$, where

[TABLE]

is the expansion of the positive rational number $a/b$ as a continued fraction, with $a_{1},\ldots,a_{k}>0$.

Lemma 1**.**

Let A=\left(\begin{array}[]{cc}\alpha&\beta\\ \gamma&\delta\end{array}\right)\in\textup{GL}^{-}_{2}(\mathbb{Z}) be a normalized matrix.

• •

If $A=\pm H$ then $c_{A}=d(A\Delta_{-},\Delta_{-})=d(A\Delta_{+},\Delta_{+})=0$.

• •

If $A\neq\pm H$ then $c_{A}=d(A\Delta_{-},\Delta_{+})=S(\beta/\delta)-1$.

Proof.

The first statement is straightforward since $\pm H\Delta_{\pm}=\Delta_{\pm}$. To prove the second one let A=\left(\begin{array}[]{cc}\alpha&\beta\\ \gamma&\delta\end{array}\right)\neq\pm H and so $|\beta|>1$ (see Remark 1). Denote by $\mathcal{D}_{\beta/\delta}$ the set of triangles of the Farey triangulation having a vertex in $\beta/\delta$. By [9, Lemma 4.3] we have $\textup{min}\{d(\Delta,\Delta_{+})\mid\Delta\in\mathcal{D}_{\beta/\delta}\}=S(\beta/\delta)-1$. If $A$ is normalized then:

[TABLE]

Indeed, since $\alpha\delta-\beta\gamma=-1$, we have:

[TABLE]

So,

[TABLE]

where we suppose $\delta-\gamma\neq 0$, otherwise the last inequality is straightforward. Clearly $A\Delta_{-}=\Delta_{\frac{\alpha}{\gamma},\frac{\beta}{\delta},\frac{\beta-\alpha}{\gamma-\delta}}$, $A\Delta_{+}=\Delta_{\frac{\alpha}{\gamma},\frac{\beta}{\delta},\frac{\beta+\alpha}{\gamma+\delta}}$ and the relative position of the triangles is represented in Figure 2, where for convenience we use the Poincaré disk model of $\mathbb{H}^{2}$. All triangles of $\mathcal{D}_{\beta/\delta}$ different from $A\Delta_{-}$ and $A\Delta_{+}$ are contained in the two hyperbolic half-planes depicted in gray. As a consequence, we have $\textup{min}\{d(\Delta,\Delta_{+})\mid\Delta\in\mathcal{D}_{\beta/\delta}\}=d\left(A\Delta_{-},\Delta_{+}\right)$. Since the path in $\mathbb{F}^{*}$ going from $v_{A\Delta_{+}}$ to $v_{\Delta_{-}}$ contains $v_{A\Delta_{-}}$ and $v_{\Delta_{+}}$, we have $c_{A}=d(A\Delta_{-},\Delta_{+})=S(\beta/\delta)-1$. ∎

We end this section by introducing a notation to distinguish edges corresponding to $\pm H$ from the others. Given a decomposition graph $(V,E,\iota)$ with $V=\{v_{i}\mid i\in I\}$ and $E=\{e_{j}\mid j\in J\}$, let $E^{\prime}=\{e_{j}\mid A_{j}=\pm H\}$, $E^{\prime\prime}=E-E^{\prime}$ and $J^{\prime}=\{j\in J\mid e_{j}\in E^{\prime}\}$, $J^{\prime\prime}=J-J^{\prime}$. Moreover, set $d^{+}_{i}=|\{j\in J^{\prime\prime}\mid i^{\prime}_{j}=i\}|$ and $d^{-}_{i}=|\{j\in J^{\prime\prime}\mid i^{\prime\prime}_{j}=i\}|$. Observe that, if $d^{0}_{i}=|\{j\in J^{\prime}\mid i^{\prime}_{j}=i\}|+|\{j\in J^{\prime}\mid i^{\prime\prime}_{j}=i\}|$ then $d_{i}=d^{+}_{i}+d^{-}_{i}+d^{0}_{i}$.

3 Complexity upper bounds

In this section we provide an upper bound for the complexity of graph manifolds. Before stating the general result (Theorem 3), we deal with two special classes of graph manifolds: the first one (Theorem 1) is the class of manifolds having decomposition graphs in which all the edges are associated to matrices different from $\pm H$, that is $J^{\prime}=\emptyset$, while the second one (Theorem 2) concerns manifolds whose decomposition graphs admit a spanning tree containing all the edges associated to matrices of type $\pm H$.

In all cases we obtain a spine for a graph manifold starting from skeletons of its Seifert pieces. The construction of these skeletons is essentially the one described in [2], specialized to our case (i.e., orientable Seifert manifolds) and adapted to take care of the fact that the boundary components of the Seifert pieces will be glued together to obtain a closed graph manifold. Anyway, for the sake of the reader we recall how to construct a skeleton for the Seifert pieces. The construction and the number of true vertices of the resulting skeletons depend on some choices: we will discuss in the proof of the theorems how to fix them in order to minimize the number of true vertices of the spine.

