| Izvestiya: Mathematics |
|
|
|
|
|
On the local fundamental group of the complement of a curve in a normal surface Vik. S. Kulikov Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Bibliographic databases:
     IntroductionLet $(S,C)$ be a pair, where $S$ is a normal complex (not necessary compact) surface and $C\subset S$ is an effective reduced Weil divisor (possibly zero) on $S$ (hereinafter referred to as a curve lying in $S$) with the following properties: (i) $C$ is a connected subset of $S$, (ii) if $C$ is a zero divisor, then $C=o\in S$ is a point, (iii) if an irreducible component of $C$ is not compact (in this case, such components will be called curve germs), then it is homeomorphic to the disc $\mathbb D_1=\{ z\in\mathbb C\mid |z|< 1\}$, has at most one singular point, and this point is the image of the centre of $\mathbb D_1$ (the image of the centre of $\mathbb D_1$ will be called the centre of this component), (iv) each non-compact irreducible component of $C$ intersects other irreducible components of $C$ only at its centre. Denote by $C_0$ the union of all compact irreducible components of $C$ (if $C$ has not a one-dimensional compact component, then $C_0$ is a point common to all irreducible components of $C$). Let $U\subset S$ be an open neighbourhood (relative to the complex analytic topology) of the curve $C_0$. A proper holomorphic mapping $\nu\colon X\to U$ will be called the resolution of singularities of $C$ if 1) $X$ is a non-singular surface and $\nu\colon X\setminus \nu^{-1}(C)\to U\setminus C$ is a bi-holomorphic mapping, 2) $\widetilde C=\nu^{-1}(C)$ is a divisor with normal crossings in $X$, 3) each irreducible component of $\widetilde C$ is a non-singular curve, 4) any two different irreducible components of $\widetilde C$ can meet at most in a single point, 5) if $C=o$ is a point or all irreducible components of $C$ are not compact, then $\nu\colon X\to U$ is not a bi-holomorphic mapping. We set $\widetilde C_0=\nu^{-1}(C_0)\subset X$. It follows from condition 5) that $\widetilde C_0$ is a non-empty union of compact curves, and by applying the Stein Factorization Theorem (see [1]), to the mapping $\nu$ and since $C$ is connected it follows that $\widetilde C$ and $\widetilde C_0$ are connected sets. Based on existence of a polynomial real-valued function $\alpha \colon X\to \mathbb R$ such that $\alpha(x)\geqslant 0$ for $x\in X$ and $\alpha(\widetilde C_0)=0$, in [2] (see also [3]) a “tubular” neighbourhood of the curve $C_0\subset S$ was defined. In the present paper, we use a slightly modified definition of a set $\mathcal U_C$ of “tubular” neighbourhoods $U_{\varepsilon}\subset S$ of $C_0$ based on existence of good (with respect to the curve $\widetilde C$) Hermitian metrics (that is, positive definite Hermitian quadratic forms) $(ds)^2$ on $X$. The set $\mathcal U_C$ has the following properties: – the set $\mathcal U_C$ is a base of open subsets of $S$ containing the curve $C_0$, – for all $U_{\varepsilon}\in \mathcal U_C$, the fundamental groups $\pi_1(U_{\varepsilon}\setminus C)$ are naturally isomorphic. We call $\pi_1^{\mathrm{loc}}(S,C):=\pi_1(U_{\varepsilon}\setminus C)$, $U_{\varepsilon}\in \mathcal U_C$, the local fundamental group of the complement of $C$ in $S$. We say that a Hermitian metric $(ds)^2$ defined in a neighbourhood $\widetilde U\subset X$ of $\widetilde C_0$ is good (with respect to $\widetilde C$) if, for each singular point $p$ of $\widetilde C$, there is a neighbourhood $V_p\subset \widetilde U$ such that (i) in each $V_p$, there are local coordinates $(z_1,z_2)$ such that $p=(0,0)$ and
$$
\begin{equation*}
V_p\simeq\mathbb B_2=\{(z_1,z_2)\in \mathbb C^2\mid \sqrt[2]{|z_1|^2+|z_2|^2}<2\},
\end{equation*}
\notag
$$
(ii) $z_1z_2=0$ is the equation of $\widetilde C\cap V_p$, (iii) $(ds)^2$ is given in $V_p$ by $(ds)^2=dz_1\,d\overline z_1+dz_2\,d\overline z_2$, (iv) $V_{p_1}\cap V_{p_2}=\varnothing$ for $p_1\neq p_2$. To define the set of “tubular” neighbourhoods $\mathcal U_C$, consider a neighbourhood $U\subset S$ of $C_0$ such that: first, if a point $p\in \operatorname{Sing} S\cap U$, then $p\in C$, and, second, the pair $(U,U\cap C)$ satisfies conditions (i)–(iv). In § 1.1, it is proved the existence of good metrics in compactly imbedded in $X$ neighbourhoods $\widetilde U$ of $\widetilde C_0$, and, in § 2, it is shown that, for given good metric $(ds)^2$ in $\widetilde U\subset X$, there is a positive $\varepsilon_0:=\varepsilon_0((ds)^2)\ll 1$ such that the sets $\widetilde U_{\varepsilon_1}\setminus \widetilde C_0$ and $\widetilde U_{\varepsilon_2}\setminus \widetilde C_0$ are homeomorphic to each other if $\varepsilon_1,\varepsilon_2 <\varepsilon_0$, where
$$
\begin{equation*}
\widetilde U_{\varepsilon}= \{ p\in \widetilde U\mid \operatorname{dist}_{(ds)^2}(\widetilde C_0, p)<\varepsilon\}.
\end{equation*}
\notag
$$
We denote by $\mathcal U_{\widetilde C,\nu, (ds)^2}:= \{ \widetilde U_{\varepsilon}\mid \varepsilon<\varepsilon_0((ds)^2)\}$ the set of all neighbourhoods $\widetilde U_{\varepsilon}$ of $\widetilde C_0$ with $\varepsilon<\varepsilon_0$; $\widetilde U_{\varepsilon}\in \mathcal U_{\widetilde C,\nu, (ds)^2}$ (see Definition 2 in § 1.2) will be called a tubular neighbourhood of $\widetilde C_0\subset X$ (defined via the good metric $(ds)^2$). Denote by $\mathcal U_{\widetilde C, \nu}:= \bigsqcup_{(\widetilde ds)^2}\mathcal U_{\widetilde C,\nu, (ds)^2}$ the disjunct union of sets $\mathcal U_{\widetilde C,\nu, (ds)^2}$ over all good metrics $(ds)^2$, by $\mathcal U_{\widetilde C}:=\bigsqcup_{\nu}\mathcal U_{\widetilde C,\nu}$ the disjunct union of the sets $\mathcal U_{\widetilde C,\nu}$ over all resolutions $\nu\colon X\to U$ of singularities of $C\subset S$. The open (in S) sets $U_{\varepsilon}$ lying in one of the sets $\mathcal U_{C,\nu}:=\{ U_{\varepsilon}= \nu(\widetilde U_{\varepsilon})\subset S\mid \widetilde U_{\varepsilon}\in \mathcal U_{\widetilde C,\nu}\}$ are called “tubular” neighbourhoods of the curve $C_0\subset S$. Let $\mathcal U_{C}:=\bigsqcup_{\nu}\mathcal U_{C,\nu} $ and let ${\mathcal U}:=\bigsqcup_{C}\mathcal U_{\widetilde C}$ be the disjunct union of the sets $\mathcal U_{\widetilde C}$ over all pairs $(S,C)$. The boundary $\partial U_{\varepsilon}\subset S$ of $U_{\varepsilon}\in \mathcal U_{C}$ is a compact connected three dimensional $C^0$-manifold without boundary. In § 2, we will prove the following result. Theorem 1. Let $C\subset S$ be a compact curve in a normal complex surface $S$. Then 1) for $U_{\varepsilon} \in \mathcal U_{C,\nu, (ds)^2}$, there is a homeomorphism $\rho_{\varepsilon}\colon U_{\varepsilon}\setminus C\to \partial U_{\varepsilon}\times (0,\varepsilon)$ (here $(0,\varepsilon)= \{ t\in \mathbb R\mid 0< t< \varepsilon\})$ such that $\rho^{-1}_{\varepsilon}(\partial U_{\varepsilon}\times \{\varepsilon_1\}) = \partial U_{\varepsilon_1}\subset U_{\varepsilon}$ for $0<\varepsilon_1<\varepsilon$, and, in particular, $\pi_1(U_{\varepsilon}\setminus C)\simeq \pi_1(\partial U{_\varepsilon})$; 2) the groups $\pi_1(U_{\varepsilon}\setminus C)$ and $\pi_1(U'_{\varepsilon'}\setminus C)$ are isomorphic for all $U_{\varepsilon}$, $U'_{\varepsilon'}\in \mathcal U_{C}$. Let $\Gamma(\widetilde C)$ be the dual partially bi-weighted graph of $\widetilde C=\nu^{-1}(C)= C_1\cup \dots\cup C_{m+k}$, where $C_1,\dots,C_m$ are the irreducible compact components of the graph $\widetilde C$, and $C_{m+1},\dots, C_{m+k}$ are the irreducible non-compact components. The vertex set of $\Gamma(\widetilde C)$ is $\{ v_{1,0},\dots, v_{m,0}\}\cup \{ v_{m+1,1},\dots, v_{m+k,1}\}$. The vertices $v_{j,0}$, $j=1,\dots, m$, of $\Gamma(\widetilde C)$ correspond to the compact components $C_j$, and their bi-weights are $(w_{1,j},w_{2,j})$, where $w_{1,j}=(C_{j}^2)_{X}$ is the self-intersection number and $w_{2,j}=g_j$ is the genus of $C_j$. The vertices $v_{j,1}$, $j=m+1,\dots, n=m+k$, correspond to the non-compact components of $\widetilde C$, and they have no bi-weights. Two vertices $v_{j_1,\delta_{j_1}}$ and $v_{j_2,\delta_{j_2}}$ are connected by the edge $e_{j_1,j_2}:=(v_{j_1,\delta_{j_1}},v_{j_2,\delta_{j_2}})$ if and only if $C_{j_1}\cap C_{j_2}\neq \varnothing$. Note that $\Gamma(\widetilde C)$ is a connected graph, since so is $\widetilde C$. Denote by $\mathcal G$ the set of all connected partially bi-weighted finite graphs $\Gamma$ with the following properties:
