The boundary of the Milnor fiber of the singularity f(x,y) + zg(x,y) = 0

TL;DR

This paper provides an explicit algorithm to construct the plumbing graph of the boundary of the Milnor fiber for a specific class of complex surface singularities, using common resolutions of the defining functions.

Contribution

It introduces a novel algorithm that explicitly constructs the plumbing graph for the Milnor fiber boundary based on common resolutions of the functions defining the singularity.

Findings

Provides an explicit plumbing graph construction method.

Connects resolution data to Milnor fiber boundary topology.

Enhances understanding of singularity boundaries in complex surfaces.

Abstract

Let $f,g\in\mathbb{C}\{x,y\}$ be germs of functions defining plane curve singularities without common components in $(\mathbb{C}^2,0)$ and let $\Phi(x,y,z) = f(x,y) + zg(x,y)$. We give an explicit algorithm producing a plumbing graph for the boundary of the Milnor fiber of $\Phi$ in terms of a common resolution for $f$ and $g$.

Equations

$$\Phi (x,y,z)=f(x,y)+zg(x,y)=0,$$

$${\overline{T}}_{f,g}=({\overline{T}}_{f,1}\setminus T^{\prime })\cup {\overline{T}}_{\epsilon }\cup ({\overline{T}}_2\setminus (T^{\prime }\cup T_{g,2})).$$

$$M_v\cap M_w={⨿}_{i=1}^{e_{v,w}}{S}_{v,w,i},$$

$$S_{v,w,i}\to B_{v,w,i}\times B_{w,v,i},p\mapsto ({\pi }_v(p),{\pi }_w(p))$$

$$-b[C]=[C^{\prime }].$$

$${\epsilon }_e=([C_v],[C_w]{)}_{S_{v,w,i}}.$$

$$({\pi }_{v,\ast }([N_v]),[\xi ]{)}_{{\Sigma }_v}=0.$$

$$-b_w{m}_w+\sum _{e\in {\mathcal{E}}_w}{\epsilon }_e{m}_e=n_v.$$

$$\left(-n_w-b_w{m}_w+\sum _{e\in {\mathcal{E}}_w}{\epsilon }_e{m}_e\right)[C_w]=0.$$

$$([K_e\cap S_e],[C_w]{)}_{S_e}=m_v,([K_e\cap S_e],[C_v]{)}_{S_e}=m_v.$$

$$([C_w],[C_w]{)}_{S_e}=([C_v],[C_v]{)}_{S_e}=0,([C_w],[C_v]{)}_{S_e}={\epsilon }_e,$$

$$\begin{array}{rl}0=[\partial K_e]& =-n_w[C_w]+\sum _{e\in {\mathcal{E}}_w}{\epsilon }_e(m_v[C_w]-m_w[C_v])\\ & =\left(-n_w-b_w{m}_w+\sum _{e\in {\mathcal{E}}_w}{\epsilon }_e{m}_v\right)[C_w].\end{array}$$

$$\frac{a}{b}=k_1-\frac{1}{k_2-\frac{1}{\cdots -\frac{1}{k_s}}}$$

$${\mu }_0=0,{\mu }_1=1,{\tilde{\mu }}_0=-1,{\tilde{\mu }}_1=0.$$

$${\mu }_{i+1}=k_i{\mu }_i-{\mu }_{i-1},{\tilde{\mu }}_{i+1}=k_i{\tilde{\mu }}_i-{\tilde{\mu }}_{i-1}.$$

$$\left|\begin{array}{cc}{\mu }_i& {\mu }_{i+1}\\ {\tilde{\mu }}_i& {\tilde{\mu }}_{i+1}\end{array}\right|=1,i=0,\dots ,s.$$

$$M_1=D_{\frac{1}{s}}\times S^1,M_i=({\bar{D}}_{\frac{i+1}{s}}\setminus D_{\frac{i}{s}})\times S^1,i=2,\dots ,s.$$

$${\pi }_s(t^{{\tilde{\mu }}_{s+1}},t^{-{\mu }_{s+1}})=t^{{\mu }_s{\tilde{\mu }}_{s+1}-{\tilde{\mu }}_s{\mu }_{s+1}}=t.$$

$$d_{{\Gamma }^{\prime }}(2-2g_{{\Gamma }^{\prime }})$$

$$d_{{\Gamma }^{\prime }}(-b_{{\Gamma }^{\prime }})$$

$$m_{a,i,+}=(m_w-l_w){\mu }_i-m_a{\tilde{\mu }}_i,m_{a,i,-}=l_w{\mu }_i$$

$$m_{e,i,+}=(m_{v_1}-l_{v_1}){\mu }_i-(m_{v_2}-l_{v_2}){\tilde{\mu }}_i,m_{e,i,-}=l_{v_1}{\mu }_i-l_{v_2}{\tilde{\mu }}_i$$

$$r_{i-1}-b_i{r}_i+r_{i+1}=0,i=1,\dots ,\hat{j},\dots ,s.$$

$$d_{{\Gamma }^{\prime }}=3,g_{{\Gamma }^{\prime }}=0,-b_{{\Gamma }^{\prime }}=-1,d_{{\Gamma }^{\prime \prime }}=2,g_{{\Gamma }^{\prime \prime }}=0,-b_{{\Gamma }^{\prime \prime }}=-1.$$

$$\begin{array}{rl}f(x,y)& =(x^2+{\lambda }_1{y}^3)((x^2+y^3{)}^2+{\mu }_1{x}^5{)}^2,\\ g(x,y)& =(x^2+{\lambda }_2{y}^3{)}^3((x^2+y^3{)}^2+{\mu }_2{x}^5),\end{array}$$

$$\begin{array}[]{rclrcl}d_{\Gamma_{1}}&=&\gcd\{10,30,15,1,34\},&d_{\Gamma_{1}}&=&1,\\ 1\cdot(2-2g_{\Gamma_{1}})&=&1\cdot 15+1\cdot 10+(-2)\cdot 30+1+2,&g_{\Gamma_{1}}&=&17,\\ 1\cdot(-b_{\Gamma_{1}})&=&(30\cdot 0-1\cdot 30)+(30\cdot 32-30\cdot 34),&-b_{\Gamma_{1}}&=&-90.\end{array}$$

$$m_{a,0}=-30,m_{a,1,+}=0,m_{a,1,-}=30,n_{a,1,+}=1.$$

$$\begin{array}[]{lcrlcrlcr}m_{v_{1}}&=&30,\quad\quad\tilde{m}_{1}&=&15,\quad\quad l_{v_{1}}&=&30,\\ m_{v_{2}}&=&34,\quad\quad\tilde{m}_{2}&=&17,\quad\quad l_{v_{2}}&=&32,\end{array}$$

$$m_{e,0}=-30,m_{e,1,+}=0,m_{e,1,-}=30.$$

$${\overline{T}}_{f,g}\cap T_1\setminus (T_{f,1}\cup T_2).$$

$$F_{f,1}:=F_f\cap T_1\setminus (T_{f,1}\cup T_2),$$

Full text

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**The boundary of the Milnor fiber of the singularity $f(x,y)+zg(x,y)=0$ **

Baldur Sigurðsson 111Baldur Sigurðsson, Basque Center for Applied Mathematics, Bilbao, Spain; bsigurdsson@bcamath.org

1 Introduction

It is known that the boundary of any hypersurface singularity in $(\mathbb{C}^{3},0)$ is a plumbed manifold. This was stated by Michel and Pichon in [6] and proved by separate methods by Némethi and Szilárd [12] and Michel and Pichon [7]. A stronger statement for certain real analytic map germs was proved by de Bobadilla and Neto [4]. As these theorems rely on resolution of singularities, they do not easily provide an explicit description of a plumbing graph describing the boundary. Calculations have been carried out, however for some particular singularities and families: for Hirzebruch singularities [8], suspensions of isolated plane curves [9] and in the many examples of [12].

In the case of a hypersurface singularity given by the equation

[TABLE]

where $f,g$ are singular germs with no common factors (but not necessarily reduced), we give an explicit algorithm producing a plumbing graph for the boundary of the Milnor fiber in terms of the graph associated with an embedded resolution of the plane curve singularities defined by $f$ and $g$. This is obtained from an explicit description of the Milnor fiber by the author [14]. Singularities of the form $f(x,y)+zg(x,y)$ are closely related to the deformation theory of sandwitched singularities, see [5]. The article is organized as follows.

In section 2 we recall the results of [14] and fix notation related to the resolution graph of $f$ and $g$.

In section 3 we define plumbed manifolds and prove some useful lemmas related to them.

In section 4 we introduce families of multiplicities and dual multiplicities assigned to a complex valued function on a plumbed $3$-manifold, satisfying certain conditions. In the case of a fibration over $S^{1}$, these multiplicities coincide with the multiplicities used in [3, 12].

In section 5 we prove a useful lemma relating the negative continued fraction expansion of a rational number to a plumbing construction.

In section 6 and section 7 we provide the details of the construction of the plumbing graph for the boundary of the Milnor fiber of $\Phi$ and the families of multiplicities and dual multiplicities for the coordinate function $z$. These statements can be read after only reading section 2.

In section 8 we provide some examples. First we give the simple plumbing graph describing boundary of the Milnor fiber of a $T_{a,b,*}$ singularity given by the equation $x^{a}+y^{b}+xyz=0$. This example is discussed in [12] 22.2.

Section 9 contains proofs of theorem 6.1 and theorem 7.1.

\thenotconv** Notation and conventions****.**

• (i)

We denote by $D\subset\mathbb{C}$ the open unit disk and by $\overline{D}$ the closed unit disk. We also set $S^{1}=\partial\overline{D}$. For any $r>0$, let $D_{r},\overline{D}_{r},S^{1}_{r}$ be the corresponding disks and circle with radius $r$.