Skeletons of Seifert pieces Let $S=(g,d,(p_{1},q_{1}),\ldots,(p_{r},q_{r}),b)\in\mathcal{S}$. Denote with $S_{0}=(g,d,(p_{1},q_{1}),\ldots,(p_{r},q_{r}),0)$ and let $S_{0}^{\prime}=(g,d+r+1,0)$ be the space obtained from $S_{0}$ by removing $r+1$ open fibred solid tori (with disjoint closures) which are regular neighborhoods of the exceptional fibres of $S_{0}$ and of a regular fibre of type $(1,0)$ contained in $\textup{int}(S_{0})$. Then $S_{0}\setminus\textup{int}(S^{\prime}_{0})=\Phi_{0}^{\prime}\sqcup\Phi_{1}\sqcup\cdots\sqcup\Phi_{r}$, where $\Phi_{k}$ (resp. $\Phi_{0}^{\prime}$) is a closed solid torus having the $k$-th exceptional fibre (resp. a regular fibre) as core. Let $p_{0}:S_{0}\to B_{0}$ and $p:S\to B$ be the projection maps and set $s=d+r$. Note that if $g\geq 0$ (resp. $g<0$) then $B^{\prime}_{0}=p_{0}(S^{\prime}_{0})$ is a disk with $2g+s$ orientable (resp. $-g$ non-orientable and $s$ orientable) handles attached. We set $h=2g$ if $g\geq 0$ and $h=-g$ if $g<0$.

Let $D=p_{0}(\Phi_{0}^{\prime})$ and let $A_{0}$ be the union of the disjoint arcs properly embedded in $B^{\prime}_{0}$ depicted by thick lines in Figure 3. Then $A_{0}$ is non-empty and is composed by $h$ edges with both endpoints in $\partial D$ and $s$ edges with an endpoint in $\partial D$ and the other one in a different component of $\partial B^{\prime}_{0}$. By construction, $B^{\prime}_{0}\setminus\left(A_{0}\cup\partial B^{\prime}_{0}\right)$ is homeomorphic to an open disk and the number of points of $A_{0}$ belonging to $\partial D$ is at least three, since the conditions on the class $\mathcal{S}$ ensure that $s+2h>2$.

Let $s_{0}^{\prime}:B^{\prime}_{0}\to S^{\prime}_{0}$ be a section of $p_{0}$ restricted to $S^{\prime}_{0}$. If $b\neq 0$, it is convenient to replace the fibre of type $(1,b)$ with $|b|$ fibres of type $(1,\textup{sign}(b))$. In this way the manifold $S$ is obtained from $S_{0}$ by removing $|b|$ open trivially fibred solid tori (with disjoint closures) $\textup{int}(\Phi^{\prime}_{1}),\ldots,\textup{int}(\Phi^{\prime}_{|b|})$, each being a fiber-neighborhood of regular fibres $\phi_{1},\ldots\phi_{|b|}$ contained in $\textup{int}(S_{0})$, and by attaching back $|b|$ solid tori $D^{2}\times S^{1}$ via homeomorphisms $\psi_{l}:\partial(D^{2}\times S^{1})\to\partial\Phi^{\prime}_{l}$ such that $\psi_{l}(\partial D^{2}\times\{*\})$ is a curve of type $(1,\textrm{sign}(b))$ on $\partial\Phi^{\prime}_{l}$, with respect to a positive basis $(\mu_{l},\lambda_{l})$ of $H_{1}(\partial\Phi_{l}^{\prime})$, where $\mu_{l}=s_{0}^{\prime}(p_{0}(\partial\Phi^{\prime}_{l}))$ and $\lambda_{l}$ is the fibre over a point $*^{\prime}\in p_{0}(\partial\Phi^{\prime}_{l})$, for $l=1,\ldots,|b|$. Referring to Figure 4, it is convenient to take the fibre $\phi_{l}$ corresponding to an internal point $Q_{l}$ of $A_{0}$ and suppose that $p_{0}(\Phi^{\prime}_{l})$ is a “small” disk intersecting the component $\delta_{l}$ of $A_{0}$ containing $Q_{l}$ in an interval and being disjoint from $\partial B^{\prime}_{0}$ and from the other components of $A_{0}$. In this way $\delta_{l}\setminus\textrm{int}(p_{0}(\Phi^{\prime}_{l}))$ is the disjoint union of two arcs $\delta^{\prime}_{l}$ and $\delta^{\prime\prime}_{l}$. Let $A=A_{0}\setminus\cup_{l=1}^{|b|}\textrm{int}(p_{0}(\Phi^{\prime}_{l}))$ and note that $p$ and $p_{0}$ coincide on $S_{0}\setminus\cup_{l=1}^{|b|}\textrm{int}(\Phi^{\prime}_{l})$.