The following result can be proved quite easily (see § 2.1). Proposition 1. The mapping $\gamma\colon (\widetilde U_{\varepsilon},\widetilde C) \in \mathcal U\mapsto \Gamma(\widetilde C)\in\mathcal G$ is a surjection. Let $\Gamma_0$ be the subgraph of $\Gamma\in \mathcal G$ with vertex set $V(\Gamma_0)=\{ v_{1,0},\dots, v_{m,0}\}$ and the edge set formed of the edges of the graph $\Gamma$ that connect vertices from $V(\Gamma_0)$. The fundamental groups $\pi_1(\Gamma(\widetilde C),v_{1,0})$ and $\pi_1(\Gamma(\widetilde C_0),v_{1,0})$ are free groups of the same rank $r\geqslant 0$, since the valences $\operatorname{v}_j$ of the vertices $v_{j,1}$ are equal to $1$ for $j>m$. We will consider a graph $\Gamma\in \mathcal G$ as a geometric graph. Let us choose $r$ edges $E_1,\dots, E_r$ of $\Gamma_0$ such that $\pi_1\bigl(\Gamma\setminus \bigcup_{l=1}^sE_{l},v_{1,0}\bigr)$ are free groups of rank $r-s$ for $s=1,\dots, r$. Let vertices $v_{j(E_s,1),0}$ and $v_{j(E_s,2),0}$ be connected by edges $E_s$. For each $s=1,\dots,r$, let us choose two points $v_{j(E_s,1),1}$ and $v_{j(E_s,2),1}$ in $E_s$ such that $v_{j(E_s,2),1}$ is met, first, when moving from $v_{j(E_s,1),0}$ towards $v_{j(E_s,2),0}$ along $E_s$. Let $\widetilde E=\{ E_1,\dots, E_r\}$ and ${\Gamma}_{\widetilde E}$ be the graph (called a tree of $\Gamma$) obtained from $\Gamma$ by adding the points $v_{j(E_s,1),1}$ and $v_{j(E_s,2),1}$, $s=1,\dots,r$, to the vertex set of $\Gamma$ and deleting the edges $(v_{j(E_s,2),1},v_{j(E_s,2),1})$ that connect the vertices $v_{j(E_s,2),1}$ and $v_{j(E_s,2),1}$ (see Figure 1). By definition, the vertices $v_{j(E_s,i),1}$ have no bi-weights for $i=1,2$. For each pair of vertices $v_{i,\delta_i}$ and $v_{j,\delta_j}$ of ${\Gamma}_{\widetilde E}$, we define
$$
\begin{equation*}
\Delta_{(i,\delta_i),(j,\delta_j)}=\begin{cases} 1 &\text{if } v_{i,\delta_i}\text{ and }v_{j,\delta_j} \text{ are connected by an edge in } {\Gamma}_{\widetilde E}, \\ 0 &\text{if }v_{i,\delta_i}\text{ and }v_{j,\delta_j} \text{ are not connected by an edge in } {\Gamma}_{\widetilde E}, \\ 0 &\text{if }v_{i,\delta_i}=v_{j,\delta_j}. \end{cases}
\end{equation*}
\notag
$$
Let $\operatorname{v}_{j_0}:=\operatorname{v}(v_{j_0,0})$ be the valence of a vertex $v_{j_0,0}\in {\Gamma}_{\widetilde E}$ and let
$$
\begin{equation*}
\Upsilon_{j_0}=\{ v_{i_1,\delta_{i_1}},\dots, v_{i_{\operatorname{v}_{j_0}},\delta_{i_{\operatorname{v}_{j_0}}}}\}
\end{equation*}
\notag
$$
be the set of vertices $v_{j,\delta_j}\in {\Gamma}_{\widetilde E}$ connected with $v_{j_0,0}$ by edges. Let $I_{j_0}$ be the set of bi-indices $(j,\delta_j)$ of $v_{j,\delta_j}\in \Upsilon_{j_0}$. Consider the bijection
$$
\begin{equation*}
\overline o_{j_0}=(o_{j_0},d_{j_0})\colon \{ 1,\dots, \operatorname{v}_{j_0}\}\to I_{j_0}, \qquad (o_{j_0}(i),d_{j_0}(i))\in I_{j_0}.
\end{equation*}
\notag
$$
With the vertices $v_{j,0}$, $j=1,\dots, m$, with bi-weights $(w_1,w_2)=(k_j,g_j)$, we associate the words
$$
\begin{equation}
\mathcal W_j:= x_{j,0}^{k_j}\prod_{i=1}^{g_{j}}[\mu_{j,i},\lambda_{j,i}] \prod_{i=1}^{\operatorname{v}_{j}}x_{\overline{o}_{j}(i)}
\end{equation}
\tag{1}
$$
in the alphabet $\mu_{j,1},\lambda_{j,1},\dots, \mu_{j,g_{j}},\lambda_{j,g_{j}}$, $x_{j,0}$, $x_{\overline{o}_{j}(1)},\dots,x_{\overline{o}_{j}(\operatorname{v}_{j})}$, where $[\mu_{j,i},\lambda_{j,i}]= \mu_{j,i}\lambda_{j,i}\mu_{j,i}^{-1}\lambda_{j,i}^{-1}$. In the following definition, we will use the above notation. Definition 1. For a tree $\Gamma_{\widetilde E}$ of $\Gamma\in\mathcal G$, denote by $\pi^w_1(\Gamma_{\widetilde E})$ a group with the following presentation: it is generated by $m+k+3r+2\sum_{j=1}^mg_j$ elements $(\mathrm{g}_1)$ $x_{j,0}$, $\mu_{j,i}$, $\lambda_{j,i}$, $1\leqslant j\leqslant m$, $1\leqslant i\leqslant g_j$, $(\mathrm{g}_2)$ $x_{m+1,1},\dots, x_{m+k,1}$, $(\mathrm{g}_3)$ $x_{j(E_s,1),1}$, $x_{j(E_s,2),1}$, $y_s$, $1\leqslant s\leqslant r$, which are related by $(\mathrm{r}_1)$ $\mathcal W_{j} =1$, $1\leqslant j \leqslant m$, $(\mathrm{r}_2)$ $[x_{j,0}, \mu_{j,i}]=[x_{j,0}, \lambda_{j,i}] =1$, $1\leqslant j\leqslant m$, $1\leqslant i\leqslant g_j$, $(\mathrm{r}_3)$ $[x_{j_1,\delta_1}, x_{j_2,\delta_2}] =1$, $\Delta_{(j_1,\delta_1),(j_2\delta_2)}=1$, $(\mathrm{r}_4)$ $x_{j(E_s,1),0}^{-1}y_sx_{j(E_s,1),1}y_s^{-1} =1$, $1\leqslant s\leqslant r$, $(\mathrm{r}_5)$ $x_{j(E_s,2),1}^{-1}y_s^{-1}x_{j(E_s,2),0}y_s =1$, $1\leqslant s\leqslant r$, where the words $\mathcal W_{j}$ are defined in (1). Theorem 2. Let $\Gamma_{\widetilde E_1}$ and $\Gamma_{\widetilde E_2}$ be two trees of $\Gamma\in\mathcal G$. Then $\pi^w_1(\Gamma_{\widetilde E_1})$ and $\pi^w_1(\Gamma_{\widetilde E_2})$ are isomorphic groups. Let $\Gamma_{\widetilde E}$ be a tree of $\Gamma\,{\in}\,\mathcal G$. In view of Theorem 2, the group $\pi_1^w(\Gamma)\,{:=}\,\pi_1^w(\Gamma_{\widetilde E})$ will be called the fundamental group of the partially bi-weighted graph $\Gamma$, and the group $\pi_1(\Gamma,v_{1,0})$ will be simply called the fundamental group of the graph $\Gamma$. Theorem 3. For $U_{\varepsilon}\in \mathcal U_{C,\nu}$ and a tree ${\Gamma}_{\widetilde E}(\widetilde{C})$ of $\Gamma(\widetilde C)$, and $\widetilde C=\nu^{-1}(C)$, the group $\pi_1(U_{\varepsilon}\setminus C)$ is isomorphic to $\pi_1^w(\Gamma_{\widetilde E})$. The groups $\pi_1(U_{\varepsilon}\setminus C)$ and $\pi_1(U'_{\varepsilon'}\setminus C)$ are isomorphic for all $U_{\varepsilon}$, $U'_{\varepsilon'}\in \mathcal U_{C}$. In view of Proposition 1, it is easy to see that Theorem 2 is a direct consequence of Theorem 3. In the case $C=o$ is a singular point of a normal surface $S$ and $\Gamma(\widetilde C)$ is a tree, Theorem 3 was proved by Mumford in [2] if $\widetilde C$ is a union of rational curves; Wagreich [4] proved the general case, that is, where the irreducible components of the curve $\widetilde C$ are not necessarily rational.1 A proof of Theorem 3 is also contained in [5] in the case $C$ is a compact curve; it is contained in [6] if $C_0$ is a singular point of a normal surface. Note also that Theorem 3 was formulated in the case when $C_0$ is a singular point of a normal surface, the graph $\Gamma(\widetilde C)$ is a tree, and the irreducible components of $\widetilde C_0$ are rational curves; this result was used in [7] and [8] in order to describe a connection between the set of rational Belyi pairs and the set of rigid germs of finite morphisms of smooth surfaces branched at curve germs with $\mathit{ADE}$ type singular points. Let $\mathcal S$ be a subset of $\mathcal G$. Denote by $\Pi_w(\mathcal S)=\{\pi^w_1(\Gamma)\mid \Gamma\in \mathcal S\}$ the set of the fundamental groups of the partially bi-weighted graphs $\Gamma\in \mathcal S$ (considered up to isomorphisms of groups). The following theorem (a proof of which will be given in § 3) is a corollary to Theorem 3. Theorem 4. 1. Let $\mathcal Ch^0_i\subset \mathcal G$, $i=0,1,2$, be the set of partially bi-weighted graphs that are chains with $i$ vertices without bi-weights, and the vertices $v_{j,0}$ of which have bi-weights $(k_j,0)$. Then 2. Let $\mathcal L\subset \mathcal G$ be the set of partially bi-weighted graphs that are loops and whose vertices $v_{j,0}$ have bi-weights $(k_j,0)$. Then