• (ii)

If $X$ is a manifold, and $C\subset X$ is a submanifold of dimension $d$, then we denote by $[C]\in H_{d}(M,\mathbb{Z})$ the associated homology class. If $X$ is a compact oriented compact manifold, possibly with boundary, we denote by $(\cdot,\cdot)_{X}$ the intersection pairing between $H_{i}(X,\mathbb{Z})$ and $H_{n-i}(X,\partial X,\mathbb{Z})$, where $n=\dim X$. In particular, if $\partial X=\emptyset$ and $i=n/2$, then $(\cdot,\cdot)_{X}$ is the intersection form on the middle homology.

• (iii)

The boundary of an oriented manifold is oriented by the usual outward-pointing-vector first rule. Note that if a codimension one submanifold $N\subset M$ splits $M$ into two pieces, this rule induces opposite orientations according to which piece is chosen.

• (iv)

A locally trivial differential fiber bundle with a chosen orientation on the total space and the base space induces an orientation on each fiber by the following requirement. A lifting of a positive basis of the tangent space of the base space, followed by a positive basis of the tangent space of the fiber yields a positive orientation of the total space. In fact, this rule induces an orientation on the fibers, the total space or the base space, given orientations on the other two.

Acknowledgements**.**

I would like to thank Némethi András for suggesting this problem to me and for the many helpful discussions we have had.

2 The Milnor fiber

In [14] the author gives a description of the Milnor fiber of the singularity $f(x,y)+zg(x,y)=0$. We will now recall that result and fix some notation.

\thedefinition** Definition****.**

Let $\phi:V\to\mathbb{C}^{2}$ be a common resolution of the functions $f$ and $g$ with exceptional divisor $E$, decomposing into irreducible components as $E=\cup_{v\in\mathcal{W}}E_{v}$ and denote by $\Gamma$ the associated embedded resolution graph. The set of vertices in $\Gamma$ is $\mathcal{V}=\mathcal{W}\amalg\mathcal{A}$, where $\mathcal{W}$ corresponds to components of the exceptional divisor, while elements of $\mathcal{A}$ are arrowheads, corresponding to components of the strict transforms of $f$ and $g$. For any $a\in\mathcal{A}$ there is a $w_{a}\in\mathcal{W}$ so that $\{w_{a},a\}$ is an edge in $\Gamma$. Write $\mathcal{A}=\mathcal{A}_{f}\amalg\mathcal{A}_{g}$, where elements of $\mathcal{A}_{f}$ and $\mathcal{A}_{g}$ correspond to components of the strict transform of $f$ and $g$. For $v\in\mathcal{V}$, we denote by $m_{v}$ and $l_{v}$ the multiplicities of $f$ and $g$, respectively. In particular, $m_{v}=0$ if and only if $v\in\mathcal{A}_{g}$ and, similarly, $l_{v}=0$ if and only if $v\in\mathcal{A}_{f}$.

Define $\mathcal{W}_{1}=\left\{w\in\mathcal{W}\,\middle|\,m_{w}\leq l_{w}\right\}$ and $\mathcal{W}_{2}=\left\{w\in\mathcal{W}\,\middle|\,m_{w}>l_{w}\right\}$. Write $\mathcal{A}_{i}=\left\{a\in\mathcal{A}\,\middle|\,w_{a}\in\mathcal{W}_{i}\right\}$ for $i=1,2$. Similarly, take $\mathcal{A}_{f,i},\mathcal{A}_{g,i}\subset\mathcal{A}_{i}$ so that $\mathcal{A}_{f}=\mathcal{A}_{f,1}\amalg\mathcal{A}_{f,1}$ and $\mathcal{A}_{g}=\mathcal{A}_{g,1}\amalg\mathcal{A}_{g,2}$.

\thedefinition** Definition****.**

For $w\in\mathcal{W}$, let $T_{w}$ be a tubular neighbourhood around $E_{w}$ in $V$ and let $T=\cup_{w\in\mathcal{W}}T_{w}$. Set also $T_{i}=\cup_{w\in\mathcal{W}_{i}}T_{w}$ for $i=1,2$. For a given $0<\varepsilon\ll 1$, let $F_{f}=f^{-1}(\varepsilon)$ be the Milnor fiber of $f$, and $F^{\prime}_{f}=\phi^{-1}(F_{f})$ its pullback to $V$. Let $T_{\varepsilon}$ be a small tubular neighbourhood around $F^{\prime}_{f}$ in $T$. We also choose tubular neighbourhoods $T_{a}\subset T$ around $E_{a}$ for any $a\in\mathcal{A}$. With these choices fixed, choose a small tubular neighbourhood $T^{\prime}=\cup_{w\in\mathcal{W}}T^{\prime}_{w}$ around the exceptional divisor inside $T$. This is chosen small enough that $T^{\prime}\cap T_{\varepsilon}=\emptyset$.

Set $T_{f,i}=\cup_{a\in\mathcal{A}_{f,i}}T_{a}$, and $T_{g,i}=\cup_{a\in\mathcal{A}_{g,i}}T_{a}$ for $i=1,2$ and let

[TABLE]

\thedefinition** Definition****.**

Let $X$ be a four dimensional manifold with boundary and $\iota:\overline{D}\to X$ an embedding of the closed disk into $X$ such that $\iota$ sends $S^{1}=\partial\overline{D}$ to $\partial X$ and the image of $\overline{D}$ is transversal to $\partial X$. Then there exists a map $\psi:\overline{D}\times\overline{D}\to X$ parametrizing a tubular neighbourhood of $\iota(D)$ in $X$ so that $\psi(0,z)=\iota(z)$ for $z\in\overline{D}$ and $\psi(x,z)\in\partial X$ for $x\in\overline{D}$ and $z\in S_{1}=\partial D$. For $k\in\mathbb{Z}$, the $k^{\mathrm{th}}$ twist along $\iota$ is defined as $(X\setminus\psi(D\times\overline{D}))\amalg_{t_{k}}\overline{D}\times\overline{D}$ where the glueing map $t_{k}:S^{1}\times\overline{D}\to(X\setminus\psi(D\times\overline{D}))$ is defined by $t_{k}(x,z)=\psi(x,x^{k}z)$ and is denoted by $X_{\iota,k}$. We also say that $X_{\iota,k}$ is obtained from $X$ by twisting $X$ $k$ times along $\iota(\overline{D})$.

\thedefinition** Definition****.**

In [14], the author shows that for any $a\in A_{g}$, the intersection $E_{a}\cap\overline{T}_{f,g}$ is a disjoint union of $m_{w_{a}}$ disks embedded in $T_{f,g}$. Let $F_{f,g}$ be the manifold obtained from $\overline{T}_{f,g}$ by twisting each of these disks $l_{a}$ times for all $a$.

2.1 Theorem** ([14]).**

The Milnor fiber $F_{\Phi}$ is diffeomorphic to $F_{f,g}$. ∎

\thedefinition** Definition****.**

Let $M_{f,g}=\partial F_{f,g}$. We also set $M^{\prime}_{f,g}=\partial\overline{T}_{f,g}$.

3 Plumbed 3-manifolds

In this section we give an introduction to plumbed three manifods and plumbing graphs, along with some useful properties. Throughout this text, an $S^{1}$-bundle will mean a principal $S^{1}$-bundle. In particular, we assume that there is a consistent choice for orientation on each fiber. In fact, all our $S^{1}$-bundles will have as base space an oriented real surface. This determines a consistent choice of orientation on fibers as described in section 1iii.

We note that apart from our restriction on orientability, our definition of a plumbed manifold is equivalent to the definition in [13]. This can be seen from section 3. We note, however, that our construction differs slightly to the standard one. This is explicated in section 3. The main reason for this is that in our construction in section 9, we identify the three dimensional plumbed pieces directly, but the result can in no natural way be seen as the boundary of a four dimensional plumbed manifold (as is the case for links of isolated surface singularities).

\thedefinition** Definition****.**

A plumbed manifold is a three dimensional compact manifold $M$, possibly with boundary, given as a union of submanifolds with boundary $M=\cup_{v\in\mathcal{W}}M_{v}$ having the following properties.

1. (i)

For each $v,w\in\mathcal{W}$, $v\neq w$ we have an $e_{v,w}\in\mathbb{N}$ so that

[TABLE]

with $S_{v,w,i}$ an embedded torus $M\supset S_{v,w,i}\cong S^{1}\times S^{1}$. Thus, $S_{v,w,i}$ is a component of $\partial M_{v}$ and inherits an orientation. Since $e_{v,w}=e_{v,w}$, we can assume that as sets, we have $S_{w,v,i}=S_{v,w,i}$ for $i=1,\dots,e_{v,w}$. 2. (ii)

For each $v$ we have a compact connected surface $\Sigma_{v}$ (possibly with boundary) and a locally trivial $S^{1}$ bundle $\pi_{v}:M_{v}\to\Sigma_{v}$. If $1\leq i\leq e_{v,w}$ for some $w\neq v$, then $S_{v,w,i}=\pi^{-1}(B_{v,w,i})$ where $B_{v,w,i}\cong S^{1}$ is a component of the boundary of $\Sigma$. 3. (iii)

Assume that $1\leq i\leq e_{v,w}$ for some $v\neq w$. The map

[TABLE]

is a diffeomorphism. 4. (iv)

For each $v\in\mathcal{V}$, let $B_{v,1},\ldots,B_{v,e_{v}}$ be the components of $\partial\Sigma$ not of the form $B_{v,w,i}$ for some $w,i$. We assume given a section $s_{v,i}:B_{v,i}\to\Sigma$ to the reduced bundle $\pi_{v}|_{B_{v,i}}$.

\theblock**.**

We orient $S_{v,w,i}$ by considering it as a subset of the boundary of $M_{v}$. This way, $S_{v,w,i}=S_{w,v,i}$ as sets, but $S_{v,w,i}=-S_{w,v,i}$ as oriented manifolds. We also orient the boundary of $\Sigma_{w}$ by the same rule, for any $w\in\mathcal{W}$.