Let $\bar{s}:\left(B^{\prime}_{0}\setminus\left(\cup_{l=1}^{|b|}\textup{int}(p_{0}(\Phi^{\prime}_{l}))\right)\right)\cup D\to S$ be a section of $p_{|_{\textup{Im}(\bar{s})}}$ and consider the polyhedron $P=\textup{Im}(\bar{s})\cup p^{-1}(A)\cup\partial\Phi^{\prime}_{0}\cup_{l=1}^{|b|}\left(\partial\Phi_{l}^{\prime}\cup_{\psi_{l}}\left(D^{2}\times\{*\}\right)\right)\subset S$. As represented in the central picture of Figure 5, the set $\textup{int}(\bar{s}(A))$ is a collection of quadruple lines in the polyhedron (the link of each point is homeomorphic to a graph with two vertices and four edges connecting them), and a similar phenomenon occurs for $\bar{s}(\partial D\setminus A)$. Therefore we change the polyhedron $P$ performing “small” shifts by moving in parallel the disk $\bar{s}(D)$ along the fibration and the components of $p^{-1}(A)$ as depicted in the left and right pictures of Figure 5. It is convenient to think the shifts of $p^{-1}(A)$ as performed on the components of $A$. Moreover, the shifts on $\delta^{\prime}_{l}$ and $\delta_{l}^{\prime\prime}$ can be chosen independently.

As shown by the pictures, the shift of any component of $p^{-1}(A)$ may be performed in two different ways that are not usually equivalent in term of complexity of the final spine. On the contrary, the two possible parallel shifts for $\bar{s}(D)$ are equivalent as it is evident from Figure 6, which represents the torus $\partial\Phi^{\prime}_{0}$. Let

[TABLE]

be the polyhedron obtained from $P$ after the shifts, where $D^{\prime}$ and $W^{\prime}$ are the results of the shifts of $\bar{s}(D)$ and $p^{-1}(A)$, respectively.

It is easy to see that $P^{\prime}\cup\partial S\cup_{k=1}^{r}\partial\Phi_{k}$ is simple, $P^{\prime}$ intersects each component of $\partial S$ and each torus $\partial\Phi_{k}$ in a non-trivial theta graph and the manifold $S\setminus\left(P^{\prime}\cup\partial S\cup_{k=1}^{r}\Phi_{k}\right)$ is the disjoint union of $|b|+2$ open balls. So in order to obtain a skeleton $P^{\prime\prime}$ for $S\setminus\left(\cup_{k=1}^{r}\textup{int}(\Phi_{k})\right)$ it is enough to remove a suitable open 2-cell from the torus $T_{0}=\partial\Phi^{\prime}_{0}$ and one from each torus $T_{l}=\partial\Phi^{\prime}_{l}$, for $l=1,\ldots,|b|$, connecting in this way the balls.

The graph $\Gamma_{l}=T_{l}\cap\left(\bar{s}\left(B^{\prime}_{0}\setminus\textup{int}(p_{0}(\Phi^{\prime}_{l}))\right)\cup W^{\prime}\cup\psi_{l}(\partial D^{2}\times\{*\})\right)$ (resp. $\Gamma_{0}=T_{0}\cap\left(\bar{s}\left(B^{\prime}_{0}\right)\cup D^{\prime}\cup W^{\prime}\right)$) is cellularly embedded in $T_{l}$ (resp. $T_{0}$) and its vertices with degree greater than 2 are true vertices of $P^{\prime}\cup\partial S\cup_{k=1}^{r}\partial\Phi_{k}$: we will remove the region $R_{l}$ (resp. $R_{0}$) of $T_{l}\setminus\Gamma_{l}$ (resp. $T_{0}\setminus\Gamma_{0}$) having in the boundary the greatest number of vertices of $\Gamma_{l}$, for $l=1,\ldots,|b|$ (resp. $\Gamma_{0}$).

Referring to Figure 6, the graph $\Gamma_{0}$ is composed by two horizontal parallel loops $\xi=\partial(\bar{s}(D))$ and $\xi^{\prime}=\partial D^{\prime}$, and an arc with both endpoints on $\xi$ for each boundary point of $A$ belonging to $\partial D$. Changing the shift of a component of $A$ has the same effect as performing a symmetry along $\xi$ of the correspondent arc(s). A region of $T_{0}\setminus\Gamma_{0}$ has 4 or 6 vertices when the non-horizontal arcs belonging to its boundary are not parallel or 5 vertices otherwise. So, except the case in which all the arcs are parallel there is always a region with 6 vertices.

When $b\neq 0$, the graph $\Gamma_{l}$, for $l=1,\ldots,|b|$, is depicted in Figure 7 (resp. Figure 8) for a fibre of type $(1,1)$ (resp. $(1,-1)$), just labeled by $+$ (resp. $-$) inside the disk. If we take for $\delta^{\prime}_{l}$ and $\delta^{\prime\prime}_{l}$ the shifts induced by that of $\delta_{l}$, then we can choose as region $R_{l}$ the gray one, containing in its boundary all vertices of $\Gamma_{l}$ belonging to $\partial\Phi_{l}^{\prime}$ except one (the thick points in the first two pictures). On the contrary, if one of the two shifts is changed as in the third draw of Figures 7 and 8, then $R_{l}$ can be chosen containing in its boundary all the vertices of $\Gamma_{l}$ belonging to $\partial\Phi_{l}^{\prime}$.