$$
\begin{equation*}
\Pi_w(\mathcal L)= \{ \mathbb Z^2\ltimes_M\mathbb F_1\mid M\in \operatorname{SL}(2,\mathbb Z)\},
\end{equation*}
\notag
$$
where $\mathbb Z^2\ltimes_{M}\mathbb F_1=\langle (z_1,z_2),t\mid (z_1,z_2)\in \mathbb Z^2,\, t^{-1}(z_1,z_2)t=(z_1,z_2)M\rangle$ are the semidirect products of $\mathbb Z^2$ and $\mathbb F_1\simeq \mathbb Z$, and $M\in \operatorname{SL}(2,\mathbb Z)$ on $\mathbb Z^2$ from the right. 3. Let $\Gamma\in\mathcal G$ contain $n_i$ vertices $v_{j,0}$ whose bi-weights are $(k_j,i)$, let $i_0$ be a number such that $n_i=0$ for $i>i_0$, and let $\pi_1(\Gamma,v_{1,0})\simeq \mathbb F_r$ be the free group of rank $r$. Then there is an epimorphism to the free product of the fundamental groups $\mathcal R_g=\pi_1(R_g)$ of Riemann surfaces $R_g$ of genus $g$ and $\mathbb F_r$.Further we freely use the above notation. § 1. Proof of Theorem 31.1. Good metricsSince the curve $\widetilde C_0$ is compact, we can choose a partition of unity $\{\rho_i\}$ subordinate to a finite open in $X$ cover $\{ W_i\}$ of $\widetilde C_0$ such that $\sum_i \rho_i(p)= 1$ for all $p\in\widetilde C_0$. Let $(z_{i,1},z_{i,2})$ be local coordinates in $W_i$. Then
$$
\begin{equation*}
(d\widetilde s)^2 =\sum_{i}\rho_i(dz_{i,1}\, d\overline z_{i,1}+ dz_{i,2}\, d\overline z_{i,2})
\end{equation*}
\notag
$$
is a Hermitian metric on a neighbourhood $\widetilde U\subset \bigcup_iW_i$ of $\widetilde C_0$. Let $W\subset \widetilde U$ be a neighbourhood of a point $p\in \widetilde U$ with local coordinates $z_1$, $z_2$ such that $W$ is bi-holomorphic to the ball $\mathbb B_{r}=\{ (z_1,z_2)\in\mathbb C^2\mid \sqrt{|z_1|^2+|z_2|^2}<r\}$ of radius $r$ and centred at $p$, $\mathbb B_{r}\simeq W$. Then $(d\widetilde s)^2$ is given in $W$ by
$$
\begin{equation}
(d\widetilde s)^2=h_{1}\,dz_1\, d\overline z_1+h_{2}\, dz_2\, d\overline z_2+ (a+ib)\, dz_1\, d\overline z_2 +(a-ib)\, d\overline z_1\, dz_2,
\end{equation}
\tag{2}
$$
where $h_{1}:=h_{1}(z_1,z_2)$, $h_{2}:=h_{2}(z_1,z_2)$, $a:=a(z_1,z_2)$, and $b:=b(z_1,z_2)$ are real-valued functions in $W$. In addition, it follows from Sylvester’s criterion of positive definiteness of a quadratic form that the form given by (2) is positive definite if and only if the functions $h_{1}$, $h_{2}$, $a$, $b$ satisfy, ar each point of $W$, Without loss of generality, we can assume that $r=4$, and, for $\varepsilon\leqslant 4$, we consider the balls $\mathbb B_{\varepsilon}=\{ (z_1,z_2)\in\mathbb C^2\mid \sqrt{|z_1|^2+|z_2|^2}<\varepsilon\}$ as open subsets of $W\simeq \mathbb B_{4}$. Proof. Let us choose monotone $C^{\infty}$-functions $f_1(t)$ and $g_1(t)$ such that
$$
\begin{equation*}
f_1(t)= \begin{cases} 2 &\text{if }t\leqslant 3, \\ 1 &\text{if }t\geqslant \dfrac{7}{2}, \end{cases} \qquad g_1(t)= \begin{cases} 0 &\textrm{if }t\leqslant 3, \\ 1 &\textrm{if }t\geqslant \dfrac{7}{2}. \end{cases}
\end{equation*}
\notag
$$
Applying (3) it is easy to see that
$$
\begin{equation*}
(ds_1)^2= \begin{cases} f_1(\sqrt{|z_1|^2+|z_2|^2})(h_{1}\, dz_1\, d\overline z_2+h_{2}\, dz_2\, d\overline z_2) \\ \ +\,g_1(\sqrt{|z_1|^2+|z_2|^2})((a+ib)\, dz_1\, d\overline z_2+ (a-ib)\, d\overline z_1\, dz_2) &\text{on }\mathbb B_{7/2}\subset W, \\ (d\widetilde s)^2 &\text{on }\widetilde U\setminus \mathbb B_{7/2} \end{cases}
\end{equation*}
\notag
$$
is a Hermitian metric on $\widetilde U$ such that $(ds_1)^2=2h_{1}\, dz_1\, d\overline z_1+2h_{2}\, dz_2\, d\overline z_2$ on $\mathbb B_{3}\,{\subset}\,W$.
Next, let us choose monotone $C^{\infty}$-functions $f_0(t)$ and $g_0(t)$ such that
$$
\begin{equation*}
f_0(t)= \begin{cases} 1 &\text{if }t\leqslant 2, \\ 0 &\textrm{if } t\geqslant \dfrac{5}{2}, \end{cases}\qquad g_0(t)= \begin{cases} 0 &\text{if }t\leqslant 2, \\ 1 &\text{if }t\geqslant \dfrac{5}{2}. \end{cases}
\end{equation*}
\notag
$$
Then
$$
\begin{equation*}
(ds_0)^2= \begin{cases} \bigl(f_0(\sqrt{|z_1|^2+|z_2|^2})+2g_0(\sqrt{|z_1|^2+|z_2|^2}) h_{1}\bigr)\, dz_1\, d\overline z_2 \\ \ + \bigl(f_0(\sqrt{|z_1|^2+|z_2|^2})+2g_0(\sqrt{|z_1|^2+|z_2|^2}) h_{2}\bigr)\, dz_2\, d\overline dz_2 &\text{on }\mathbb B_{5/2}\subset W, \\ (ds_1)^2 &\textrm{on }\widetilde U\setminus \mathbb B_{5/2} \end{cases}
\end{equation*}
\notag
$$
is a Hermitian metric on $\widetilde U$ such that $(d\widetilde s_0)^2=dz_1\, d\overline z_1+dz_2\, d\overline z_2$ on $\mathbb B_{2}\subset W$. Lemma 1 is proved. For the points $p_{j_1,j_2}=C_{j_1}\cap C_{j_2}\subset \operatorname{Sing} \widetilde C$, $1\leqslant j_1 < j_2\leqslant n$, let us choose pairwise non-intersecting neighbourhoods $W_{j_1,j_2}\subset \widetilde U\subset X$ of the points $p_{j_1,j_2}$ bi-holomorphic to the ball $\mathbb B_{4}=\{ (z_1,z_2) \in\mathbb C^2\mid \sqrt{|z_1|^2+|z_2|^2}< 4\}$, and such that, in the coordinates $(z_1,z_2)$ in $W_{j_1,j_2}$, the curve $W_{j_1}:=C_{j_1}\cap W_{j_1,j_2}$ (respectively, the curve $W_{j_2}:=C_{j_2}\cap W_{j_1,j_2}$) is given by the equation $z_1=0$ (respectively, $z_2=0$). Applying Lemma 1, in each neighbourhood $W_{j_1,j_2}$ we change the metric $(d\widetilde s)^2$ by $(ds_0)^2$ and obtain a good metric ($ds)^2$ (with respect to $\widetilde C$) on $\widetilde U\subset X$. 1.2. Tubular neighbourhoodsLet us fix $\widetilde U\subset X$ and a good metric $(ds)^2$ on $\widetilde U$. For positive $\varepsilon\ll 1$ and for $j=1,\dots,m$, denote by
$$
\begin{equation*}
U_{j,\varepsilon}= \{ p\in X\mid \operatorname{dist}_{(ds)^2}(C_j, p)<\varepsilon\}
\end{equation*}
\notag
$$
the $\varepsilon$-neighbourhood of the compact curve $C_j$, let $\overline U_{j,\varepsilon}$ be its closure in $X$, and let
$$
\begin{equation*}
\partial U_{j,\varepsilon}= \{ p\in X\mid \operatorname{dist}_{(ds)^2}(C_j, p)=\varepsilon\}
\end{equation*}
\notag
$$
be its boundary. Obviously, there is positive $\varepsilon_j\ll 1$ such that if $\varepsilon<\varepsilon_j$, then $\overline U_{j,\varepsilon}\subset \widetilde U$ and the set $\{ U_{j,\varepsilon}\}$ is a base of open subsets of $X$ containing $C_j$. Consider the restriction $T_{X\mid C_j}$ of the tangent bundle $T_X$ of $X$ to $C_j$ as a bundle of four dimensional vector spaces over $\mathbb R$. Then $(ds)^2$ defines a splitting of $T_{X\mid C_j}$ into the direct sum $T_{C_j}\bigoplus N_{C_j,(ds)^2}$ of the tangent bundle $T_{C_j}$ of $C_j$ and the normal bundle
$$
\begin{equation*}
N_{C_j,(ds)^2}= \{ (p,v) \mid p\in C_j,\, v\in T_{X\mid C_j,p},\, v\perp T_{C_j,p}\}
\end{equation*}
\notag
$$
on $C_j$ of the vector spaces transversal to $T_{C_j,p}$ in $T_{X\mid C_j,p}$ with respect to the scalar product
$$
\begin{equation*}
(v_1,v_2)=\frac{1}{2}[(ds)^2(v_1+v_2)-(ds)^2(v_1)-(ds)^2(v_2)].