\theblock**.**

For a closed surface $\Sigma$, the Euler number classifies the $S^{1}$ bundles over $\Sigma$. However, every $S^{1}$ bundle $\pi:M\to\Sigma$ over a compact surface with nonempty boundary is trivial. But given a trivialization, or, equivalently, a section $s:\partial\Sigma\to\partial M$, over the boundary, a relative Euler number is well defined, and invariant under homotopy of the section. This is a complete invariant in the following sense. Let $\Sigma$ be a compact surface with boundary and take two $S^{1}$ bundles $M,M^{\prime}\to\Sigma$ with sections $s:\partial\Sigma\to M$ and $s^{\prime}:\partial\Sigma\to M^{\prime}$ and an isomorphism of bundles $\psi:M|_{\partial\Sigma}\to M^{\prime}|_{\partial\Sigma}$ sending $s$ to $s^{\prime}$. Then $\psi$ extends to an isomorphism of bundles $M\to M^{\prime}$ if and only if the relative Euler numbers coincide. We will refer to the relative Euler number simply as the Euler number.

The relative Euler number is defined as follows. Let $D\subset\Sigma$ be an open disk. We can extend the section $s:\partial\Sigma\to\partial M$ to a section $\overline{s}:\Sigma\setminus D\to M\setminus\pi^{-1}(D)$. Given an orientation preserving diffeomorphism $\varphi:\partial D\to S^{1}$, there is a unique number $b\in\mathbb{Z}$ so that the twisted section $\overline{s}^{b}:\partial D\to\pi^{-1}(\partial D)$, $x\mapsto\varphi(x)^{b}\overline{s}(x)$ extends over the disk $D$. The relative Euler number is defined as $-b$.

\thelemma** Lemma****.**

Let $M\to\Sigma$ be an $S^{1}$ bundle over a compact surface with boundary. Let $-b$ be its the Euler number relative to a section $s:\partial\Sigma\to M$. Let $C\subset M$ be a fiber of the bundle and $C^{\prime}$ the image of $s$ (as oriented submanifolds). Then, in $H_{1}(M,\mathbb{Z})$

[TABLE]

Proof.

This follows from the definition of the relative Euler number. Indeed, let $\overline{s}:\Sigma\setminus D\to M$ be a section as above. It follows that $[C^{\prime}]-\overline{s}_{*}[\partial D]=0$. The sign comes from the fact that $\partial D$ is oriented as the boundary of the disk $D$, which is the opposite to the orientation inherited from the complement of the disk. Since the section $\overline{s}^{b}_{*}$ extends over $D$ and $D$ is null-homotopic, the map $\overline{s}^{b}:\partial D\to M$ is homotopic to a constant map $\partial D\to M$. It follows that $\overline{s}_{*}[\partial D]=-b[C]$. ∎

\therem** Remark****.**

If $\partial\Sigma\neq\emptyset$, then eq. 3.1 can be taken as an alternative definition of the (relative) Euler number. Indeed, it follows from the Künneth formula that the $[C]$ is not a torsion element of $H_{1}(M,\mathbb{Z})$.

\thedefinition** Definition****.**

A plumbing graph is a decorated graph $G$ (with no loops) with vertex set $\mathcal{V}=\mathcal{W}\amalg\mathcal{A}$, where each vertex $a\in\mathcal{A}$ has a unique neighbour $w_{a}$ and $w_{a}$. We refer to vertices in $\mathcal{A}$ as arrowhead vertices. $G$ is decorated as follows.

• ❀

For each $w\in\mathcal{W}$, we have integers $-b_{w}\in\mathbb{Z}$ and $g_{v}\in\mathbb{Z}_{\geq 0}$. These are referred to as the associated Euler number (or sometimes selfintersection number) and the genus.

• ❀

Each edge $e$ connecting two vertices in $\mathcal{W}$ is given a sign $\varepsilon_{e}\in\{+,-\}$.

In a drawing of a graph, the genus $g_{v}$ is written within square brackets as $[g_{v}]$ to be distinguished from the Euler number. If it is omitted, it is assumed to be [math]. A negative edge will be indicated by the symbol $\circleddash$, whereas if indication is omitted, the sign is assumed to be positive. An edge connecting $w\in\mathcal{W}$ and an arrowhead $a\in\mathcal{A}$ is drawn as a dashed edge, see e.g. fig. 2.

Let $M=\cup_{v\in\mathcal{W}}M_{i}$ be a plumbed manifold and use the notation introduced in section 3. The associated plumbing graph $G$ has vertex set $\mathcal{V}=\mathcal{W}\amalg\mathcal{A}$ where $\mathcal{A}=\amalg_{v\in\mathcal{W}}\mathcal{A}_{v}$, where the elements of $\mathcal{A}_{v}$ correspond to the boundary components $B_{v,1},\ldots,B_{v,e_{v}}$ of $\Sigma_{v}$. It has $e_{v,w}$ edges connecting $v$ and $w$ if $v,w$ are distinct elements of $\mathcal{W}$ and a single edge connecting any $a\in\mathcal{A}_{w}$ with $w$ if $w\in\mathcal{W}$, and no other edges. Denote by $\mathcal{E}$ this set of edges.

The genus $g_{v}$ is the genus of the surface $\Sigma_{v}$. The Euler number $-b_{v}$ is the Euler number of the $S^{1}$ bundle $M_{v}\to\Sigma_{v}$, trivialized on the boundary components $B_{v,i}$ by the given section, and on the components $B_{v,w,i}$ by any fiber of $S_{v,w,i}=S_{w,v,i}\to B_{w,v,i}$.

Any edge $e\in\mathcal{E}$ connecting $v,w\in\mathcal{W}$ corresponds to a component $S_{v,w,i}=S_{w,v,i}$ of the intersection $M_{v}\cap M_{w}$. Take fibers $C_{v}$ and $C_{w}$ of $\pi_{v}$ and $\pi_{w}$, respectively, contained in $S_{v,w,i}$. The sign $\varepsilon_{e}$ is defined as the intersection number of $C_{v}$ and $C_{w}$ in $S_{v,w,i}$, that is,

[TABLE]

It follows from definition that this intersection number is $\pm 1$. This sign depends on the orientation on $S_{v,w,i}$, which, we recall, is obtained by viewing $S_{v,w,i}$ as a subset of $\partial M_{v}$.

\thelemma** Lemma****.**

Let $v,w$ be vertices connected by an edge $e$ in a plumbing graph associated to a plumbed manifold $M$. Let $C_{w}$ be a fiber of $\pi_{w}$ contained in the torus $S_{v,w,i}$ corresponding to $e$. Then the sign $\varepsilon_{e}$ is positive if and only if $-C_{w}$ is an oriented section to the map $S_{v,w,i}\to B_{v,w,i}$.

Proof.

Let $C_{v}\subset S_{v,w,i}$ be a fiber of $\pi_{v}$. We have $\varepsilon_{e}=([C_{v}],[C_{w}])_{S_{v,w,i}}$. Therefore, if $B$ is the oriented image of some section of $\pi_{v}|_{S_{v,w,i}}$, then it suffices to show that $([C_{v}],[B])_{S_{v,w,i}}=-1$. By construction, $C_{v}$ and $B$ intersect in a single point, say $x\in S_{v,w,i}$, and we can assume that this intersection is transverse. Let $c,b\in T_{x}S_{v,w,i}$ be tangent vectors inducing positive bases of $T_{x}C_{v}$ and $T_{x}B_{v}$. Let $a\in T_{x}M_{v}$ be an outward pointing tangent vector. By definition, $(\pi_{v}(a),\pi_{v}(b))$ is a positive basis of $T_{\pi_{v}(x)}B_{v,w,i}$. Therefore, $(a,b,c)$ is a positive basis of $T_{x}M_{v}$, and so $(b,c)$ is a positive basis of $T_{x}S_{v,w,i}$. This means that $([B],[C_{v}])_{S_{v,w,i}}=1$ and so $([C_{v}],[B])_{S_{v,w,i}}=-1$. ∎

\therem** Remark****.**

The above lemma may seem contrary to the usual definition of plumbing [13, 12]. There, the authors start with $S^{1}$-bundles over a closed surfaces. The glueing of two pieces, corresponding to an edge $e$, is made by removing a tubular neighbourhood around a fiber in each piece and identifying the boundaries by switching meridians and fibers, multiplied with a sign $\varepsilon_{e}$. The output of the two constructions is identical, but the submanifold $B$ in the proof above, is a meridian, but with the opposite orientation to that of a standard meridian.

\theexample** Example****.**

[10, 12] Let $\tilde{X}$ be a smooth complex surface and let $E\subset\tilde{X}$ be a compact normal crossing divisor. This means that $E$ is a compact reduced analytic subspace of pure dimension one, decomposing as $E=\cup_{v\in\mathcal{V}}E_{v}$ into irreducible components, with the condition that each $E_{v}$ is a submanifold of $\tilde{X}$, that each $E_{v}$ and $E_{w}$ intersect transversally, and that any singularity of $E$ is a double point. If $T\subset\tilde{X}$ is a suitable small neighbourhood of $E$, then $M=\partial T$ is a plumbed manifold, whose plumbing graph $G$ is given by the intersection matrix of $E$, that is, $G$ has vertex set $\mathcal{V}$, the genus $g_{v}$ is the genus of $E_{v}$, the Euler number $-b_{v}$ is the selfintersection number $(E_{v},E_{v})$, equivalently, it is the Euler number of the normal bundle of the embedding $E_{v}\hookrightarrow\tilde{X}$, and the number of edges between $v,w\in\mathcal{V}$ is the cardinality $|E_{v}\cap E_{w}|$. Furthermore, $\varepsilon_{e}=+$ for any edge $e$.

\theblock**.**

A plumbed manifold $M$ can be recovered from its (decorated) plumbing graph $G$ as follows. As before, denote by $\mathcal{V}$ and $\mathcal{E}$ the set of vertices and edges in $G$, and by $g_{v}$ and $-b_{v}$ the genus and the selfintersecion number of a vertex $v$ and by $\varepsilon_{e}$ the sign of an edge. For each $v\in\mathcal{V}$, let $\Sigma_{v}$ be a compact surface of genus $g_{v}$ with $e_{v}+\sum_{w\in\mathcal{V}\setminus\{v\}}e_{v,w}$ boundary components, give names $B_{v,w,i}$, for $w\in\mathcal{V}\setminus\{v\}$ and $1\leq e_{v,w}$ and $B_{v,i}$ for $1\leq i\leq e_{v}$. Let $\pi_{v}:M_{v}\to\Sigma_{v}$ be an $S^{1}$ bundle with sections $s_{v,w,i}$ and $s_{v,i}$ over the boundary inducing Euler number $-b_{v}$. The section $s_{v,w,i}$ induces a trivialization $\phi_{v,w,i}:S^{1}\times S^{1}\to\pi^{-1}_{v}(B_{v,w,i})$.