We remark that changing the shift of a component of $A$ changes the intersection between the corresponding element of $W^{\prime}$ and $\partial S$ (which is a non-trivial theta graph) by a flip move (see Figure 1). We denote with $P^{\prime\prime}$ the skeleton obtained by removing the regions $R_{0}$ and $R_{l}$ from $P^{\prime}$, for $l=1,\ldots,|b|$.

In order to construct a skeleton for $\Phi_{k}$, for $k=1,\ldots,r$, consider the skeleton $P_{F}$ depicted in Figure 9: it is a skeleton for $T^{2}\times[0,1]$ with one true vertex and such that $\theta_{0}=P_{F}\cap(T^{2}\times\{0\})$ (the graph in the upper face) and $\theta_{1}=P_{F}\cap(T^{2}\times\{1\})$ (the graph in the bottom face) are two theta graphs differing for a flip move. Denote with $\Theta_{p_{k}/q_{k}}$ the subset of $\Theta(T^{2})$, consisting of the theta graphs containing the slope corresponding to $p_{k}/q_{k}\in\mathbb{Q}\cup\{\infty\}$. Let $\theta_{p_{k}/q_{k}}$ be the theta graph in $\Theta_{p_{k}/q_{k}}$ that is closest to $\theta_{+}$. The skeleton $X_{k}$ for $\Phi_{k}$ is obtained by assembling several skeletons of type $P_{F}$ connecting the theta graph $P^{\prime\prime}\cap\Phi_{k}$ to a theta graph which is one step closer to $\theta_{+}$ than $\theta_{p_{j}/q_{j}}$,with respect to the distance on $\Theta(T^{2})$ (see [6]). The number of the required flips is either $S(p_{j},q_{j})-2$ or $S(p_{j},q_{j})-1$ depending on the shift chosen for the corresponding component of $A$ used in the construction of the skeleton $P^{\prime\prime}$. We call the shift regular in the first case and singular in the second one (see Figure 10).

The skeleton $P_{S}$ of $S$ is obtained by assembling $P^{\prime\prime}$ with $X_{k}$, via the identity, for $k=1,\ldots,r$.

We end this section by introducing a function that will be useful in the following: let $f_{m,M}:\mathbb{Z}\to\mathbb{N}$ be defined by

[TABLE]

for $m,M\in\mathbb{Z}$, $m<M$, $m\leq 1$ and $M\geq-1$ (see the graph in Figure 11).

Moreover, let ${\mathcal{T}}_{G}=\{T=(V,E_{T})\mid T\textrm{ is a spanning tree of }G\}$ and for $T\in\mathcal{T}_{G}$ set $J_{T}=\{j\in J\mid e_{j}\in E_{T}\}$, $E^{\prime}_{T}=E_{T}\cap E^{\prime}$, $E^{\prime\prime}_{T}=E_{T}\setminus E^{\prime}_{T}$, $J^{\prime}_{T}=\{j\in J\mid e_{j}\in E^{\prime}_{T}\}$, and $J^{\prime\prime}_{T}=J_{T}\setminus J^{\prime}_{T}$.

3.1 The case $\mathbf{J^{\prime}=\emptyset}$

Before stating our first result, we need to discuss how to fix the choices in the construction of the skeleton $P_{S}$ previously described, when the Seifert fibre space $S=(g,d,(p_{1},q_{1}),\ldots,(p_{r},q_{r}),b)$ is a piece of a graph manifold having all gluing matrices different from $\pm H$. According to the notations introduced at the end of Section 2, we have $d=d^{+}+d^{-}$ since $J^{\prime}=\emptyset$.

Remark 2**.**

Let $\theta_{+}$ and $\theta_{-}$ be the theta graphs corresponding, respectively, to $\Delta_{+}$ and $\Delta_{-}$ in the Farey triangulation. The intersection of each boundary component of $S$ with the skeleton $P_{S}$ is either $\theta_{+}$ or $\theta_{-}$, depending whether the shift of the corresponding component $\delta$ of $A$ has been chosen as depicted in the left or right part of Figure 12, respectively.

We always choose these shifts such that exactly $d^{+}$ (resp. $d^{-}$) components have $\theta_{-}$ (resp. $\theta_{+}$) as intersection with $P_{S}$. Suppose that $m\leq b\leq M$, where $m=-r-h-d^{-}+1$ and $M=h+d^{+}-1$. If $b\leq-1$ we can choose $p_{0}(\Phi^{\prime}_{1}),\ldots,p_{0}(\Phi_{|b|}^{\prime})$ as $|b|$ disks between those marked with $-$ in Figure 13. In this way: (i) we can remove a region from $T_{0}$ containing 6 vertices of $\Gamma_{0}$, (ii) we can remove from $T_{l}$ a region $R_{l}$ containing in its boundary all the vertices of the graph $\Gamma_{l}$ (as in the third draw of Figure 8), for each $l=1,\ldots,|b|$, and (iii) we can take all regular shifts in the skeletons $X_{k}$ corresponding to the exceptional fibres, for $k=1,\ldots,r$. If $b=0$ we do not have to remove any regular neighborhood $\Phi^{\prime}_{l}$ of regular fibres but still (i) and (iii) hold. An analogue situation happens if $b\geq 1$, but in this case in order to satisfy (i), (ii) and (iii) the fibres of type $(1,1)$ correspond to some of the disks marked with $+$ in Figure 14. As a result, when $m\leq b\leq M$ the polyhedron $P_{S}$ has $3(d+r+2h-2)+\sum_{k=1}^{r}(S(p_{k}/q_{k})-2)$ true vertices.