\end{equation*}
\notag
$$
Denote by $\operatorname{pr}_j\colon N_{C_j,(ds)^2}\to C_j$ the projection of $N_{C_j,(ds)^2}$ to $C_j$ and denote by $S_{j,0}=\{ (p,0)\in N_{C_j,(ds)^2}\mid p\in C_j\}$ the zero section. For $p\in C_j$, consider the restriction $\operatorname{Exp}_{p\mid N_{C_j,(ds)^2}}\colon N_{C_j,(ds)^2}\to \widetilde U\subset X$ to $N_{C_j,(ds)^2}$ of the exponential mapping $\operatorname{Exp}_p\colon T_{X,p}\to \widetilde U\subset X$ which sends segments of one dimensional vector spaces of $N_{C_j,(ds)^2}$ to geodesic lines perpendicular to $C_j$ at the point $p$. It is well-known that, for each point $p\in C_j$, there are neighbourhoods $W_p\subset C_j$ of $p$ and $V_p\subset \operatorname{pr}_j^{-1}(W_p)\subset N_{C_j,(ds)^2}$ of $S_{j,0}\cap \operatorname{pr}_j^{-1}(W_p)$ such that the mapping $\varphi_{j,p}\colon V_p \to \widetilde U$, as obtained at each point $(q,v)\in V_p$ from the exponential mapping $\operatorname{Exp}_{q\mid N_{C_j,(ds)^2}}\colon T_{X,q}\to \widetilde U$, is a diffeomorphism of $V_p$ and its image $\varphi_{j,p}(V_p)$. Since for $1\leqslant j\leqslant m$ the curves $C_j$ are compact, it is easy to see that there exist neighbourhoods $V_j\subset N_{C_j,(ds)^2}$ of the zero sections $S_{j,0}$ such that $\varphi_j\colon V_j\to \widetilde U$, as given by $\varphi_j((p,v))=\varphi_{j,p}((p,v))$, are diffeomorphisms of $V_j$ to their images. Without loss of generality, we can assume that $U_{j,\varepsilon}\subset \varphi_j(V_j)$ for $\varepsilon<\varepsilon_j$. We set $N_{C_j,(ds)^2}(\varepsilon):=\varphi_j^{-1}(U_{j,\varepsilon})$. Below, we will identify $N_{C_j,(ds)^2}(\varepsilon)\subset N_{C_j,(ds)^2}$ with $U_{j,\varepsilon}\subset X$ if it does not lead to a misunderstanding. In particular, the projection $\operatorname{pr}_j$ defines a structure of a $C^{\infty}$-locally trivial bundle of closed discs of radius $\varepsilon$ on $\overline U_{j,\varepsilon}$ and defines a structure of a $C^{\infty}$-locally trivial bundle of circles of radius $\varepsilon$ on $\partial U_{j,\varepsilon}\subset \overline U_{j,\varepsilon}$. For $p\in C_j$, the set $\operatorname{pr}_j^{-1}(p)\cap \overline U_{j,\varepsilon}$, which is diffeomorphic to the disc $\mathbb D_{\varepsilon}=\{ z\in\mathbb C\mid |z|\leqslant\varepsilon\}$, is the union of segments of length $\varepsilon$ of geodesic lines lying in $\overline {U}_{j,\varepsilon}$ perpendicular to $C_j$ and emanating from the point $p\in C_j$. We set $d_0=\min_{1\leqslant j\leqslant m} \varepsilon_j> 0$ and define
$$
\begin{equation*}
d_j = \operatorname{dist}_{ds^2}\biggl(C_j\setminus \biggl(\bigcup_{i=1}^{\operatorname{v}_j} V_{j,o_j(i)}\biggr), \widetilde C\setminus C_j\biggr)>0
\end{equation*}
\notag
$$
for $1\,{\leqslant}\, j\,{\leqslant}\, m$, where the neighbourhoods $V_{j,o_j(i)}:=V_{p_{j,o_j(i)}}$ are the neighbourhoods of the points $p_{j,o_j(i)}\,{=}\, C_j\cap C_{o_j(i)}$, involved in the definition of the good metric $(ds)^2$ (see the introduction). We set $\varepsilon_0=\frac{1}{2}\min(1,d_0,d_1,\dots,d_m)>0$. Definition 2. The open subsets $\widetilde U_{\varepsilon}=\bigcup_{j=1}^mU_{j,\varepsilon}$ of $X$, $\varepsilon<\varepsilon_0$, are called the tubular neighbourhoods of $\widetilde C_0\subset X$. Consider a curve $C_j$, $j\leqslant m$, and a curve $C_{o_j(i)}$ for some $i\leqslant \operatorname{v}_j$. By definition of the good metric $(ds)^2$, there is a neighbourhood $V:=V_{p_{j,o_j(i)}}$ of the point $p_{j,o_j(i)}=C_{j}\cap C_{o_j(i)}$ bi-holomorphic to the ball $\mathbb B_2\,{=}\,\{ (z_1,z_2)\,{\in}\,\mathbb C\mid \sqrt{|z_1|^2\,{+}\,|z_2|^2}\,{<}\,2\}$ and such that in the coordinates $(z_1,z_2)$ in $V_{p_{j,o_j(i)}}$ we have: $(ds)^2= dz_1\, d\overline z_1+dz_2\, d\overline z_2$, and, in addition, $C_{j}\cap V_{p_{j,o_j(i)}}$ is given by $z_1=0$ and $C_{o_j(i)}\cap V_{p_{j,o_j(i)}}$ is given by $z_2=0$. Remark 1. Below, if $j_1=o_j(i)\leqslant m$, we will assume that $V\,{=}\,V_{p_{j_1,j}}\,{=}\,V_{p_{j,j_1}}\subset X$ and the identifications of the neighbourhood $V$ with the ball $\mathbb B_2=\{ (z_1,z_2)\in\mathbb C\mid \sqrt{|z_1|^2+|z_2|^2}<2\}$ define the bi-holomorphic automorphism of $\mathbb B_2$ sending $(z_1,z_2)$ to $(z_2,z_1)$. It is easy to see that the following result holds. Claim 1. For $1\leqslant j_i\leqslant m$, $i=1,2$, and for $\varepsilon<\varepsilon_0$, we have: 1) $U_{j_1,j_2,\varepsilon}:=U_{j_1,\varepsilon} \cap U_{j_2,\varepsilon}\subset V_{p_{j_1,j_2}}=V$,
$$
\begin{equation*}
U_{j_1,j_2,\varepsilon}\simeq \mathbb D^2_{\varepsilon}= \{\mathbb (z_1,z_2)\in \mathbb B_2\mid |z_1|<\varepsilon,\,\, |z_2|<\varepsilon\}\subset\mathbb B_2\simeq V_{p_{j_1,j_2}},
\end{equation*}
\notag
$$
and in the coordinates $(z_1,z_2)$ the projections $\operatorname{pr}_{j_i\mid U_{j_1,j_2,\varepsilon}}\colon U_{j_1,j_2,\varepsilon}\to W_{j_i,\varepsilon}$, $i=1,2$, are given by $\operatorname{pr}_{j_i\mid U_{j_1,j_2,\varepsilon}} \colon (z_1,z_2)\mapsto z_{\overline i}$, where $W_{j_i,\varepsilon}=C_{j_i}\cap U_{j_1,j_2,\varepsilon}$ and $\{ i,\overline i\}=\{1,2\}$, and, in particular, $U_{j_i,\varepsilon}\cap C_{j_{\overline i}}= \operatorname{pr}_{j_i\mid U_{j_1,j_2,\varepsilon}}^{-1}(p_{j_1,j_2})$; 2) the segments of geodesic lines lying in $V_{p_{j_1,j_2}}$ perpendicular to $C_{j_1}$ (respectively, $C_{j_2})$ and emanating from a point $p=(0,z_{2,0})\in C_{j_1}$ (respectively, $p=(z_{1,0},0)\in C_{j_2}$) are where $z_1\in\mathbb C$, $|z_1|=1$ (respectively, $z_2\in\mathbb C$, $|z_2|=1$).1.3. On normal bundles of curvesConsider a curve $\mathcal C:=C_{j}$ corresponding to a vertex $v_{j,0}\in \Gamma(\widetilde C)$, $j=1,\dots ,m$. If $X$ and $\mathcal C$ are considered as complex manifolds, then their tangent bundles $T_X$ and $T_{\mathcal C}$ are vector bundles defined over $\mathbb C$ and the normal bundle $N_{\mathcal C}$ of $\mathcal C$ in $X$ is defined via the exact sequence
$$
\begin{equation}
0\to T_{\mathcal C}\to T_{X \mid \mathcal C} \xrightarrow{\psi} N_{\mathcal C}\to 0.
\end{equation}
\tag{4}
$$
We set $\operatorname{v}:=\operatorname{v}_j$ and define $p_i:=p_{j,o_{j}(i)}=C_{j}\cap C_{o_{j}(i)}$. Let $M:=M_j$, $K:=K_j$ be two non-negative integers such that $(\mathcal C^2)_X=M-K$ and a divisor
$$
\begin{equation*}
D:=D_j=(p_{\operatorname{v}+1}+\dots + p_{\operatorname{v}+M})- (p_{\operatorname{v}+M+1}+\cdots+p_{\operatorname{v}+M+K})\in \operatorname{Pic}(\mathcal C)
\end{equation*}
\notag
$$
is equivalent to the restriction $\mathcal C_{\mid \mathcal C}$ of the divisor $\mathcal C\in \operatorname{Pic}(X)$ to the curve $\mathcal C$. Then $N_{\mathcal C}=L_{\mathcal C,D}$, where $L_{\mathcal C,D}$ is the line bundle associated with the divisor $D$. We can assume that $p_{i}\notin \operatorname{Supp}(D)$ for $i=1,\dots,\operatorname{v}$. Let us choose neighbourhoods $W_i\subset \mathcal C$ of the points $p_i$, $i=1,\dots,\operatorname{v}+M+K$, bi-holomorphic to the disc $\mathbb D_2=\{ w\in\mathcal C\mid |w|< 2\}$ and such that $W_{i_1}\cap W_{i_2}=\varnothing$ for $i_1\neq i_2$. Let us identify $\mathcal C$ with the zero section $S_0$ of the bundle $L_{\mathcal C,D}$. We can compactify $L_{\mathcal C,D}$ by adding the section at “infinity” $S_{\infty}$, thereby obtaining a relatively minimal ruled surface $\overline L_{\mathcal C,D}$ over $\mathcal C$ with fibres $F_p=\operatorname{pr}^{-1}(p)\simeq\mathbb P^1$ over the points $p\in\mathcal C=S_0$ of the projection $\operatorname{pr}\colon \overline L_{\mathcal C,D}\to \mathcal C$. In $\operatorname{Pic}(\overline L_{\mathcal C,D})$, we have
$$
\begin{equation}
S_0= S_{\infty}+ \sum_{i=1}^MF_{p_{\operatorname{v}+i}} - \sum_{i=1}^KF_{p_{\operatorname{v}+M+i}}.
\end{equation}
\tag{5}
$$
For $\mathcal C=C_j$, let us add the imbedding $\iota_j\colon N_{C_j,(ds)^2}\to T_{X\mid C_j}$ to the exact sequence (4): Note that $\psi_j\circ \iota_j\colon N_{C_j,(ds)^2}\to N_{C_j}$ is a $\mathbb R$-linear isomorphism of line bundles $\mathbb C$-linear over $W_i$ for $i=1,\dots,\operatorname{v}_j$. We set $\widetilde U_{j,\varepsilon}:= \psi_j\circ \iota_j(N_{C_j,(ds)^2}(\varepsilon))\subset N_{C_j}$. Obviously, the set $\widetilde{\mathcal U}_{j}= \{ \widetilde U_{j,\varepsilon}\mid \varepsilon< \varepsilon_0 \}$ is a base of open (in complex analytic topology) subsets of the normal bundle $N_{C_j}$ containing the zero section $S_{j,0}\subset N_{C_j}$. Applying Claim 1, we arrive at the following result.Claim 2. The diffeomorphisms $\psi_j\circ \iota_j\circ \varphi_j^{-1}\colon U_{j,\varepsilon}\to \widetilde U_{j,\varepsilon}$ determine diffeomorphisms between $U_{j,\varepsilon}\setminus \widetilde C$ and $\widetilde U_{j,\varepsilon}\setminus \bigl(S_{j,0}\cup \bigcup_{i=1}^{\operatorname{v}_j}F_{p_{j,o_j(i)}}\bigr)$. In particular, we have isomorphisms of fundamental groups $\pi_1(U_{j,\varepsilon}\setminus \widetilde C)\simeq \pi_1\bigl(\widetilde U_{j,\varepsilon}\setminus \bigl(S_{j,0}\,\cup\, \bigcup_{i=1}^{\operatorname{v}_j}F_{p_{j,o_j(i)}}\bigr)\bigr)$. 1.4. Elementary transformationsFor $p\in S_0$ (respectively, $\widetilde p=F_p\cap S_{\infty}$), we denote by $\operatorname{elm}_{p}\colon \overline L_{\mathcal C,D} \dashrightarrow \overline L_{\mathcal C,D'}$ (respectively, $\operatorname{elm}_{\widetilde p}\colon \overline L_{\mathcal C,D} \dashrightarrow \overline L_{\mathcal C,\widetilde D'}$) a bi-rational map (called the elementary transformation of $\overline L_{\mathcal C,D}$) consisting of the blowup of the point $p$ (respectively, $\widetilde p$) and after that the blowdown of the proper inverse image of the fibre $F_p$. Lemma 2. Let $\operatorname{elm}_{p}\colon \overline L_{\mathcal C,D} \dashrightarrow \overline L_{\mathcal C,D'}$ (respectively, $\operatorname{elm}_{\widetilde p}\colon \overline L_{\mathcal C,D} \dashrightarrow \overline L_{\mathcal C,\widetilde D'}$) be an elementary transformation of $\overline L_{\mathcal C,D}$. Then $D'=D-p$ (respectively, $\widetilde D'=D+p$). Proof. We will prove the lemma only in the case when the elementary transformation is $\operatorname{elm}_{p}$, since the proof in the second case is similar. Without loss of generality, we can assume that $p\not\in \operatorname{Supp} D$.