We then have $M\cong\amalg_{v\in\mathcal{V}}M_{v}/\sim$ where $\sim$ is the equivalence relation on $\amalg_{v\in\mathcal{V}}M_{v}$ generated by $\phi_{v,w,i}(\theta_{1},\theta_{2})\sim\phi_{w,v,i}(\theta_{2}^{-\varepsilon_{e}},\theta_{1}^{-\varepsilon_{e}})$ where $e$ is the $i^{\mathrm{th}}$ edge connecting $v$ and $w$. The negative sign in the exponents in the glueing map is explained by section 3.

4 Multiplicities associated with complex valued functions

In this section we give a definition of multiplicities of a complex valued function on a plumbed manifold under some restrictions (see section 4). This definition coincides with the multiplicities associated with fibred links in section 18 of [3], if the function is a fibration over $S^{1}$. These multiplicities are useful as they can be obtained by local computation, but can be used to determine Euler numbers, see section 4.

\theblock**.**

Let $M=\cup_{v\in\mathcal{V}}M_{v}$ be a plumbed manifold with graph $G$, with vertex set $\mathcal{V}=\mathcal{W}\cup\mathcal{A}$ and let $\zeta:M\to\mathbb{C}$ be a differentiable function having [math] as a regular value. Furthermore, assume that $\zeta$ does not vanish on $\partial M_{v}$ for all $v\in\mathcal{W}$. Thus, $N_{v}=\zeta^{-1}(0)\cap M_{v}$ is a closed submanifold of $M_{v}$ which does not intersect its boundary. Assume also that $N_{v}$ is homologous to a multiple of $[C_{v}]$ in $M_{v}$, that is, $[N_{v}]=n_{v}[\pi_{v}^{-1}(p)]$ for some (well defined) $n_{v}\in\mathbb{Z}$.

For any $x\in\Sigma_{v}\setminus\pi_{v}(N_{v})$, there is a unique $m_{x}\in\mathbb{Z}$ so that $\zeta_{*}([\pi^{-1}(x)])=m_{x}[S^{1}]\in H_{1}(\mathbb{C}^{*})$. This number is a locally constant function of $x$. In fact, let $\xi:[0,1]\to\sigma$ be a $1$-chain connecting $x=\xi(0)$ and $y=\xi(1)$. We can assume that $\xi$ is an embedding, and by a small perturbation, we can assume that the map $N_{v}\to\Sigma$, induced by $\pi_{v}$, is an immersion, transverse to $\xi$. At any intersection point of $\xi$ and $\pi_{v}(N_{v})$, one sees that $m_{\xi(\cdot)}$ changes by $\pm 1$, depending on the sign of the intersection. In particular, if $x,y\in\partial\Sigma_{v}$, then $\xi$ is a cycle inducing an element $[\xi]\in H_{1}(\Sigma_{v},\partial\Sigma_{1},\mathbb{Z})$. It follows from the assumptions that we made that $\pi_{v,*}([N_{v}])=0\in H_{1}(\Sigma,\mathbb{Z})$, and so

[TABLE]

It follows that for $x,y\in\partial\Sigma$, the number $m_{x}=m_{y}$ is a number which well defined by the map $\zeta$; we denote it by $m_{v}$.

\thedefinition** Definition****.**

Let $\zeta:M\to\mathbb{C}$ be as in section 4. We refer to the families $(m_{v})_{v\in\mathcal{V}}$ and $(n_{w})_{w\in\mathcal{W}}$ (defined above) as the family of multiplicities and dual family of multiplicities associated with $\zeta$, respectively. In a drawing of a plumbing graph, a multiplicity is written within parenthesis, whereas a dual multiplicity is written in parenthesis next to an arrow emanating from the vertex.

\thelemma** Lemma****.**

Let $\zeta:M\to\mathbb{C}$ be as in section 4, and let $(m_{v})_{v\in\mathcal{V}}$ and $(n_{v})_{v\in\mathcal{W}}$ be the associated families of multiplicities and dual multiplicities. Let $w\in\mathcal{W}$. If $e\in\mathcal{E}_{w}$ connects $w$ and $v$, set $m_{e}=m_{v}$. We then have

[TABLE]

Proof.

Let $C_{w}$ be a fiber of $\pi_{w}$. Since $M_{w}\cong\Sigma_{w}\times S^{1}$, the element $[C_{w}]\in H_{1}(M_{w},\mathbb{Z})$ is nontorsion. It therefore suffices to show that

[TABLE]

We can assume that $\zeta$ is transversal to the submanifold with boundary $\mathbb{R}_{\geq 0}\subset\mathbb{C}$ so that $\zeta|_{M_{w}}^{-1}(\mathbb{R}_{\geq 0})$ is a submanifold with boundary $K_{w}$ in $M_{w}$. Furthermore, we can assume that $K_{w}$ is transversal to $\partial M_{w}$. This way, $[\partial K_{w}]=-[N_{w}]+\sum_{e\in\mathcal{E}_{w}}[K_{w}\cap S_{e}]\in H_{1}(M_{w},\mathbb{Z})$. Let $e\in\mathcal{E}_{w}$, connecting $w$ and $v\in\mathcal{V}$. Assume that the fiber $C_{w}$ was chosen so that $C_{w}\subset S_{e}$. Furthermore, let $C_{v}$ be a fiber of $\pi_{v}$ contained in $S_{e}$ if $v\in\mathcal{W}$, otherwise, let $C_{v}$ be the image of $s_{v}$. It follows from definition that

[TABLE]

Since $[C_{v}]$ and $[C_{w}]$ form a basis of $H_{1}(S_{e},\mathbb{Z})$, and we have

[TABLE]

we get $[K_{e}\cap S_{e}]=\varepsilon_{e}(m_{v}[C_{w}]-m_{w}[C_{v}])$. This yields

[TABLE]

Here, the variable $v$ inside the sum depends on $e$. The last equality follows from section 3 ∎

\theexample** Example****.**

Let $\tilde{X}$ and $E=\cup_{v\in\mathcal{V}}E_{v}$ be as in section 3, and let $h:\tilde{X}\to\mathbb{C}$ be a holomorphic function. Decompose the divisor of $h$ as $(h)=(h)_{\mathrm{exc}}+(h)_{\mathrm{str}}$ so that $(h)_{\mathrm{exc}}$ is supported on $E$, and $(h)_{\mathrm{str}}$ has no components with nonzero coefficient included in $E$. We can then write $(h)_{\mathrm{exc}}=\sum_{v\in\mathcal{V}}m_{v}E_{v}$, and $(h)_{\mathrm{str}}=\sum_{D}n_{D}D$ with $n_{D}=0$ if $D=E_{v}$ for some $v\in\mathcal{V}$. Assume that the support of $(h)_{\mathrm{str}}$ does not contain any intersection points in $E$, that is, if $n_{D}\neq 0$, then $D\cap E_{v}\cap E_{w}=\emptyset$ for $v,w\in\mathcal{W}$, $v\neq w$. If $T\subset\tilde{X}$ is a small tubular neighbourhood around $E$, then $M=\partial T$ is a plumbed manifold and $h|_{M}$ satisfies the conditions in section 4. The associated family of multiplicities is $(m_{v})_{v\in\mathcal{W}}$. Furthermore, the family $(n_{v})_{v\in\mathcal{V}}$ of dual multiplicities is given as the intersection $n_{v}=(E_{v},(h)_{\mathrm{str}})$.

Note that here we do not assume $(h)_{\mathrm{str}}$ to be smooth, only that its intersection points with $E$ lie in the regular part of $E$.

5 Negative continued fractions

In this section we discuss negative continued fractions and a plumbing construction related to them. Some of the notation introduced in this section follows [2, III.5].

\theblock**.**

Let $a,b$ be relatively prime integers, $b>0$. The fraction $a/b$ can be written in a unieque way as a (negative) continued fraction

[TABLE]

where $k_{i}\geq 2$ for $i\geq 2$. Further, we have $k_{1}\geq 2$ if and only if $a>b$ and $k_{1}>0$ if and only if $a>0$.

\thedefinition** Definition****.**

The rational number $a/b$ is called the (negative) continued fraction associated with the sequence $k_{1},\ldots,k_{s}$ and is denoted by $[k_{1},\ldots,k_{s}]$. The sequence $k_{1},\ldots,k_{s}$ is called the (negative) continued fraction expansion of the rational number $a/b$.

\theblock**.**

Given $a/b=[k_{1},\ldots,k_{s}]$ as above, define integers $\mu_{i}$ and $\tilde{\mu}_{i}$ for $0\leq i\leq s+1$ as follows. Start by setting

[TABLE]

Then, assuming that we have defined $\mu_{j},\tilde{\mu}_{j}$ for $0\geq j\geq i$ for some $i>0$, define

[TABLE]

Using induction, one finds

[TABLE]

Furthermore, the numbers $\mu_{i}$ and $\tilde{\mu}_{i}$ are positive for $i>1$ if $a>0$. A simple induction on $s$ also proves $\mu_{s+1}=a$ and $\tilde{\mu}_{s+1}=b$.