If $b<m\leq 0$ (resp. $b>M\geq 0$) (i) and (iii) hold and there are exactly $m-b$ (resp. $b-M$) tori in which we remove a region $R_{l}$ containing in its boundary all the vertices of $\Gamma_{l}$ except one (see the first two pictures of Figure 7 and 8). Finally, if either $b<m=1$ or $b>M=-1$ then (i) does not hold so we remove a region from $T_{0}$ containing 5 vertices of $\Gamma_{0}$. Moreover, there are exactly $|b|$ torus in which we remove a region $R_{l}$ containing all the vertices of $\Gamma_{l}$ except one and (iii) holds. Summing up, if $b<m$ (resp. $b>M$) then the number of true vertices of $P_{S}$ increases by $m-b$ (resp. $b-M$) with respect to the case $m\leq b\leq M$.

As a consequence $P_{S}$ has $3(d+r+2h-2)+\sum_{k=1}^{r}(S(p_{k}/q_{k})-2)+f_{m,M}(b)$ true vertices.

Theorem 1**.**

Let $M$ be a graph manifold associated to a decomposition graph $G=(V,E,\iota)$ such that $A_{j}\neq\pm H$ for any $j\in J$ (i.e., $J^{\prime}=\emptyset$), then

[TABLE]

where $m_{i}=-r_{i}-h_{i}-d^{-}_{i}+1$ and $M_{i}=h_{i}+d^{+}_{i}-1$.

Proof.

Let $T=(V,E_{T})$ be a spanning tree of $G$ and consider the graph manifold $M_{T}$ (with boundary if $T\neq G$). We will construct a skeleton $P_{M_{T}}$ for $M_{T}$ by assembling skeletons of its Seifert pieces (constructed as above) with skeletons of thickened tori corresponding to edges of $T$. More precisely, for each $j\in J_{T}$, let $T^{\prime}_{j}\subset\partial S_{i^{\prime}_{j}}$ and $T^{\prime\prime}_{j}\subset\partial S_{i^{\prime\prime}_{j}}$ be the boundaries attached by $A_{j}$. We construct a skeleton $P_{A_{j}}$ for $T^{\prime\prime}_{j}\times I$, and assemble $P_{S_{i^{\prime}_{j}}}$ with $P_{A_{j}}$ using the map $A_{j}:T^{\prime}_{j}\to T^{\prime\prime}_{j}=T^{\prime\prime}_{j}\times\{0\}$ and $P_{A_{j}}$ with $P_{S_{i^{\prime\prime}_{j}}}$ with the identification $T^{\prime\prime}_{j}\times\{1\}=T^{\prime\prime}_{j}$.

Given $j\in J_{T}$, let $\theta_{j}$ be the theta graph corresponding to $A_{j}\Delta_{-}$. We construct the skeleton $P_{A_{j}}$ by assembling flip blocks (see Figure 9) so that (i) $P_{A_{j}}\cap(T^{\prime\prime}_{j}\times\{0\})=\theta_{j}$ and (ii) $P_{A_{j}}\cap(T^{\prime\prime}_{j}\times\{1\})=\theta_{+}$. By Lemma 1 we have $c_{A_{j}}=d(A_{j}\Delta_{-},\Delta_{+})=S(\beta_{j}/\delta_{j})-1$, so the number of flip blocks required to construct $P_{A_{j}}$, as well as the number of true vertices of $P_{A_{j}}$, is $S(\beta_{j}/\delta_{j})-1$.

By Remark 2, we can construct the skeleton $P_{S_{i}}$ having $3(d_{i}+r_{i}+2h_{i}-2)+\sum_{k=1}^{r_{i}}(S(p_{k}/q_{k})-2)+f_{m_{i},M_{i}}(b_{i})$ true vertices. So $P_{M_{T}}$ has

[TABLE]

true vertices.