We have $\operatorname{elm}_{p}= \sigma\circ\sigma^{-1}_p\colon \overline L_{\mathcal C,D}\dashrightarrow \overline L_{\mathcal C,D'}$, where $\sigma_p\colon Y\to \overline L_{\mathcal C,D}$ is the $\sigma$-process with centre at the point $p= S_0\cap F_p$ and $\sigma\colon Y\to \overline L_{\mathcal C,D'}$ is the $\sigma$-process contracting the proper inverse image $\sigma^{-1}_p(F_p)$ of the fibre $F_p$. We set $F'_p=\sigma^{-1}_p(p)$ and denote the proper inverse images (and respectively, after that, the images under the mapping $\sigma$) of the sections $S_0$, $S_{\infty}$, and of the fibres $F_q$ over the points $q\in \operatorname{Supp} D$ by the same letters. We have $\sigma^*_p(S_0)=S_0+F'_p$. Therefore, by (5),
$$
\begin{equation*}
\sigma^*_p(S_0)=S_0+F'_p= S_{\infty}+\sum_{i=1}^MF_{p_{\operatorname{v}+i}} - \sum_{i=1}^KF_{p_{\operatorname{v}+M+i}}
\end{equation*}
\notag
$$
in $\operatorname{Pic}(Y)$, and
$$
\begin{equation*}
\sigma_*(S_0)=S_0= S_{\infty}+\sum_{i=1}^MF_{p_{\operatorname{v}+i}} - \sum_{i=1}^KF_{p_{\operatorname{v}+M+i}}-F'_p
\end{equation*}
\notag
$$
in $\operatorname{Pic}(\overline L_{\mathcal C,D'})$. Hence, $D'=D-p\in \operatorname{Pic}(\mathcal C)$. Lemma 2 is proved. By Lemma 2, we have the bi-rational mapping
$$
\begin{equation*}
T=\operatorname{elm}_{p_{\operatorname{v}+1}}\cdots \operatorname{elm}_{p_{\operatorname{v}+M}} \operatorname{elm}_{\widetilde p_{\operatorname{v}+M+1}}\cdots \operatorname{elm}_{\widetilde p_{\operatorname{v}+M+K}}\colon \overline L_{\mathcal C,D}\dashrightarrow \overline L_{\mathcal C,0}\simeq \mathbb P^1\times \mathcal C
\end{equation*}
\notag
$$
and the inverse mapping
$$
\begin{equation}
T^{-1}=\operatorname{elm}_{\widetilde p_{\operatorname{v}+1}}\cdots \operatorname{elm}_{\widetilde p_{\operatorname{v}+M}} \operatorname{elm}_{p_{\operatorname{v}+M+1}}\cdots \operatorname{elm}_{p_{\operatorname{v}+M+K}}\colon \overline L_{\mathcal C,0}\dashrightarrow \overline L_{\mathcal C,D}.
\end{equation}
\tag{6}
$$
Consider a local case of elementary transformations, that is, the case
$$
\begin{equation*}
\operatorname{elm}_p\colon \mathbb P^1\times \mathbb D_2\dashrightarrow \mathbb P^1\times \mathbb D_2\quad (\text{respectively,} \operatorname{elm}_{\widetilde p}\colon \mathbb P^1\times \mathbb D_2 \dashrightarrow \mathbb P^1\times \mathbb D_2),
\end{equation*}
\notag
$$
where $\mathbb D_2=\{ w\in \mathbb C\mid |w|< 2\}$ is the disc in $\mathbb C$ and $\mathbb P^1$ is the projective line. Let $(z_1:z_2)$ be homogeneous coordinates in $\mathbb P^1$ and $p=\{ (0,1)\}\times \{ w=0\}$ (respectively, $\widetilde p=\{ (1,0)\}\times \{ w=0\}$). As above, denote by $S_0=\{ z_1=0\}\times \mathbb D_2$ the zero section, and, by $S_{\infty}=\{ z_2=0\}\times \mathbb D_2$, the section at “infinity” of the projection to $\mathbb D_2$. The mapping
$$
\begin{equation*}
\operatorname{elm}_p \text{ (respectively, } \operatorname{elm}_{\widetilde p}) \colon \mathbb P^1\times \mathbb D_2\setminus (S_0\cup S_{\infty}\cup F_p)\to \mathbb P^1\times \mathbb D_2\setminus (S_0\cup S_{\infty}\cup F_p)
\end{equation*}
\notag
$$
is a bi-holomorphic mapping. Therefore, the mapping $\operatorname{elm}_p$ (respectively, $\operatorname{elm}_{\widetilde p}$) induces an isomorphism
$$
\begin{equation*}
\begin{aligned} \, &\operatorname{elm}_{p*} \text{ (respectively, } \operatorname{elm}_{\widetilde p*}) \colon \pi_1\bigl(\mathbb P^1\times \mathbb D_2\setminus (S_0\cup S_{\infty}\cup F_p)\bigr) \\ &\qquad\qquad\qquad\to \pi_1\bigl(\mathbb P^1\times \mathbb D_2 \setminus (S_0\cup S_{\infty}\cup F_p)\bigr). \end{aligned}
\end{equation*}
\notag
$$
Denote by $z=z_1/z_2$ the coordinate in $\mathbb C=\{ (z_1:z_2)\in \mathbb P^1\mid z_2\neq 0\}$. The fundamental group $\pi_1(\mathbb P^1\times \mathbb D_2\setminus (S_0\cup S_{\infty}\cup F_p))= \pi_1(\mathbb C\times \mathbb D_2\setminus (S_0\cup F_p), q)$ is a free abelian group generated by two elements $x_0$, $x_1$, represented, respectively, by the loops
$$
\begin{equation*}
\begin{aligned} \, \gamma_0 &=\{ z=e^{2\pi\sqrt{-1}\, t},\, 0\leqslant t\leqslant 1 \}\times \{ w=1\}, \\ \gamma_1 &=\{ z=1 \}\times \{ w=e^{2\pi\sqrt{-1}\,t},\, 0\leqslant t\leqslant 1\}. \end{aligned}
\end{equation*}
\notag
$$
This result is a direct consequence of Lemma 7 in s[9]. 1.5. Presentation of the fundamental group of an irreducible curve with puncturesWe set $N:=\operatorname{v}+M+K$, and as above, let
$$
\begin{equation*}
W_i\simeq \mathbb D_2=\{ w_i\in \mathbb C\mid |w_i|<2\}
\end{equation*}
\notag
$$
be pairwise non-intersecting $N$ neighbourhoods of the points $p_i=\{ w_i=0\}\in W_i$, $i=1,\dots, N$, in the curve $\mathcal C=C_j$. The curve $\mathcal C$ of genus $g$, considered as a Riemann surface, is a sphere with $g$ handles. It is well-known that we can choose $g$ “meridians” $\widetilde{\mu}'_i:=\widetilde{\mu}'_{j,i}$ and $g$ “parallels” $\widetilde{\lambda}'_i:=\widetilde{\lambda}'_{j,i}$, $i=1,\dots, g$, that is, $\widetilde{\mu}'_1,\dots, \widetilde{\mu}'_g$, $\widetilde{\lambda}'_1,\dots, \widetilde{\lambda}'_g$ are oriented smooth loops intersecting only at a point $q'_0:=q'_{j,0}\in \mathcal C$ such that if we cut $\mathcal C$ along these loops, then we obtain a $2g$-polygon $P_{\widetilde{\mu}',\widetilde{\lambda}'}(\mathcal C)$. As is well known, the fundamental group $\pi_1\bigl(\mathcal C\setminus \bigl(\bigcup_{i=1}^N p_i\bigr),q'_0\bigr)$ of the Riemann surface $\mathcal C$ with $N$ punctures has the presentation
$$
\begin{equation}
\pi_1\biggl(\mathcal C\setminus \biggl(\bigcup_{i=1}^N p_i\biggr),q'_0\biggr) = \biggl\langle\mu'_{1},\lambda'_{1},\dots,\mu'_g,\lambda'_{g},x'_{1}, \dots,x'_{N} \biggm| \prod_{i=1}^{g}[\mu'_{i},\lambda'_{i}] \prod_{i=1}^{N}x'_{i}=1 \biggr\rangle,
\end{equation}
\tag{7}
$$
where $\mu'_i$ and $\lambda'_i$ are represented by the loops $\widetilde{\mu'_i}$ and $\widetilde{\lambda}'_i$ and $x'_{i}$ are represented by the simple loops $\widetilde x'_i$ consisting of paths $l'_i$ from the point $q'_0$ to the points $q'_i=\{ w_i=1\}\in W_i$, the circles $\gamma'_i=\{ w_i=e^{2\pi\sqrt{-1}\, t},\, 0\leqslant t\leqslant 1\}\subset \mathbb D_2=\{ w_i\in \mathbb C\mid |w_i|<2\}\simeq W_i$ and the returns to the point $q'_0$ along the paths $l'_i$ (see Figure 2). 1.6. Presentation of the fundamental group of the complement of a union of zero section and fibres in the normal bundle of an irreducible curveThe surface $\overline L_{\mathcal C,0}\setminus \bigl(S_0 \cup S_{\infty} \cup \bigl( \bigcup_{i=1}^NF_{p_i}\bigr)\bigr)\subset \overline L_{\mathcal C,0}$ is isomorphic to $\mathbb C^*_{z}\times \bigl(\mathcal C\setminus \bigl(\bigcup_{i=1}^N p_i\bigr)\bigr)$, where $\mathbb C^*_{z}=\{ z\in\mathbb C \mid z\neq 0\}$. Therefore,
$$
\begin{equation*}
\pi_1\biggl(\overline L_{\mathcal C,0}\setminus \biggl(S_0 \cup S_{\infty}\cup \biggl( \bigcup_{i=1}^NF_{p_i}\biggr)\biggr),q_0\biggr)\simeq \pi_1(\mathbb C^*_{z},1)\times\pi_1\biggl(\mathcal C\setminus \biggl(\bigcup_{i=1}^N p_i\biggr),q'_0\biggr),
\end{equation*}
\notag
$$
$q_0=\{z=1\}\times q'_0$, and the fundamental group $\pi_1\bigl(\overline L_{\mathcal C,0}\setminus \bigl(S_0 \cup S_{\infty}\cup \bigl(\bigcup_{i=1}^NF_{p_i}\bigr)\bigr),q_0\bigr)$ has the following presentation:
$$
\begin{equation}