\thelemma** Lemma****.**

Let $a,b$ be positive integers with no common factors, and let $k_{i},\mu_{i},\tilde{\mu}_{i}$ be defined as above. The manifold $M=\bar{D}\times S^{1}$ is a plumbed manifold, given as $M=\cup_{i=1}^{s}M_{i}$ where

[TABLE]

We set $\Sigma_{1}=D_{\frac{1}{s}}$ and $\pi_{1}:M_{1}\to\Sigma_{1}$, $(rt_{1},t_{2})\mapsto rt_{1}$ where $r\in\mathbb{R}_{\geq 0}$ and $t_{i}\in S^{1}$, as well as $\Sigma_{i}=\bar{D}_{\frac{i}{s}}\setminus D_{\frac{i-1}{s}}$ for $i>1$ and $\pi_{i}:M_{i}\to\Sigma_{i}$, $(rt_{1},t_{2})\mapsto rt_{1}^{\mu_{i}}t_{2}^{\tilde{\mu}_{i}}$ for $i>1$. The section over $S^{1}\subset\Sigma_{s}$ is given by $t\mapsto(t^{\tilde{\mu}_{s+1}},t^{-\mu_{s+1}})$. The associated plumbing graph is shown in fig. 2.

Proof.

It is clear that the given components intersect in tori. Furthermore, eq. 5.2 gives $\gcd(\tilde{\mu}_{i},\mu_{i})=1$. It follows that $\pi_{i}$ is an $S^{1}$ fibration for all $i$. Another consequence of eq. 5.2 is that for $1\leq i<s$, the map $\pi_{i}\times\pi_{i+1}:M_{i}\cap M_{i+1}\to S^{1}\times S^{1}$ is a diffeomorphism and that fibers of $\pi_{i}$ and $\pi_{i+1}$ intersect positively in the torus $M_{i}\cap M_{i+1}$. The same equation shows that the map $t\mapsto(t^{\tilde{\mu}_{s+1}},t^{-\mu_{s+1}})$ is really a section:

[TABLE]

Therefore, $M$ is a plumbed manifold with $s$ components. What is left to show is that the Euler number for the $i^{\mathrm{th}}$ vertex, call it $v_{i}$, in the graph is $-k_{i}$. To see this, consider the function $\zeta:M\to\mathbb{C}$, $(z,t)\mapsto t$. The function does not vanish on $M$, and so the dual set of multiplicities vanish. We have parametrizations $S^{1}\to M_{i}$, $t\mapsto(rt^{-\tilde{\mu}_{i}},t^{\mu_{i}})$ of a fiber of $\pi_{i}$ for a suitable $r$. Thus, the multiplicities of $\zeta$ are given by $m_{v_{i}}=\mu_{i}$, and similarly, $m_{a}=\mu_{s+1}$, where $a$ is the arrowhead. Thus, by section 4, we have $-b_{v_{i}}\mu_{i}+\mu_{i-1}+\mu_{i+1}=0$ for $1\leq i\leq s$. Since the same equation holds with $b_{v_{i}}$ replaced with $k_{i}$ (and $\mu_{i}\neq 0$), we get $-b_{v_{i}}=-k_{i}$. ∎

6 Construction

In this section we state our main result in details. We construct a plumbing graph $G$ from the resolution graph $\Gamma$ along with the multiplicities $m_{v}$ and $l_{v}$ of $f$ and $g$. Theorem 6.1 says that this construction describes the boundary of the Milnor fiber of the hypersurface singularity given by $\Phi(x,y,z)=f(x,y)+zg(x,y)$.

\thedefinition** Definition****.**

• (i)

Let $\Gamma^{\prime}$ be a connected component of $\Gamma_{1}$. Let $\mathcal{V}(\Gamma^{\prime})$ be the vertex set of $\Gamma^{\prime}$ and, for $v\in\mathcal{V}(\Gamma^{\prime})$, let $\hat{\mathcal{E}}_{v}(\Gamma^{\prime})$ be the set of edges connecting $v$ and a vertex in $\mathcal{A}_{f,1}\cup\mathcal{W}_{2}$. Set also $\hat{\mathcal{E}}(\Gamma^{\prime})=\cup_{v\in\mathcal{V}(\Gamma^{\prime})}\hat{\mathcal{E}}_{v}(\Gamma^{\prime})$. For any edge $e\in\hat{\mathcal{E}}(\Gamma^{\prime})$ connecting $v\in\mathcal{V}(\Gamma^{\prime})$ and $w\in\mathcal{A}_{f,1}\cup\mathcal{W}_{2}$, set $v_{e}=v$ and $w_{e}=w$ and $m_{e}=\gcd(m_{v},m_{w})$. For $v\in\mathcal{V}(\Gamma^{\prime})$, let $\hat{\delta}_{v}$ be the number of edges connecting $v$ and some vertex in $\mathcal{V}(\Gamma^{\prime})$ or $\hat{\mathcal{V}}(\Gamma^{\prime})$. Let $d_{\Gamma^{\prime}}=\gcd_{v\in\mathcal{V}(\Gamma^{\prime})\cup\hat{\mathcal{V}}(\Gamma^{\prime})}m_{v}$ and define $g_{\Gamma^{\prime}},-b_{\Gamma^{\prime}}$ by the equations

[TABLE]

Since $d_{\Gamma^{\prime}}\neq 0$, these are well defined. As we will see later, we have $g_{\Gamma^{\prime}},-b_{\Gamma^{\prime}}\in\mathbb{Z}$.

The graph $G_{\Gamma^{\prime}}$ has vertex set $\mathcal{W}(G_{\Gamma^{\prime}})=\{v_{\Gamma^{\prime},1},\ldots,v_{\Gamma^{\prime},d_{\Gamma^{\prime}}}\}$, with each vertex decorated by the selfintersection number $-b_{\Gamma^{\prime}}$ and genus $[g_{\Gamma^{\prime}}]$. No two of these vertices are connected by an edge. Define $G_{1}$ as the disjoint union of the graphs obtained in this way.

• (ii)

Let $a\in\mathcal{A}_{f,1}$ and write $m_{a}/m_{w}=[k_{1},\ldots,k_{s}]$. The graph $G_{a}$ has $2s+1$ vertices $v_{a,1,+},\ldots,v_{a,s,+},v_{a,1,-},\ldots,v_{a,s,-},v_{a,0}$. There is an edge with sign $\pm$ connecting $v_{a,i,\pm}$ and $v_{a,i,\pm}$ for each $1\leq i\leq s-1$, as well as positive edges connecting $v_{a,0}$ and $v_{a,s,\pm}$. All these vertices have genus zero. The vertex $v_{a,i,\pm}$ has selfintersection $-b_{a,i,\pm}=\mp k_{i}$ and $v_{a,0}$ has selfintersection number $-b_{a,0}=0$.

Define $G_{f,1}$ as the disjoint union of these graphs.

• (iii)

Let $v_{1}\in\mathcal{W}_{1}$ and $v_{2}\in\mathcal{W}_{2}$ be vertices of $\Gamma$ connected by an edge $e$ and write $m_{v_{2}}/m_{v_{1}}=[k_{1},\ldots,k_{s}]$. The graph $G_{e}$ has vertices $v_{e,1,+},\ldots,v_{e,s,+}$, $v_{e,1,-},\ldots,v_{e,s,-},v_{e,0}$, each with genus zero. The vertex $v_{e,i,\pm}$ has the selfintersection number $-b_{e,i,\pm}=\mp k_{i}$ and $v_{e,0}$ has selfintersection $b_{e,0}=0$. We have an edge with sign $\pm$ connecting $v_{e,i,\pm}$ and $v_{e,i+1,\pm}$, as well as positive edges connecting $v_{e,s,\pm}$ and $v_{e,0}$.

Let $G_{\mathrm{b}}$ be the disjoint union of graphs obtained in this way.

• (iv)

The graph $G_{2}$ is defined as follows. For each $w\in\mathcal{W}_{2}$, we have two vertices $v_{w,+},v_{w,-}$ in $G_{2}$ and these are all the nonarrowhead vertices of $G_{2}$. They are decorated by genus zero and have selfintersection number $-b_{w,\pm}=\mp b_{w}$, where $-b_{w}$ is the selfintersection number of $w$ in $\Gamma$.

• (v)

Let $a\in\mathcal{A}_{g,2}$. The graph $G_{a}$ has nonarrowhead vertices $v_{a,0},\ldots,v_{a,m_{w}}$ where $w=w_{a}$, each of genus zero. The vertex $v_{a,0}$ has selfintersection number $-b_{a,0}=0$, whereas $v_{a,i}$ has selfintersection $-b_{a,i}=-l_{a}$. For each $1\leq i\leq m_{w}$ there is a negative edge connecting $v_{a,0}$ and $v_{a,i}$.

Let $G_{g,2}$ be the disjoint union of graphs obtained in this way.

\thedefinition** Definition****.**

The graph $G$ is the disjoint union of the graphs $G_{1}$, $G_{f,1}$, $G_{\mathrm{b}}$, $G_{2}$, $G_{g,2}$, with the following additional edges.

• (i)

For $\Gamma^{\prime}\subset\Gamma_{1}$ a connected component, $1\leq i\leq d_{\Gamma^{\prime}}$ and $a\in\mathcal{A}_{f,1}$, connect $v_{\Gamma^{\prime},i}$ and $v_{a,0}$ with $m_{e}/d_{\Gamma^{\prime}}$ negative edges, where $m_{e}$ is as in section 6i.

• (ii)

Similarly, assuming that $\Gamma^{\prime}\subset\Gamma_{1}$ is a connected component, $1\leq i\leq d_{\Gamma^{\prime}}$ and that $v_{1}\in\mathcal{W}(\Gamma^{\prime})$ and $v_{2}\in\mathcal{W}_{2}$ are connected by an edge $e\in\mathcal{E}(\Gamma)$. Connect $v_{\Gamma^{\prime},i}$ and $v_{e,0}$ with $m_{e}/d_{\Gamma^{\prime}}$ negative edges and connect $v_{w,\pm}$ and $v_{a,1,\pm}$ with an edge with sign $\pm$.

• (iii)

Let $a\in\mathcal{A}_{g,2}$ and $w=w_{a}$. The vertex $v_{a,0}$ is connected to both $v_{w,+}$ and $v_{w,-}$ by a positive edge.

6.1 Theorem**.**

The boundary of the Milnor fiber of the singularity $f(x,y)+zg(x,y)=0$ at the origin is a plumbed manifold with plumbing graph $G$.