For each $j\in J\setminus J_{T}$, the matrix $A_{j}$ identifies two boundary components of $M_{T}$. Denote with $M_{T\cup e_{j}}$ the resulting manifold. Construct $P_{A_{j}}$ such that (i) $P_{A_{j}}\cap(T^{\prime\prime}_{j}\times\{0\})=\theta_{j}$ and (ii) $P_{A_{j}}\cap(T^{\prime\prime}_{j}\times\{1\})=\theta^{\prime}$, where $\theta^{\prime}$ is at distance one from $\theta_{+}$ and is closer to $\theta_{j}$ than $\theta_{+}$. The graphs $\theta^{\prime}$ and $\theta_{+}$ differ for a flip, since they correspond to adjacent triangles, and therefore, as shown in Figure 15, we have $i(\theta^{\prime},\theta_{+})=2$, where $i(\cdot,\cdot)$ denotes the geometric intersection (i.e., the minimum number of intersection points up to isotopy). Consider the polyhedron $P_{M_{T}}\cup P_{A_{j}}\cup(T_{j}^{\prime\prime}\times\{1\})$: it is a skeleton for $P_{M_{T}\cup e_{j}}$. Since $P_{A_{j}}$ consists of $S(\beta_{j}/\delta_{j})-2$ flip blocks and the graph $\theta_{+}\cup\theta^{\prime}$ has 6 vertices of degree greater than 2, the new polyhedron has $5+S(\beta_{j}/\delta_{j})-1$ true vertices more than $P_{M_{T}}$. By repeating this construction for any $j\in J\setminus J_{T}$ and observing that $|J\setminus J_{T}|=|E\setminus E_{T}|=|E|-|V|+1$, we get the statement. ∎

3.2 An intermediate case

Now we deal with the class of graph manifolds having a decomposition graph with a spanning tree containing all the edges associated to the matrices $\pm H$. This case seems to be rather technical, but it is quite interesting since, up to complexity 12, about $99\%$ of prime graph manifolds belong to this class (they are exactly 14346 out of 14502).

Before stating our result we introduce some notations. If $J^{\prime}\neq\emptyset$ let $\Psi=\left\{\psi:J^{\prime}\to\{+,-\}\right\}$ and set:555Clearly, $d_{i}=d^{+}_{i}+d^{-}_{i}+d^{+}_{i,\psi}+d^{-}_{i,\psi}$.

• •

$d^{+}_{i,\psi}=\left|\{j\in J^{\prime}\mid i^{\prime}_{j}=i,\psi(j)=+\}\right|+\left|\{j\in J^{\prime}\mid i^{\prime\prime}_{j}=i,\psi(j)=+\}\right|$;

• •

$d^{-}_{i,\psi}=\left|\{j\in J^{\prime}\mid i^{\prime}_{j}=i,\psi(j)=-\}\right|+\left|\{j\in J^{\prime}\mid i^{\prime\prime}_{j}=i,\psi(j)=-\}\right|$.

Theorem 2**.**

Let $M$ be a graph manifold associated to a decomposition graph $G=(V,E,\iota)$. If there exists a spanning tree $T\in\mathcal{T}_{G}$ of $G$ such that $A_{j}\neq\pm H$ for any $j\in J\setminus J_{T}$, then

[TABLE]

where $m_{i}=-r_{i}-h_{i}-d^{-}_{i}-d^{-}_{i,\psi}+1$ and $M_{i}=h_{i}+d^{+}_{i}+d^{+}_{i,\psi}-1$.

Proof.

As in the proof of Theorem 1, we start by constructing a skeleton $P_{M_{T}}$ for the graph manifold $M_{T}$ (with boundary if $T\neq G$). By Lemma 1 we have $0=c_{\pm H}=d(\pm H\Delta_{-},\Delta_{-})=d(\pm H\Delta_{+},\Delta_{+})$, so whenever $j\in J^{\prime}_{T}=J^{\prime}$, no flip block is required in $P_{A_{j}}$ and we can assemble directly $P_{S_{i^{\prime}_{j}}}$ with $P_{S_{i^{\prime\prime}_{j}}}$. So, if $T_{j}^{\prime}$ and $T_{j}^{\prime\prime}$ denote, respectively, the boundary components of $S_{i^{\prime}_{j}}$ and $S_{i^{\prime\prime}_{j}}$ glued by $A_{j}$, we require that either (i) $P_{S_{i^{\prime}_{j}}}\cap T_{j}^{\prime}=\theta_{+}$ and $P_{S_{i^{\prime\prime}_{j}}}\cap T_{j}^{\prime\prime}=\theta_{+}$ or (ii) $P_{S_{i^{\prime}_{j}}}\cap T_{j}^{\prime}=\theta_{-}$ and $P_{S_{i^{\prime\prime}_{j}}}\cap T_{j}^{\prime\prime}=\theta_{-}$. In order to take care of these two possibilities we use a function $\psi:J^{\prime}\to\{+,-\}$ of the edges of $E^{\prime}$. If the shifts in the construction of $P_{S_{i^{\prime}_{j}}}$ and $P_{S_{i^{\prime\prime}_{j}}}$ are chosen so that (i) holds (resp. (ii) holds) set $\psi(j)=-$ (resp. $\psi(j)=+$). Following the construction of Remark 2, with $d^{+}=d^{+}_{i}+d^{+}_{i,\psi}$ and $d^{-}=d^{-}_{i}+d^{-}_{i,\psi}$, we obtain a skeleton $P_{S_{i}}$ with $3(d_{i}+r_{i}+2h_{i}-2)+\sum_{k=1}^{r_{i}}(S(p_{k}/q_{k})-2)+f_{m_{i},M_{i}}(b_{i})$ true vertices. As a consequence, the minimum number of true vertices of the skeleton $P_{M_{T}}$ of $M_{T}$ is $\sum_{j\in J^{\prime\prime}_{T}}(S(\beta_{j}/\delta_{j})-1)+\sum_{i\in I}\left(3(d_{i}+r_{i}+2h_{i}-2)+\sum_{k=1}^{r_{i}}(S(p_{k}/q_{k})-2)\right)+$ $+\min_{\psi\in\Psi}\left\{\sum_{i\in I}f_{m_{i},M_{i}}(b_{i})\right\}$.