\begin{aligned} \, &\pi_1\biggl(\overline L_{\mathcal C,0}\setminus \biggl(S_0 \cup S_{\infty}\cup \biggl( \bigcup_{i=1}^NF_{p_i}\biggr)\biggr),q_0\biggr) \nonumber \\ &\qquad= \biggl\langle \mu_{1},\lambda_{1},\dots,\mu_g,\lambda_{g},x_0, x_{1},\dots,x_{N} \biggm| \prod_{i=1}^{g}[\mu_{i},\lambda_{i}] \prod_{i=1}^{N}x_{i}=1, \nonumber \\ &\qquad\qquad [x_0,\mu_j]=[x_0,\lambda_j]=1 \text{ for } 1\leqslant j\leqslant g, \ [x_0,x_i]=1 \text{ for }1\leqslant i\leqslant N \biggr\rangle, \end{aligned}
\end{equation}
\tag{8}
$$
where $x_0$ is represented by the circle $\widetilde x_0= \bigl\{ z=e^{2\pi\sqrt{-1}\,t},\, 0\leqslant t\leqslant 1\bigr\}\times q'_0$; $\mu_{i}$ and $\lambda_i$ are represented by the loops $\widetilde{\mu}_i=\{ z=1\}\times \widetilde{\mu}'_i$ and $\widetilde{\lambda}_{i}=\{ z=1\}\times \widetilde{\lambda}'_i$ obtained as a result of lifting in the section $S_1=\{ z=1\}\times \mathcal C\subset \overline L_{\mathcal C,0}$ of the loops representing the elements $\mu'_i$ and $\lambda'_{i}$; and $x_i$ are elements represented by simple loops $\widetilde x_i$ consisting of paths $l_i=\{ z=1\}\times l'_i$ from the point $q_0$ to the points $q_i=\{ z=1\}\times q'_i$, the circles $\gamma_i =\{ z=1\}\times \gamma'_i$ in $\{ z=1\}\times W_i$,
$$
\begin{equation*}
\gamma_i =\{ z=1\}\times \bigl\{ w_i= e^{2\pi\sqrt{-1}\,t},\, 0\leqslant t\leqslant 1\bigr\},
\end{equation*}
\notag
$$
and the returns to the point $q_0$ along the paths $l_i$. The bi-holomorphic mapping (see (6))
$$
\begin{equation*}
T^{-1}\colon \overline L_{\mathcal C,0}\setminus \biggl(S_0 \cup S_{\infty}\cup \biggl(\bigcup_{i=1}^NF_{p_i}\biggr)\biggr)\to \overline L_{\mathcal C,D}\setminus \biggl(S_0 \cup S_{\infty}\cup \biggl(\bigcup_{i=1}^NF_{p_i})\biggr)
\end{equation*}
\notag
$$
defines the isomorphism
$$
\begin{equation*}
\begin{aligned} \, &T^{-1}_*\colon \pi_1\biggl(\overline L_{\mathcal C,0}\setminus \biggl(S_0 \cup S_{\infty}\cup \biggl(\bigcup_{i=1}^NF_{p_i}\biggr)\biggr),q_0\biggr) \\ &\qquad \to \pi_1\biggl(\overline L_{\mathcal C,D}\setminus \biggl(S_0 \cup S_{\infty}\cup \biggl( \bigcup_{i=1}^NF_{p_i}\biggr)\biggr),\widetilde q_0\biggr), \end{aligned}
\end{equation*}
\notag
$$
where $\widetilde q_0=T^{-1}(q_0)$. We set $\widetilde l_i=T^{-1}(l_i)$. In the notation of § 1.4, let us identify the neighbourhoods $W_i\subset\mathcal C$ with the neighbourhood $W$ considered in § 1.4 ($w_i:=w$) and denote by
$$
\begin{equation*}
y_i\in \pi_1\biggl(\overline L_{\mathcal C,0}\setminus \biggl(S_0 \cup S_{\infty}\cup \biggl( \bigcup_{i=1}^NF_{p_i}\biggr)\biggr),q_0\biggr),\qquad i=\operatorname{v}+1,\,\dots,\,\operatorname{v}+(M+K),
\end{equation*}
\notag
$$
the elements represented by the loops consisting of the paths $\widetilde l_i=T^{-1}(l_i)$, the circles $\gamma_i =\{ z'=1\}\times \{ w_i=e^{2\pi\sqrt{-1}\,t},\, 0\leqslant t\leqslant 1\}$ in $\mathbb C_{z'}\times W_i=\mathbb C_{z'}\times W$, and the returns to the point $\widetilde q_0$ along the paths $\widetilde l_i$. We also denote by the same letters $\mu_1,\lambda_1,\dots,\mu_g,\lambda_g$ and $x_0,\dots, x_{\operatorname{v}}$ the images $T_*^{-1}(\mu_i)$, $T_*^{-1}(\lambda_i)$, $T_*^{-1}(x_i)$ of the elements
$$
\begin{equation*}
\mu_1,\lambda_1,\dots,\mu_g,\lambda_g,x_0,\dots, x_{\operatorname{v}}\in \pi_1\biggl(\overline L_{\mathcal C,0}\setminus \biggl(S_0 \cup S_{\infty}\cup \biggl( \bigcup_{i=1}^NF_{p_i}\biggr)\biggr),q_0\biggr).
\end{equation*}
\notag
$$
Now by Lemma 3 $ T_*^{-1}(x_i)=x_0y_i$ for $i=\operatorname{v}+1,\dots,\operatorname{v}+M$ and $T_*^{-1}(x_i)=x_0^{-1}y_i$ for $i=\operatorname{v}+M+1,\dots,\operatorname{v}+M+K$. In view of presentation (8), the fundamental group $\pi_1\bigl(\overline L_{\mathcal C,D}\setminus \bigl(S_0 \cup S_{\infty}\cup \bigl( \bigcup_{i=1}^NF_{p_i}\bigr)\bigr),\widetilde q_0\bigr)$ has the presentation, where $\omega= M-K$,
$$
\begin{equation}
\begin{aligned} \, &\biggl\langle \mu_{1},\lambda_{1},\dots,\mu_g,\lambda_{g},x_0,x_{1}, \dots,x_{\operatorname{v}}, y_{\operatorname{v}+1},\dots, y_{\operatorname{v}+M+K} \biggm| \nonumber \\ &\qquad x_0^{\omega}\prod_{i=1}^{g}[\mu_{i},\lambda_{i}] \prod_{i=1}^{\operatorname{v}}x_{i}\prod_{i=1}^{M+K} y_{\operatorname{v}+i}=1,\ [x_0,\mu_j]=[x_0,\lambda_j]=1 \text{ for }1\leqslant j\leqslant g, \nonumber \\ &\qquad [x_0,x_i]=1 \text{ for } 1\leqslant i\leqslant \operatorname{v},\ [x_0,y_{\operatorname{v}+i}]=1 \text{ for } 1\leqslant i\leqslant M+K \biggr\rangle. \end{aligned}
\end{equation}
\tag{9}
$$
The imbedding
$$
\begin{equation*}
i\colon \overline L_{\mathcal C,D}\setminus \biggl(S_0 \cup S_{\infty}\cup \biggl( \bigcup_{i=1}^NF_{p_i}\biggr)\biggr)\hookrightarrow \overline L_{\mathcal C,D}\setminus \biggl(S_0 \cup S_{\infty}\cup \biggl( \bigcup_{i=1}^{\operatorname{v}}F_{p_i}\biggr)\biggr)
\end{equation*}
\notag
$$
defines an epimorphism
$$
\begin{equation*}
i_*\colon \pi_1\biggl(\overline L_{\mathcal C,D}\setminus \biggl(S_0 \cup S_{\infty}\cup \biggl( \bigcup_{i=1}^NF_{p_i}\biggr)\biggr),\widetilde q_0\biggr) \twoheadrightarrow \pi_1\biggl(\overline L_{\mathcal C,D}\setminus \biggl(S_0 \cup S_{\infty}\cup \biggl( \bigcup_{i=1}^{\operatorname{v}}F_{p_i}\biggr) \biggr), \widetilde q_0\biggr).
\end{equation*}
\notag
$$
It is obvious that the kernel of $i_*$ is the normal closure in
$$
\begin{equation*}
\pi_1\biggl(\overline L_{\mathcal C,D}\setminus \biggl(S_0 \cup S_{\infty} \cup \biggl( \bigcup_{i=1}^NF_{p_i}\biggr)\biggr),\widetilde q_0\biggr)
\end{equation*}
\notag
$$
of the subgroup generated by elements $y_{\operatorname{v}+1},\dots, y_{\operatorname{v}+M+K}$. Therefore, by applying presentation (9) with $\mathcal C=C_j$, we obtain Proposition 2, in which we use the following notation: $x_{j,0}=i_*(x_0)$, $x_{o_j(l),1}=i_*(x_l)$ for $l=1,\dots, \operatorname{v}_j$, $\mu_{j,l}=i_*(\mu_l)$, and $\lambda_{j,l}=i_*(\lambda_l)$ for $l=1,\dots,g_j$. Proposition 2. For $N_{C_j}=L_{C_j,D}$, the group 1.7. Presentation of the fundamental group of the complement of $\widetilde C$ in tubular neighbourhoods of its compact irreducible componentsFor $\varepsilon< \varepsilon_0$ the imbeddings $i_{j,\varepsilon}\colon \widetilde U_{j,\varepsilon}\hookrightarrow N_{C_j}$ (see § 1.3) define the homomorphisms
$$
\begin{equation*}
i_{j,\varepsilon*}\colon \pi_1\biggl(\widetilde U_{j,\varepsilon}\setminus \biggl(S_{j,0}\cup \bigcup_{i=1}^{\operatorname{v}_j}F_{p_{j,o_j(i)}}\biggr)\biggr)\to \pi_1\biggl(N_{C_j}\setminus \biggl(S_{j,0}\cup \bigcup_{i=1}^{\operatorname{v}_j}F_{p_{j,o_j(i)}}\biggr)\biggr).
\end{equation*}
\notag
$$
Proposition 3. For each $j$ and for each $\varepsilon<\varepsilon_0$, the homomorphism Proof. Let us choose a finite covering $\{ W_{i}\}$ of the curve $C_j$, $C_j=\bigcup_iW_{i}$ such that the line bundle $N_{C_j}$ is trivial over each $W_{i}$. We identify $\operatorname{pr}_j^{-1}(W_{i})\subset N_{C_j}$ with $N_{C_j\mid W_{i}}=\mathbb C_z\times W_{i}$, and put $N_{C_j\mid W_{i}}(\delta):=\{ (z,p)\in \mathbb C_z\times W_{i}\mid |z|<\delta\}\subset N_{C_j}$. It is obvious that the set $\bigl\{ N_{C_j}(\delta):=\bigcup_iN_{C_j\mid W_{i}}(\delta)\bigr\}_{\delta>0}$ is a base of open subsets in $N_{C_j}$ containing the zero section $S_{j,0}$. Therefore, for each $\varepsilon<\varepsilon_0$, there is $\delta_{\varepsilon}>0$ such that $N_{C_j}(\delta)\subset \widetilde U_{j,\varepsilon}$ for each $\delta\leqslant\delta_{\varepsilon}$.