\therem** Remark****.**

• (i)

Let $a\in\mathcal{A}_{g,2}$ and assume that $l_{a}=1$. In this case, the vertices $v_{a,1},\ldots,v_{a,m}$, where $w=w_{a}$ and $m=m_{w}$, blow down to simplify the graph (see [13] for blowing down). This operation removes these vertices, and replaces the Euler number $-b_{a,0}=0$ with $-b_{a,0}=m$.

• (ii)

We can apply the operation R0(a) from [13] to the vertices $v_{\Gamma^{\prime},i}$ for a connected component $\Gamma^{\prime}\subset\Gamma_{1}$ as well as to $v_{a,i}$ for $a\in\mathcal{A}_{g,2}$ and $1\leq i\leq m_{w_{a}}$. This way, all the edges adjacent to these vertices will be positive instead of negative. Note, however, that this also changes the sign of the corresponding multiplicities given in section 7.

7 A multiplicity system for $z$

In this section, we give multiplicities and dual multiplicities for the function $z$. For simplicity, the multiplicitiy and dual multiplicity for a vertex $v_{*}$ constructed in section 6 will be denoted by $m_{*}$ and $n_{*}$. The proof of theorem 7.1 is given in section 9.

7.1 Theorem**.**

The restriction $z|_{M}$ of the coordinate function $z$ to the boundary $M=\partial F_{\Phi}$ of the Milnor fiber of $\Phi$ satisfies the conditions given in section 4. Furthermore, the associated families of multiplicities $(m_{v})_{v\in\mathcal{V}(G)}$ and dual multiplicities $(n_{v})_{v\in\mathcal{W}(G)}$ are is given as follows.

• (i)

If $\Gamma^{\prime}\subset\Gamma_{1}$ is a connected component, then $m_{v}=1$ and $n_{v}=0$ for $v=v_{\Gamma^{\prime},i}$, $i=1,\ldots,d_{\Gamma^{\prime}}$.

• (ii)

Let $a\in\mathcal{A}_{1,f}$ be an arrowhead connected to $w\in\mathcal{W}_{1}$ in $\Gamma$ and set $\tilde{m}_{w}=m_{w}/\gcd(m_{w},m_{a})$ and $\tilde{m}_{a}=m_{a}/\gcd(m_{w},m_{a})$. Write $m_{w}/m_{a}=\tilde{m}_{w}/\tilde{m}_{a}=[k_{1},\ldots,k_{s}]$ and define $\mu_{i},\tilde{\mu}_{i}$ as in section 5. The multiplicities of $z$ are given by

[TABLE]

for $i=1,\ldots,s$ and $m_{a,0}=-\tilde{m}_{a}l_{w}$. The dual multiplicities for these vertices are given by $n_{a,1,+}=m_{a}$, and [math] otherwise.

• (iii)

Let $v_{1},v_{2}$ and $e$ be as in section 6iii*. Let $\tilde{m}_{i}=m_{v_{i}}/\gcd(m_{v_{1}},m_{v_{2}})$ for $i=1,2$. Write $m_{v_{2}}/m_{v_{1}}=\tilde{m}_{2}/\tilde{m}_{1}=[k_{1},\ldots,k_{s}]$ and define $\mu_{i},\tilde{\mu}_{i}$ for $i=1,\ldots,s$ as in section 5. Then $m_{e,0}=\tilde{m}_{1}l_{v_{2}}-\tilde{m}_{2}l_{v_{1}}$ and*

[TABLE]

The dual multiplicities vanish on these vertices.

• (iv)

Let $w\in\mathcal{W}_{2}$. Then $m_{w,+}=m_{w}-l_{w}$ and $m_{w,-}=l_{w}$. The dual multiplicities are given by $n_{w,+}=\sum_{w_{a}=w}m_{a}$ and $n_{w,-}=0$.

• (v)

Let $a\in\mathcal{A}_{g,2}$ and set $w=w_{a}$. Then $m_{a,0}=-l_{a}$ and $m_{a,i}=1$ for $i=1,\ldots,m_{w}$. The dual multiplicities associated with $v_{a,i}$ vanish.

\therem** Remark****.**

Let $e,v_{1}$ be as in theorem 7.1iii. One proves that, in fact, $m_{e,1,+}=m_{v_{1}}-l_{v_{1}}$ and $m_{e,1,-}=l_{v_{1}}$.

8 Examples

\theexample** Example****.**

The singularity $T_{a,b,\infty}$ is the singularity at the origin of the hypersurface given by $x^{a}+y^{b}+xyz=0$. In the case $b=2$, the boundary of the Milnor fiber has been described in [12]. We take $f(x,y)=x^{a}+y^{b}$ and $g(x,y)=xy$. We will assume $a,b$ satisfying $a\geq b\geq 2$ and $a>2$. We claim that the boundary of the Milnor fiber of this singularity is given by the plumbing graph

Let $\phi:V\to\mathbb{C}^{2}$ be the minimal resolution of the plane curve $fg$ and let $\Gamma$ be its resolution graph. Then $\Gamma$ is a string with two arrowheads corresponding to $g$, one on each end of the string, as well as $d:=\gcd(a,b)$ arrowheads corresponding to $f$. Name the nonarrowhead vertices of the graph $v_{1},\ldots,v_{s}$ so that $v_{i},v_{i+1}$ are adjacent. Let $-b_{i}$ be the selfintersection number associated with the vertex $v_{i}$. There is a unique $j$ so that $-b_{j}=-1$. The set $\mathcal{A}_{f}$ consists of $d$ arrowheads, each connected to $v_{j}$, whereas $\mathcal{A}_{g}$ consists of two arrowheads, one connected to $v_{1}$ and the other to $v_{s}$. Write also $m_{i},l_{i}$ for the multiplicities of $f$ and $g$ on $v_{i}$.

Claim: We have $m_{1}\geq l_{1}$ and $m_{s}>l_{s}$ and $m_{i}>l_{i}$ for $i=2,\ldots,s-1$.

In fact, using [3, Lemma 20.2], one finds $m_{j}=ab/d$ and $l_{j}=a/d+b/d$. It follows from our assumptions that $m_{j}-l_{j}>0$. Now, define integers $r_{i}=m_{i}-l_{i}$ for $i=1,\ldots,s$ and $r_{0}=r_{s+1}=-1$. We then have

[TABLE]

It follows easily that this sequence increases strictly from $r_{0}=-1$ to $r_{j}=m_{j}-l_{j}$, and then decreases strictly from $r_{j}$ to $r_{s+1}=-1$. Since these are integers, the claim follows.

We leave to the reader to show that the equality $m_{i}=l_{i}$ holds for $i=1$ or $i=s$ if and only if $b=2$, the case already covered by Némethi and Szilárd [12]. This can be achieved by calculating $m_{i}$ and $l_{i}$ explicitly using Lemma 20.2 of [3].

We start by showing how the above graph is obtained from the output of the algorithm in the case when $l_{i}>m_{i}$ for all $i$. Since $\mathcal{W}=\mathcal{W}_{2}$, the graph $G_{2}$ consists of two strings, one of them identical to $\Gamma$, the other one having Euler numbers with opposite signs and negative edges. In addition, we have $\mathcal{A}_{g,2}=\{a_{x},a_{y}\}$, two arrowhead vertices corresponding to the strict transform of the factors $x$ and $y$ of $g$. As described in section 6, the graph $\Gamma$ can be taken as these two strings, connected on each end by vertices with Euler number $m_{x}$ and $m_{y}$. These are the multiplicities of $f$ along the components on the end of the string. It follows from [3] that these multiplicities are $a$ and $b$. Furthermore, the two strings blow down (we can blow down the vertices one by one in the opposite order in which they appear during the process of resolving $f$). Each string is replaced by an edge, the first string by a positive edge, the second one by a negative edge. Below, we explicate the case when $a=7$ and $b=5$.

In the case when either $m_{1}=l_{1}$ or $m_{s}=l_{s}$, the algorithm has, in fact, the same output. We let it suffice to clarify this principle by considering an example. Take $a=3$ and $b=2$. A resolution graph $\Gamma$, decorated with the pairs of multiplicities $(m_{v},l_{v})$ is shown in fig. 6.

We see that $\mathcal{W}_{2}$ now only contains the vertex $v_{2}$, whereas $v_{1},v_{3}\in\mathcal{W}_{1}$, each providing a connected component of $\Gamma_{1}$. We order the vertices in fig. 6 in such a way that $b_{1}=-2$ and $b_{3}=-3$. Applying section 6i to the component $\Gamma^{\prime}$ of $\Gamma_{1}$, containing only the vertex $v_{1}$, as well as $\Gamma^{\prime\prime}$ containing only $v_{3}$, we get

[TABLE]

We get five new vertices. The edges $e_{1}=\{v_{1},v_{2}\}$ and $e_{2}=\{v_{2},v_{3}\}$ are of the form described in section 6iii. The five new vertices are connected to $v_{e_{1},0}$ and $v_{e_{2},0}$ to obtain the graph in fig. 7, which also shows the multiplicities of the function $z$. After blowing down, we obtain fig. 3.

\theexample** Example****.**

Consider the plane curves

[TABLE]

where $\lambda_{1},\lambda_{2}\in\mathbb{C}\setminus\{0,1\}$ are distinct and $\mu_{1},\mu_{2}\in\mathbb{C}\setminus\{0\}$ are distinct. The resolution graph $\Gamma$, decorated with the multiplicities $m$ and $l$ is given in fig. 8. The set $\mathcal{W}_{1}$ consists of the first three vertices appearing during the resolution process, corresponding to the first Puiseux pair, where $f$ and $g$ have equal multiplicities. But, as $f$ has more components with two Puiseux pairs, compared with $g$, the multiplicities of $f$ are higher along the second part of the resolution process, that is, the three vertices appearing last.

The graph $\Gamma_{1}$ is connected, and the invariants from section 6i are easily computed using eq. 6.1 and eq. 6.2:

[TABLE]

Thus, we obtain a single vertex $v_{\Gamma_{1}}$ with genus $17$ and Euler number $-90$. Furthermore, $m_{\Gamma_{1}}=1$.