Since all the matrices associated to the edges $e_{j}\notin E_{T}$ are different from $\pm H$, starting from $P_{M_{T}}$ we can construct a spine for $M$ as described in the proof of Theorem 1. This concludes the proof. ∎

3.3 The general case

For the general case we need to consider a certain set of spanning trees of $G$.

Let $\phi:{\mathcal{T}}_{G}\to\mathbb{N}$ be the function defined by $\phi(T)=|(E-E_{T})\cap E^{\prime}|$. A spanning tree $T\in{\mathcal{T}}_{G}$ is called optimal if $\phi(T)\leq\phi(T^{\prime})$ for any $T^{\prime}\in{\mathcal{T}}_{G}$. Denote the set of the optimal spanning trees of $G$ with ${\mathcal{O}}_{G}$ and set $\Phi(G)=\min\{\phi(T)\mid T\in{\mathcal{T}}_{G}\}$ (i.e., $\Phi(G)=\phi(T)$, for any $T\in{\mathcal{O}}_{G}$). Finally, let

[TABLE]

and set:

• •

$d^{+}_{i,\psi,T}=\left|\{j\in J^{\prime}_{T}\mid i^{\prime}_{j}=i,\psi(j)=+\}\right|+\left|\{j\in J^{\prime}_{T}\mid i^{\prime\prime}_{j}=i,\psi(j)=+\}\right|$;

• •

$d^{-}_{i,\psi,T}=\left|\{j\in J^{\prime}_{T}\mid i^{\prime}_{j}=i,\psi(j)=-\}\right|+\left|\{j\in J^{\prime}_{T}\mid i^{\prime\prime}_{j}=i,\psi(j)=-\}\right|$;

• •

$d^{+}_{i,\psi^{\prime},T}=2\left|\{j\in J^{\prime}-J^{\prime}_{T}\mid i^{\prime}_{j}=i,\psi^{\prime}(j)=++\}\right|+\left|\{j\in J^{\prime}-J^{\prime}_{T}\mid i^{\prime}_{j}=i,\psi^{\prime}(j)=+\}\right|++\left|\{j\in J^{\prime}-J^{\prime}_{T}\mid i^{\prime\prime}_{j}=i,\psi^{\prime}(j)=++\}\right|+2\left|\{j\in J^{\prime}-J^{\prime}_{T}\mid i^{\prime\prime}_{j}=i,\psi^{\prime}(j)=+\}\right|++\left|\{j\in J^{\prime}-J^{\prime}_{T}\mid i^{\prime}_{j}=i,\psi^{\prime}(j)=+-\}\right|+\left|\{j\in J^{\prime}-J^{\prime}_{T}\mid i^{\prime\prime}_{j}=i,\psi^{\prime}(j)=-+\}\right|$;

• •

$d^{-}_{i,\psi^{\prime},T}=2\left|\{j\in J^{\prime}-J^{\prime}_{T}\mid i^{\prime}_{j}=i,\psi^{\prime}(j)=--\}\right|+\left|\{j\in J^{\prime}-J^{\prime}_{T}\mid i^{\prime}_{j}=i,\psi^{\prime}(j)=-\}\right|++\left|\{j\in J^{\prime}-J^{\prime}_{T}\mid i^{\prime\prime}_{j}=i,\psi^{\prime}(j)=--\}\right|+2\left|\{j\in J^{\prime}-J^{\prime}_{T}\mid i^{\prime\prime}_{j}=i,\psi^{\prime}(j)=-\}\right|++\left|\{j\in J^{\prime}-J^{\prime}_{T}\mid i^{\prime}_{j}=i,\psi^{\prime}(j)=-+\}\right|+\left|\{j\in J^{\prime}-J^{\prime}_{T}\mid i^{\prime\prime}_{j}=i,\psi^{\prime}(j)=+-\}\right|$.

We are now ready to state the general case.

Theorem 3**.**

Let $M$ be a graph manifold associated to a decomposition graph $G=(V,E,\iota)$. Then

[TABLE]

where $m_{i}=-r_{i}-h_{i}-d^{-}_{i}-d^{-}_{i,\psi,T}-d^{-}_{i,\psi^{\prime},T}+1$ and $M_{i}=h_{i}+d^{+}_{i}+d^{+}_{i,\psi,T}+d^{+}_{i,\psi^{\prime},T}-1$.

Proof.