The automorphism group $\operatorname{Aut}(N_{C_j})$ contains the subgroup
$$
\begin{equation*}
A_j=\{ a_{j,r}\in \operatorname{Aut}(N_{C_j})\mid a_{j,r}(z,w)= (rz,w) \text{ for } (z,w)\in\mathbb C_z\times \widetilde W_{j,i}\}.
\end{equation*}
\notag
$$
We have $a_{j,r}(N_{C_j}(\delta))=N_{C_j}(r\delta)$.
By (10), the fundamental group
$$
\begin{equation*}
\Pi_j:=\pi_1\biggl(N_{C_j}\setminus \biggl(S_{j,0}\cup \bigcup_{i=1}^{\operatorname{v}_j}F_{p_{j,o_j(i)}}\biggr),\widetilde q_0\biggr)
\end{equation*}
\notag
$$
is finitely presented. Therefore, for each relation $\mathcal W(\overline{\mu}_j,\overline{\lambda}_j,\overline x_j)=1$ in (10), there is a continuous mapping $\theta_{\mathcal W}\colon \overline{\mathbb D}_1\to N_{C_j}\setminus \bigl(S_{j,0}\cup \bigcup_{i=1}^{\operatorname{v}_j}F_{p_{j,o_j(i)}}\bigr)$ of the closed disc $\overline{\mathbb D}_1=\{ z\in \mathbb C\mid |z|\leqslant 1\}$ such that the loop
$$
\begin{equation}
\mathcal W\bigl(\widetilde{\mu}_{j,1}, \widetilde{\lambda}_{j,1},\dots, \widetilde{\mu}_{j,g_j}, \widetilde{\lambda}_{j,g_j}, \widetilde x_{j,0}, \widetilde x_{o_j(1),1},\dots, \widetilde x_{o_j(\operatorname{v}_j),1}\bigr) =\theta_{\mathcal W}(\partial \overline{\mathbb D}_1)
\end{equation}
\tag{11}
$$
is the image of the boundary $\partial \overline{\mathbb D}_1= \{ z=e^{2\pi \sqrt{-1}\, t}\mid 0\leqslant t\leqslant 1\}$ of the disc $\overline{\mathbb D}_1$ (here in the case $\mathcal C=C_j$ the loops $\widetilde{\mu}_{j,i}:=\widetilde{\mu}_i$, $\widetilde{\lambda}_{j,i}:=\widetilde{\lambda}_i$ for $i=1,\dots, g_j$, $\widetilde x_{j,0}:=\widetilde x_0$, and $\widetilde x_{o_j(i),1}:=\widetilde x_i$ for $i=1,\dots, \operatorname{v}_j$, where the loops $\widetilde{\mu}_{i}$, $\widetilde{\lambda}_{i}$, $\widetilde x_0$, and $\widetilde x_{i}$ from $\mathcal C$ are defined in § 1.6).
Since $\theta_{\mathcal W}(\overline{\mathbb D}_1)$ are compacts, there is a constant $r_{C_j}>0$ such that
$$
\begin{equation}
\theta_{\mathcal W}(\overline{\mathbb D}_1)\subset N_{C_j}(r_{C_j})
\end{equation}
\tag{12}
$$
for all relations $\mathcal W$ in (10). Therefore, if $\delta<\delta_{\varepsilon}/r_{C_j}$, then
$$
\begin{equation*}
\begin{aligned} \, &a_{j,\delta}\bigl(\mathcal W\bigl(\widetilde{\mu}_{j,1}, \widetilde{\lambda}_{j,1},\dots, \widetilde{\mu}_{j,g_j}, \widetilde{\lambda}_{j,g_j}, \widetilde x_{j,0},\widetilde x_{o_j(1),1}, \dots, \widetilde x_{o_j(\operatorname{v}_j),1}\bigr)\bigr) \\ &\qquad\subset \widetilde U_{j,\varepsilon}\setminus \biggl(S_{j,0}\cup \bigcup_{i=1}^{\operatorname{v}_j}F_{p_{j,o_j(i)}}\biggr), \end{aligned}
\end{equation*}
\notag
$$
and it easily follows from (11) and (12) that $i_{\varepsilon*}$ is an isomorphism for each $\varepsilon<\varepsilon_0$. This proves Proposition 3. 1.8. Presentation of the fundamental group of the complement of $\widetilde C$ in a tubular neighbourhood of $\widetilde C_0$Let $\widetilde{\mathcal U}_{\nu,\varepsilon}= \{ \widetilde U_{1,\varepsilon},\dots,\widetilde U_{m,\varepsilon}\}$ be the set of neighbourhoods $\widetilde U_{j,\varepsilon}\subset N_{C_j}$ of zero sections $S_{j,0}\subset N_{C_j}$. Denote by
$$
\begin{equation*}
E'=\{ (v_{j_1,0},v_{j_2,0}) \mid 1\leqslant j_1<j_2\leqslant m, \, \Delta_{(j_1,0),(j_2,0)}=1\}
\end{equation*}
\notag
$$
the subset of the set of edges of the graph $\Gamma_{\widetilde E}(\widetilde C)$, and, for $(v_{j_1,0},v_{j_2,0})\in E'$, consider the neighbourhoods $U_{j_1,j_2,\varepsilon}\subset U_{j_1,\varepsilon}$ and $U_{j_2,j_1,\varepsilon}\subset U_{j_2,\varepsilon}$ defined in Claim 1. By Remark 1, $U_{j_1,j_2,\varepsilon}=U_{j_2,j_1,\varepsilon}$ (as subsets of $X)$. Therefore, applying the identifications of $U_{j,\varepsilon}$ with $\widetilde U_{j,\varepsilon}$ given by diffeomorphisms $\psi_j\circ\iota\circ\varphi_j^{-1}\colon U_{j,\varepsilon}\to \widetilde U_{j,\varepsilon}$, the neighbourhoods $\widetilde U_{j_1,\varepsilon}$ and $\widetilde U_{j_2,\varepsilon}$, $(v_{j_1,0},v_{j_2,0})\in E'$ can be glued along $U_{j_1,j_2,\varepsilon}\subset U_{j_1,\varepsilon}$ and $U_{j_2,j_1,\varepsilon}\subset U_{j_2,\varepsilon}$. As a result, we obtain a complex manifold $U'_{\varepsilon}$. In the notation used in the proof of Proposition 3 (with identification of $\widetilde U_{j,\varepsilon}$ and $U_{j,\varepsilon}$), we set $Q_{j,0}:=a_{j,\epsilon}(\widetilde q_{j,0})$ and $Q_{j,o_j(i)}:=a_{j,\epsilon}(\widetilde q_{j,o_j(i)})\in U_{j,\varepsilon}\subset U'_{\varepsilon}$, $i= 1,\dots,\operatorname{v}_j$. In addition, without loss of generality, we can assume that $Q_{j_1,j_2}=Q_{j_2,j_1}\in U_{j_1,j_2,\varepsilon}$ if $C_{j_1}\cap C_{j_2}\neq\varnothing$. Also let $L_{j,o_j(i)}:=a_{j,\epsilon}(\widetilde l_{j,o_j(i)})$, $i=0,1,\dots,\operatorname{v}_j$, be the paths in $U_{j,\varepsilon}$ connecting the point $Q_{j,0}$ with the points $Q_{j,o_j(i)}$. Since $\Gamma_{\widetilde E}(\widetilde C)$ is a tree, we can assume that the paths $\widetilde l_{j,o_j(i)}$ are chosen so that
$$
\begin{equation*}
L=\bigcup_{j=1}^m\bigcup_{i=1}^{\operatorname{v}_j}L_{j,o_j(i)}
\end{equation*}
\notag
$$
is a tree. For $j=1,\dots,m$ and $i=1,\dots, \operatorname{v}_j$, let $L_{0,j,o_j(i)}$ be the paths in $L$ from $Q_{1,0}$ to the points $Q_{j,o_j(i)}$. Further, if this does not lead to a misunderstanding, we will denote again by $\mu_{j,i}$ and $\lambda_{j,i}$, and by $x_{j,0}$ for $j=1,\dots, m$ and $i=1,\dots,g_j$, and also we denote by $x_{o_j(i),1}$ for $i=1,\dots \operatorname{v}_j$, the elements in $\pi_1(U'_{\varepsilon}\setminus \widetilde C,Q_{1,0})$ represented by loops. 1. Go from $Q_{1,0}$ to $Q_{j,0}$ along $L_{0,j,0}$. 2. Go once along $a_{j,\epsilon}(\widetilde \mu_{j,i})$ (respectively, $a_{j,\epsilon}(\widetilde \lambda_{j,i})$, $a_{j,\epsilon}(\widetilde x_{j,0})$, and $a_{j,\epsilon}(\widetilde x_{j,o_j(i)}$). 3. Go back to $Q_{1,0}$ along $L_{0,j,0}$. The choice of paths $L_{0,j,0}$ and the imbeddings $\alpha_j\colon U_{j,\varepsilon}\setminus \widetilde C\hookrightarrow U'_{\varepsilon}\setminus \widetilde C$ define the homomorphisms
$$
\begin{equation*}
\alpha_{j*}\colon \pi_1(U_{j,\varepsilon}\setminus \widetilde C,Q_{j,0}) \to \pi_1(U'_{\varepsilon}\setminus \widetilde C,Q_{1,0})
\end{equation*}
\notag
$$
and the equalities
$$
\begin{equation*}
\begin{gathered} \, \alpha_{j*}(a_{j,\epsilon}(\mu_{j,i}))=\mu_{j,i},\qquad \alpha_{j*}(a_{j,\epsilon}(\lambda_{ji}))=\lambda_{j,i},\qquad \alpha_{j*}(a_{j,\epsilon}(x_{j,0}))=x_{j,0}, \\ \alpha_{j*}(a_{j,\epsilon}(x_{j,o_j(i)}))=x_{o_j(i),1}. \end{gathered}
\end{equation*}
\notag
$$
Note that, for $U_{j,o_j(i),\varepsilon}\subset U_{j,\varepsilon}$, the group $\alpha_{j*}(\pi_1(U_{j,o_j(i),\varepsilon}\setminus\widetilde C))$ is generated by $x_{j,0}=\alpha_{j*}(a_{j,\epsilon}(x_{j,0}))$ and $x_{o_j(i),1}=\alpha_{j*}(a_{j,\epsilon}(x_{j,o_j(i)}))$, and it is easy to see that, for $1\leqslant j_i\leqslant m$, $i=1,2$, such that $\Delta_{(j_1,0),(j_2,0)}=1$, we have
$$
\begin{equation*}
\alpha_{j_1*}\bigl(\pi_1(U_{j_1,j_2,\varepsilon}\setminus\widetilde C)\bigr)= \alpha_{j_2*}\bigl(\pi_1(U_{j_2,j_1,\varepsilon}\setminus\widetilde C)\bigr),
\end{equation*}
\notag
$$