The set $\mathcal{A}_{f,1}$ contains one element. Using the notation in section 6ii, we have $m_{a}=1$, $m_{w}=30$ and $l_{w}=30$. We have $m_{a}/m_{w}=[1,2,\ldots,2]$ where the number of $2$’s is $29$. Therefore, $61$ new vertices are created. The vertex $v_{a,0}$ is connected to $v_{\Gamma_{1}}$ by $m_{e}=1$ edge with a negative sign. Furthermore,

[TABLE]

Note that the vertices $v_{a,i,\pm}$ blow down, leaving only the vertex $v_{a,0}$.

There is one edge $e$ connecting $\mathcal{W}_{1}$ and $\mathcal{W}_{2}$. In the notation of section 6iii, we have $m_{e}=2$ and

[TABLE]

We have $17/15=[2,2,2,2,2,2,2,3]$, hence the creation of $17$ new vertices. The vertices $v_{e,0}$ and $v_{\Gamma_{1}}$ are connected by two edges with negative sign. Furthermore, we get

[TABLE]

The set $\mathcal{W}_{2}$ contains three elements, inducing six new vertices with genus [math] and Euler number $\mp 3,\mp 1,\mp 2$. The $\mp 3$ curves are connected with the vertices $v_{e,1,\pm}$ from the previous construction.

The set $\mathcal{A}_{g,2}$ contains one vertex, say $a\in\mathcal{A}_{g,2}$, as in section 6v. Set also $w=w_{a}$. Since $m_{w}=72$ and $l_{a}=1$, we get a vertex $v_{a,0}$ with genus [math] and Euler number [math], connected to the vertices $v_{a,1},\ldots,v_{a,72}$ with genus [math] and Euler number $-1$, via a negative edge. We have $m_{a,0}=-1$ and $m_{a,i}=(1)$ for $1\leq i\leq 72$.

9 Proofs

To prove theorems 6.1 and 7.1, we start by defining pieces $M_{v}\subset M$ and projections $\pi_{v}:M_{v}\to\Sigma_{v}$ for all vertices $v\in\mathcal{V}$, using the description in theorem 2.1. From the construction, it will be clear that $M=\cup_{v\in\mathcal{V}}M_{v}$, and that individual pieces intersect according to the edges of $G$. Finally, we verify the formulas for genera and selfintersection numbers. In fact, it will be clear that the genus decoration is zero, except for in the case of $v_{\Gamma^{\prime}}$, where an argument similar to A’Campo’s formula [1] is used. Similarly as in [11, 12], nontrivial Euler numbers are determined using the multiplicities of $z$ and section 4. We note that the proof of theorem 7.1, can be carried out as soon as the projections $\pi_{v}$ are defined. In particular, this proof does not use the Euler numbers, which are computed using the multiplicities of $z$.

Proof of theorem 6.1.

We start by providing sets $M_{v}$ for each vertex of the graph $G$. We then prove that these pieces provide a plumbing structure on $M$ with the plumbing graph $G$.

• (i)

Let $X^{\prime}_{1}$ be the closure of the set

[TABLE]

By construction, this is a closed tubular neighbourhood of

[TABLE]

in particular, we have a disk bundle $X^{\prime}_{1}\to F_{f,1}$. We can assume that the intersection of this disk bundle with the divisor associated with $g$ is a set of disks. Let $X_{1}$ be the four manifold obtained from $X^{\prime}_{1}$ by twisting along these disks as in section 2 and let $M_{1}\subset\partial X_{1}$ be the associated $S^{1}$ bundle. It is then clear that we have $M_{1}\subset M_{f,g}$, that the boundary of $M_{1}$ consists of tori and that $M_{1}$ is in a natural way an $S^{1}$ bundle over $F_{f,1}$.

Let $\Gamma^{\prime}\subset\Gamma_{1}$ be a component as in section 6i. Setting

[TABLE]

we obtain correspondingly $X_{\Gamma^{\prime}}\subset X_{1}$ and $M_{\Gamma^{\prime}}\subset M_{1}$. This way, $M_{\Gamma^{\prime}}$ is an $S^{1}$ bundle over the surface $F_{f,\Gamma^{\prime}}=F_{f}\cap X^{\prime}_{\Gamma^{\prime}}$.

Firstly, we note that the number of connected components of $F_{f,\Gamma^{\prime}}$ is precisely $d_{\Gamma^{\prime}}$ and that, furthermore, the monodromy permutes these components cyclically. This follows from Proposition 2.20 of [11], see also 2.21 of the same article.

Secondly, the genus of the components of $F_{f,\Gamma^{\prime}}$ is $g_{\Gamma^{\prime}}$, satisfying eq. 6.1. This follows from a small generalization of A’Campo’s formula [1] which gives

[TABLE]

What is more, $F_{f,\Gamma^{\prime}}$ has $m_{e}$ boundary components close to the intersection of $E_{v}$ and $E_{w_{e}}$ for $e\in\hat{\mathcal{E}}(\Gamma^{\prime})$. Thus, $F_{f,\Gamma^{\prime}}$ has a total of $\sum_{e\in\hat{\mathcal{E}}(\Gamma^{\prime})}m_{e}$ boundary components.

The formula eq. 6.2 is verified below.

• (ii)

Let $a\in A_{1,f}$ and set $w=w_{a}$. Define $M_{a}=\partial\overline{T}_{f,g}\cap\overline{T}_{a}$ We have coordinates $u,v$ on $T_{a}$ so that $T_{a}\cong\left\{(u,v)\,\middle|\,|u|\leq 1,\,|v|\leq 1\right\}$ and so that $E_{a}\cap T_{a}$ and $E_{w}\cap T_{a}$ are the vanishing sets of $u$ and $v$, respectively. We can then write $M_{a}=M_{a,+}\cup M_{a,-}\cup M_{a,0}$ where for certain $0<\eta\ll\varepsilon\ll 1$, we have

[TABLE]

and $M_{a,0}$ is defined by setting

[TABLE]

where $N$ is an $\varepsilon$ neighbourhood around $F_{f}\cap M_{a,0}^{\prime}$. The projection of this picture via $(u,v)\mapsto(|u|,|v|)$ is shown in fig. 10.

We can assume that in the coordinates $u,v$, we can write $f|_{T_{a}}(u,v)=u^{m_{a}}v^{m_{w}}$. Define $\tilde{m}_{a}=m_{a}/m_{e}$ and $\tilde{m}_{w}=m_{w}/m_{e}$. We find

[TABLE]

In fact, we have an $S^{1}$ bundle projection $\pi^{\prime}_{a,0}$ mapping $M^{\prime}_{a,0}$ to an annulus by the formula $\pi^{\prime}_{a,0}(u,v)=u^{\tilde{m}_{a}}v^{\tilde{m}_{v}}$. This way, $F_{f}\cup{M^{\prime}}_{a}^{0}$ consists of $m_{e}$ fibers of $\pi^{\prime}_{0,a}$. In particular, we can assume that $\pi^{\prime}_{a,0}$ restricts to an $S^{1}$ bundle $\pi_{0,a}=\pi^{\prime}_{0,a}|_{M_{0,a}}$. We orient the fibers so that one of them is parametrized as $t\mapsto(t^{-\tilde{m}_{w}},t^{\tilde{m}_{a}})$, which induces an orientation on the target space of $\pi_{0,a}$.

By section 5, the manifold $M_{a,+}$ can be given as a plumbed manifold with plumbing graph as in fig. 2, where $[k_{1},\ldots,k_{s}]=m_{a}/m_{w}$, so that the section corresponding to the arrowhead to the right can be chosen to coincide with a fiber of $M_{a,0}$, with the opposite orientation. Furthermore, we have an orientation reversing diffeomorphism $M_{a,+}\to M_{a,-}$ given by $(u,v)\mapsto(u,\eta v)$. This way, we see $M_{a,-}$ as a plumbed manifold with the same plumbing graph, modified by changing signs on all selfintersection numbers as well as edges.

At this point, we have shown that the manifold $M_{a}$ is a plumbed manifold with plumbing graph $G_{a}$, with $m_{e}$ dashed arrows added to $v_{a,0}$, except we did not specify a section corresponding to these arrowheads. Furthermore, (and this cannot be done without the sections) we have not determined the Euler number associated with $v_{a,0}$. Since $M_{0,a}$ is obtained by removing a tubular neighbourhood around a fiber of the projection $\pi^{\prime}_{0,a}$, we can choose as a section a meridian around this fiber. Note that this section is exactly a fiber of the projection $\pi_{\Gamma^{\prime},i}$ for a suitable $1\leq i\leq d_{\Gamma^{\prime}}$, where $w$ is a vertex of the component $\Gamma^{\prime}$ of $\Gamma_{1}$. But we can be more specific. Let $\psi:S^{1}\to M_{a,+}$ be a parametrization of a fiber in the boundary component of $M_{a,s,+}$. This induces a map $S_{1}\times[\eta,1]\to{M^{\prime}}_{a,0}$ which is a global section to the fibration of $M^{\prime}_{a,0}$, restricting to a global section to the fibration of $M_{a,0}$, which again restricts to a parametrization of the fibers of $M_{a,s,\pm}$, as well as a parametrization of a meridian around $F_{f}\cap M_{a,0}$. This shows that with this choice of sections on the boundary, the Euler number of the bundle $M_{a,0}$ is [math].

Finally, we note that $M_{a,0}$ intersects $M_{\Gamma^{\prime}}$ in exactly $m_{e}$ tori, and that the number of these tori in each component of $M_{\Gamma^{\prime}}$ is the same. It follows from the construction that in each of these tori, a fiber of $M_{a,0}$ and a fiber from $M_{\Gamma^{\prime}}$ form an integral basis on homology. Furthermore, one verifies that an oriented fiber of $M_{a,0}$ in such a torus is an oriented section of $M_{\Gamma^{\prime}}$. This can be seen by noting that both wind around $E_{a}$ with multiplicity $-\tilde{m}_{a}$. Therefore, these tori yield edges with a negative sign, by section 3. These are the edges defined in section 6i.