Let $T=(V,E_{T})\in\mathcal{O}_{G}$. Given $\psi\in\Psi_{T}$, we construct a skeleton $P_{M_{T}}$ for $M_{T}$ as described in the proof of Theorem 2. If $j\in J^{\prime}\setminus J^{\prime}_{T}$, then $A_{j}=\pm H$ and it glues together two toric boundary components $T_{j}^{\prime}\subset\partial S_{i^{\prime}_{j}}$ and $T_{j}^{\prime\prime}\subset\partial S_{i^{\prime\prime}_{j}}$ of $M_{T}$.

Let $(\theta^{\prime}_{j},\theta^{\prime\prime}_{j})=(P_{S_{i^{\prime}_{j}}}\cap T^{\prime}_{j},P_{S_{i^{\prime\prime}_{j}}}\cap T^{\prime\prime}_{j})$. Since $A_{j}\Delta_{\pm}=\Delta_{\pm}$ and $i(\theta_{+},\theta_{+})=i(\theta_{-},\theta_{+})=i(\theta_{-},\theta_{-})=2$, we have to consider all possible cases of $\theta_{j}^{\prime},\theta_{j}^{\prime\prime}\in\{\theta_{+},\theta_{-}\}$. If $(\theta_{j}^{\prime},\theta_{j}^{\prime\prime})=(\theta_{+},\theta_{-})$ (resp. $(\theta_{j}^{\prime},\theta_{j}^{\prime\prime})=(\theta_{-},\theta_{+})$) then we define $\psi^{\prime}(j)=-+$ (resp. $\psi^{\prime}(j)=+-$). If $(\theta_{j}^{\prime},\theta_{j}^{\prime\prime})=(\theta_{+},\theta_{+})$ or $(\theta_{j}^{\prime},\theta_{j}^{\prime\prime})=(\theta_{-},\theta_{-})$ we can use Remark 1 in order to obtain a better estimate for $c(M)$. Indeed, if we replace the matrix $A_{j}$ with $A_{j}U^{\mp 1}$ (resp. $U^{\pm}A_{j}$), then $b_{i^{\prime}_{j}}$ (resp. $b_{i^{\prime\prime}_{j}}$) is replaced with $b_{i^{\prime}_{j}}\mp 1$ (resp. $b_{i^{\prime\prime}_{j}}\mp 1$), but anyway $i(\theta^{\prime}_{-},\theta_{-})=i(\theta^{\prime\prime}_{-},\theta_{-})=i(\theta^{\prime}_{+},\theta_{+})=i(\theta^{\prime\prime}_{+},\theta_{+})=2$, where $\theta^{\prime}_{-},\theta^{\prime\prime}_{-},\theta^{\prime}_{+},\theta_{+}^{\prime\prime}$ are the theta graphs associated to the triangles $A_{j}U^{-1}\Delta_{-},UA_{j}\Delta_{-},A_{j}U\Delta_{+},U^{-1}A_{j}\Delta_{+}$, respectively. Indeed, $A_{j}U^{-1}\Delta_{-},UA_{j}\Delta_{-}$ are adjacent to $\Delta_{-}$ and $A_{j}U\Delta_{+},U^{-1}A_{j}\Delta_{+}$ are adjacent to $\Delta_{+}$. More precisely, if we take $(\theta_{j}^{\prime},\theta_{j}^{\prime\prime})=(\theta_{-},\theta_{-})$ and replace $A_{j}$ with $A_{j}U^{-1}$ (resp. $UA_{j}$) we set $\psi^{\prime}(j)=++$ (resp. $\psi^{\prime}(j)=+$), while if we take $(\theta_{j}^{\prime},\theta_{j}^{\prime\prime})=(\theta_{+},\theta_{+})$ and replace $A_{j}$ with $A_{j}U$ (resp. $U^{-1}A_{j}$) we set $\psi^{\prime}(j)=--$ (resp. $\psi^{\prime}(j)=-$). As a result, the skeleton $P_{M_{T}}$ determined by $\psi\in\Psi_{T}$ and $\psi^{\prime}\in\Psi^{\prime}_{T}$ has

[TABLE]

true vertices.

A spine for $M$ is given by the union of $P_{M_{T}}$ with: (i) the skeleton $P_{A_{j}}\cup T_{j}^{\prime\prime}\times\{1\}$ described in the proof of Theorem 1 and having $5+(S(\beta_{j}/\delta_{j})-1)$ true vertices for each $j\in J^{\prime\prime}\setminus J^{\prime\prime}_{T}$, and (ii) the torus $T_{j}^{\prime\prime}$, containing $6$ true vertices for each $j\in J^{\prime}\setminus J^{\prime}_{T}$. Since $|J^{\prime}\setminus J^{\prime}_{T}|+|J^{\prime\prime}\setminus J^{\prime\prime}_{T}|=|E\setminus E_{T}|=|E|-|V|+1$ and $|J^{\prime}\setminus J^{\prime}_{T}|=\phi(G)$ we get the statement. ∎

The sharpness of the previous upper bound in all known cases justifies the following:

Conjecture**.**

The upper bound given in Theorem 3 is sharp for all closed connected orientable prime graph manifolds.