• (iii)

Let $e$ and $v_{i}\in\mathcal{W}_{i}$ be as in 6iii. Let $D$ be a disc in $E_{v_{1}}$ with center the intersection point of $E_{v_{1}}$ and $E_{v_{2}}$ corresponding to $e$ which is a slightly bigger than the corresponding disk in $E_{v_{1}}\cap\overline{T}_{v_{2}}$. We can add the preimage of $D$ in $\overline{T}_{v_{1}}\setminus T^{\prime}_{v_{2}}$ to $\overline{T}_{f,g}$ without changing its diffeomorphism type. From here, the proof follows similarly as in the previous case. A schematic picture is shown in fig. 11.

• (iv)

Let $w\in\mathcal{W}_{2}$ and define $M_{w}$ as the closeure of $\overline{T}_{w}\setminus\cup_{v}\overline{T}_{v}$, where the union $\cup_{v}$ ranges over $v\in\mathcal{W}\cup\mathcal{A}_{g}\setminus\{w\}$. It follows from construction that $M_{w}$ consists of two copies of an $S^{1}$ bundle over the surface $E_{w}$ with a disk removed for each neighbour in $\mathcal{W}\cup\mathcal{A}_{g}\setminus\{w\}$. Indeed, these are the corresponding subsets of the boundaries of $T_{w}$ and $T^{\prime}_{w}$ (recall section 2). Write $M_{w,+}$ and $M_{w,-}$ for the outer and inner components. These components will correspond to the vertices $v_{w,\pm}$.

These fibrations extends canonically over the disks removed from $E_{w}$ by taking $T_{w}$ and $T^{\prime}_{w}$, and we can take a meridians around central fibers as the trivializing section on the boundary. It follows immediately that the two $S^{1}$ bundles have relative Euler numbers $-b_{w,\pm}=\mp b_{w}$. Furthermore, since $E_{w}$ is a rational curve, the two vertices $v_{w,\pm}$ have associated genus [math].

As both components of $M_{w}$ are boundaries of similar tubular neighbourhoods, they can be identified, but the inner one, i.e. the boundary of $T^{\prime}_{w}$ has its orientation reversed. We will consider this part as a fibration over the same base as the outer component. Therefore, a fiber in $M_{w,+}$ is a meridian around $E_{w}$, whereas a fiber in $M_{w,-}$ is a (relatively small) meridian around $E_{w}$ with the orientation reversed.

If $w,w^{\prime}\in\mathcal{W}_{2}$ are joined by an edge, it follows easily that the two components corresponding to $w$ intersect with those of $w^{\prime}$ in the same way as prescribed by the resolution graph $\Gamma$.

• (v)

Finally, we describe what happens close to a component of the strict transform of $g$ corresponding to $a\in\mathcal{A}_{g,2}$.

Let $M^{\prime}_{a}=M^{\prime}\cap\overline{T}_{a}$ and $M_{a}$ the corresponding twisted subset of $M$. Let $M_{a,0}=M\cap\partial T_{a}$. It follows from construction that $M_{a}$ fibers by a map $\pi_{a,0}$ over the disk $E_{a}$ with $m_{a}+1$ smaller disks removed, one corresponding to $T^{\prime}_{w_{a}}$, and $m_{a}$ of them corresponding to $T_{\varepsilon}$. We orient the fiber to coincide with that of a meridian around $E_{a}$. This chooses an orientation of the base space of $\pi_{a,0}$, the opposite of the standard one on $E_{a}$. This bundle is trivialized in a similar way as in ii, yielding Euler number $-b_{a,0}=0$. It is also clear that $g_{a,0}=0$.

The closure of $M_{a}\setminus M_{a,0}$ is an $S^{1}$ bundle over $F_{f}\cap T_{a}\cong\amalg_{m_{w}}\overline{D}$. This gives $m_{w}$ pieces $M_{a,1},\ldots,M_{a,m_{w}}$, ordered arbitrarily. The fibers are meridians around $F_{f}$.

We see that $M_{a}$ is a plumbed manifold with plumbing graph $G_{a}$ (with some dashed arrows added, corresponding to the boundary). Using section 3, we see that the edges between $v_{a,0}$ and $v_{a,i}$ have a negative sign, whereas the edges connecting $v_{a,0}$ and $v_{w,\pm}$ are positive.

In i to v above we have assigned subsets $M_{v}\subset M$ to each vertex $v$ of the graph $G$ constructed in sections 6 and 6. It is clear that each piece is connected and that each boundary components of any of the pieces are tori. Furthermore, the components of intersection of two pieces correspond to the edges connecting the corresponding vertices. The base space of each fibration is a surface of the genus specified, or zero otherwise.

The only part which remains to prove is eq. 6.2. But this follows immediately from section 4 and theorem 7.1. ∎

Proof of theorem 7.1.

• (i)

Let $\Gamma^{\prime}$ be as in section 6i. It follows from the proof in [14] that $|z|$ is constant on $M_{\Gamma^{\prime},i}$ (and nonzero). It follows that the dual multiplicities $n_{\Gamma^{\prime},i}$ vanish. A fiber in $M_{\Gamma^{\prime},i}$ is an oriented meridian around $F_{f}$. The restriction of $z=(f-\varepsilon)/g$ to such a fiber is a map of degree $1$, thus $m_{\Gamma^{\prime},i}=1$.

• (ii)

Let $a\in\mathcal{A}_{1,f}$ as in section 6ii. We start by observing that the vanishing set of the function $z=(f-\varepsilon)/g$ is contained in the piece $M_{a,1,+}$. The vanishing set of $z$ is the Milnor fiber $F_{f}$ of $f$. The intersection $F_{f}\cap\partial T_{a}$ consists of two parts, contained in neighbourhoods around $E_{w}\cap\partial T_{a}$ and $E_{a}\cap\partial T_{w}$. By construction, the former is not included in $M_{a}$. The latter is homologous to a meridian around $E_{w}$ with multiplicity $m_{w}$. We can take this meridian as $E_{a}\cap\partial T_{a}$. Therefore, the dual multiplicities vanish on all vertices of $G_{a}$ except for $v_{a,1,+}$ and we have $n_{a,1,+}=m_{a}$.

It follows from the explicit calculations in ii in the proof of theorem 6.1 that the restriction of $f-\varepsilon$ to a fiber of $M_{a,0}$ has degree zero. Indeed, in the coordinates $u,v$ introduced there for the polydisk $T_{a}$, we have $f|_{T_{a}}=u^{m_{a}}v^{m_{w}}$ and a fiber in $M_{a,0}$ is parametrized in these coordinates by $[0,1]\to T_{a}$, $t\mapsto(e^{-t\tilde{m}_{w}2\pi i},e^{t\tilde{m}_{a}2\pi i})$. Since $g$ vanishes with order $l_{w}$ along $E_{w}$, and does not vanish along $E_{a}$, it follows that the multiplicity $m_{a,0}$ equals $-l_{w}\tilde{m}_{a}$.

Now, the sequence $m_{a,i,+}$, $i=1,\ldots,s$ satisfies

[TABLE]

The same equations are satisfied by the sequence $(l_{w}-\tilde{m}_{w})\mu_{i}-m_{a}\tilde{\mu}_{i}$, as is easily checked. It follows that the two sequences coincide, since the matrix associated with this system of linear equations is negative definite. A similar argument proves the statement for the multiplicities $m_{a,i,-}$.

• (iii)

Let $e$ be an edge in $G$ connecting $v_{1}$ and $v_{2}$ as in section 6iii. We start by observing that $z$ does not vanish on $M_{e}$, and so all dual multiplicities are vanish for the vertices of $G_{e}$.

Similarly as above, we find that the map $f-\varepsilon$ restricted to a fiber $C_{e,0}\subset M_{e,0}$ has degree zero, and $g$ has degree $\tilde{m}_{2}l_{v_{1}}-\tilde{m}_{1}l_{v_{2}}$. It follows that $m_{e,0}=\tilde{m}_{1}l_{v_{2}}-\tilde{m}_{2}l_{v_{1}}$. Now, similar reasoning as above determines the multiplicities $m_{e,i,\pm}$. Namely, we have linear equations

[TABLE]

The result follows as soon as we determine the multiplicities $m_{w_{2},\pm}$:

• (iv)

Let $w\in\mathcal{W}_{2}$. Above, we have determined that $C_{w,-}$ is a meridian around $E_{w}$, small with respect to $\varepsilon$ and having the opposite orientation than the standard meridian. It follows that $z=(f-\varepsilon)/g$ restricted to $C_{w,-}$ has degree $l_{w}$, i.e. $m_{w,-}=l_{w}$. Furthermore, $z$ does not vanish on $M_{w,-}$, thus $n_{w,0}=0$.

On the other hand, $C_{w,+}$ is an oriented meridian around $E_{w}$, with respect to which $\varepsilon$ is chosen small. It follws that $m_{w,+}=m_{w}-l_{w}$. Furthermore, the vanishing set of $z$ in $M_{w,+}$ is homologous to the strict transform of $f$, with multiplicities. Therefore, each $a\in\mathcal{A}_{f,2}$ contributes $m_{a}$ to $n_{w,+}$, resulting in the sum given.

• (v)

Let $a\in\mathcal{A}_{g,2}$ and set $w=w_{a}$. Similarly as in i, we find that $z$ does not vanish on $M_{a}$. Therefore, $n_{a,i}=0$ for $i=0,\ldots,m_{w}$. In v of the proof of theorem 6.1 we found that $C_{a,0}$ is a meridian around $E_{a}$. We can assume that the restriction of $f-\varepsilon$ to such a meridian has degree [math]. It is also clear that $g$ restricts to a degree $-l_{a}$ map on such a fiber. It follows that $m_{a,0}=-l_{a}$.

For $i=1,\ldots,m_{w}$, the fiber $C_{a,i}$ is a small meridian around $F_{f}$. It follows that $m_{a,i}=1$. ∎