and $x_{j_1,0}=x_{j_2,1}$, $x_{j_2,0}=x_{j_1,1}$. Since $L$ is a tree, using the paths $L_{0,j,o_j(i)}$, and applying Seifert–van Kampen Theorem (see [10], [11]) $m-1$ times, we obtain a presentation of the group $\pi_1(U'_{\varepsilon}\setminus \widetilde C)$ as the free product of the groups $\alpha_{j*}(\pi_1(U_{j_,\varepsilon}\setminus\widetilde C,Q_{j,0}))$ with amalgamation of the groups
$$
\begin{equation*}
\alpha_{j_1*}\bigl(\pi_1(U_{j_1,j_2,\varepsilon}\setminus \widetilde C, Q_{j_1,j_2})\bigr)=\alpha_{j_2*} \bigl(\pi_1(U_{j_2,j_1,\varepsilon}\setminus\widetilde C,Q_{j_2,j_1})\bigr),
\end{equation*}
\notag
$$
$\Delta_{(j_1,0),(j_2,0)}=1$. As a result, we obtain the following fact. Proposition 4. The group $\pi_1( U'_{\varepsilon}\setminus \widetilde C,Q_{1,0})$ has the following presentation: it is generated by $m+k+2\sum_{j=1}^mg_j+2r$ elements: and the defining relations are $(r_1)$–$(r_3)$ (here $x_{j,\delta}$ are in one-to-one correspondence with the vertices $v_{j,\delta}$ of the graph $\Gamma_{\widetilde E}(\widetilde C)$). To obtain the surface $\widetilde U_{\varepsilon}$ from $U'_{\varepsilon}$, we should consequently glue $U_{j(E_s,1),\varepsilon}\subset U'_{\varepsilon}$ with $U_{j(E_s,2),\varepsilon}\subset U'_{\varepsilon}$ by their “intersection” $U_{j(E_s,1),j(E_s,2),\varepsilon}\,{=}\,U_{j(E_s,1),\varepsilon}\cap\, U_{j(E_s,2)\varepsilon}\subset \widetilde U_{\varepsilon}$, $s=1,\dots,r$. The paths $L_{0,j(E_s,1),j(E_s,2)}$ and $L_{0,j(E_s,2),j(E_s,1)}$ define two imbeddings
$$
\begin{equation*}
\alpha_{j(E_s,i),j(E_s,\overline i),\varepsilon} \colon \pi_1(U_{j(E_s,1),j(E_s,2),\varepsilon}\setminus \widetilde C,Q_{j(E_s,1),j(E_s,2)})\to \pi_1(\widetilde U_{\varepsilon}\setminus \widetilde C, Q_{1,0}),
\end{equation*}
\notag
$$
where $i=1,2$ and $\{ i,\overline i\}=\{ 1,2\}$. Therefore, to obtain a presentation of $\pi_1(\widetilde U_{\varepsilon}\setminus \widetilde C)$, it suffices to apply $r$ times the $HNN$-extension of $\pi_1(U'_{\varepsilon}\setminus \widetilde C, Q_{1,0})$ relative to these imbeddings, that is, it suffices to add $r$ generators $y_1,\dots, y_r$ and relations $(i_4)$, $(r_5)$:
$$
\begin{equation*}
\begin{alignedat}{5} x_{j(E_s,1),0}^{-1}y_sx_{j(E_s,1),1}y_s^{-1} &=1 &\quad &\text{for} &\quad &1\leqslant s\leqslant r, \\ x_{j(E_s,2),0}^{-1}y_s^{-1}x_{j(E_s,2),1}y_s &=1 &\quad &\text{for} &\quad &1\leqslant s\leqslant r, \end{alignedat}
\end{equation*}
\notag
$$
to the presentation claimed in Proposition 4 (here $y_s$ are elements of $\pi_1(\widetilde U_{\varepsilon}\setminus \widetilde C)$, represented by the loops $L_{0,j(E_s,1),j(E_s,2)}\circ L^{-1}_{0,j(E_s,2),j(E_s,1)}$). As a result, we obtain the presentations claimed in Theorem 3. Therefore, to complete the proof of Theorem 3, it remains to verify that the groups $\pi_1(U_{\varepsilon}\setminus C)$ and $\pi_1(U'_{\varepsilon'}\setminus C)$ are isomorphic for all $U_{\varepsilon}$, $U'_{\varepsilon'}\in \mathcal U_{C}$. 1.9. Completion of the proof of Theorem 3Note that, for all $U_{\varepsilon}\in \mathcal U_{C,\nu}$ and a selected tree ${\Gamma}_{\widetilde E}(\widetilde{C})$ of $\widetilde C=\nu^{-1}(C)$, the presentations of the fundamental groups $\pi_1(U_{\varepsilon}\setminus C)$ from Theorem 3 are independent of the choices of $\widetilde U\subset X$, of the good metrics $(ds)^2$ on $\widetilde U$, and of $\varepsilon<\varepsilon_0$. In addition, from above we have the following remark. Remark 2. Two presentations, as obtained with two different choices of subsets $\widetilde E=\{ E_1,\dots,E_r\}$ (involved in the definition of the tree $\Gamma_{\widetilde E}(\widetilde C)$) in the edge set of $\Gamma(\widetilde C)$ (taking a part in the definition of a tree $\Gamma_{\widetilde E}(\widetilde C)$ ) are the presentations of the same group $\pi_1(U_{\varepsilon}\setminus C)$. Let $\nu\colon X\to S$ be a resolution of singularities of $C$ let and $\sigma\colon X'\to X$ be the $\sigma$-process with centre at a point $p\in\widetilde C \subset X$. Then $\widetilde C'=(\nu\circ\sigma)^{-1}(C)= \sigma^{-1}(\widetilde C)\,\cup\, C_p$, where $C_p=\sigma^{-1}(p)$, and a tree $\Gamma_{\widetilde E'}(\widetilde C')$ can be obtained from $\Gamma_{\widetilde E}(\widetilde C)$ as follows. We add one more vertex $v_{p,0}$ to the set of vertices of $\Gamma_{\widetilde E}(\widetilde C)$. The bi-weight of $v_{p,0}$ is $(w_{p,1},w_{p,2})=(-1,0)$. Next, if $p\in C_{j_0}$ and $p\notin \operatorname{Sing}(\widetilde C)$, then we add one edge connecting the vertex $v_{j_0,0}\in\Gamma_{\widetilde E}(\widetilde C)$ with $v_{p,0}$, and change the bi-weight of $v_{j_0,0}$ by $(w_{j_0,1}-1,w_{j_0,2})$. If $p=p_{j_1,j_2}=C_{j_1}\cap C_{j_2}$, then by Remark 2 we can assume that $(v_{j_1,0},v_{j_2,0})\notin \{ E_1,\dots,E_r\}$, delete this edge, add two edges $(v_{j_1,0},v_{p,0})$ and $(v_{p,0},v_{j_2,0})$, and change the bi-weights of $v_{j_i,0}$ by $(w_{j_i,1}-1,w_{j_i,2})$, $i=1,2$. Next, let $\langle \mathcal A \mid \mathcal R\rangle$ and $\langle \mathcal A' \mid \mathcal R'\rangle$ (here $\mathcal A$ and $\mathcal A'$ are the alphabets, and $\mathcal R$ and $\mathcal R'$ are the sets of defining relations) be presentations of groups obtained via the graphs $\Gamma_{\widetilde E}(\widetilde C)$ and $\Gamma_{\widetilde E'}(\widetilde C')$. In the first case, $\mathcal A'=\mathcal A\cup\{x_{p,0}\}$, and the set of defining relations $\mathcal R'$ can be obtained from $\mathcal R$ as follows: in $\mathcal R$, we change the relation $x_{j_0,0}^{\omega_{j_0,2}}\mathcal W_{g_{j_0},\operatorname{v}_{j_0} }=1$ by $x_{j_0,0}^{\omega_{j_0,2}-1}\mathcal W_{g_{j_0}, \operatorname{v}_{j_0} }x_{p,0}=1$ and add two relations $x_{p,0}^{-1}x_{j_0,0}=[x_{j_0,0},x_{p,0}]\,{=}\,1$. Obviously, $\langle \mathcal A\mid \mathcal R\rangle$ and $\langle \mathcal A' \mid \mathcal R'\rangle$ are isomorphic groups. In the second case, $\mathcal A'=\mathcal A\cup\{x_{p,0}\}$ and the set of defining relations $\mathcal R'$ can be obtained from $\mathcal R$ as follows: in $\mathcal R$, we change three relations
$$
\begin{equation}
x_{j_1,0}^{\omega_{j_1,2}}\mathcal W_{g_{j_1},\operatorname{v}_{j_1}} = x_{j_2,0}^{\omega_{j_2,2}}\mathcal W_{g_{j_2},\operatorname{v}_{j_2}} = [x_{j_1,0},x_{j_2,0}]=1
\end{equation}
\tag{14}
$$
by
$$
\begin{equation}
x_{j_1,0}^{\omega_{j_i1,2}-1}\mathcal W'_{g_{j_1},\operatorname{v}_{j_1}} = x_{j_2,0}^{\omega_{j_2,2}-1}\mathcal W'_{g_{j_2},\operatorname{v}_{j_i}} = [x_{j_1,0},x_{p,0}]=[x_{j_2,0},x_{p,0}]=1,
\end{equation}
\tag{15}
$$
where the word $\mathcal W'_{g_{j_1},\operatorname{v}_{j_1} }$ is obtained from $\mathcal W_{g_{j_1},\operatorname{v}_{j_1} }$ by substituting $x_{p,0}$ for $x_{j_2,0}$, and $\mathcal W'_{g_{j_2},\operatorname{v}_{j_2} }$ is obtained from $\mathcal W_{g_{j_2},\operatorname{v}_{j_2} }$ by substituting $x_{p,0}$ for $x_{j_1,0}$; in addition, we add one more relation By (16) and (15), we have $x_{p,0}=x_{j_1,0}x_{j_2,0}=x_{j_2,0}x_{j_1,0}$ and it is easy to see that we obtain relations (14) from (15) and (16) if in (15) we substitute $x_{j_1,0}x_{j_2,0}$ in $W'_{g_{j_1},\operatorname{v}_{j_1} }$ and $\mathcal W'_{g_{j_2},\operatorname{v}_{j_2} }$ for $x_{p,0}$. Therefore, in the second case, $\langle \mathcal A\mid \mathcal R\rangle$ and $\langle \mathcal A' \mid \mathcal R'\rangle$ are also isomorphic groups. To complete the proof of Theorem 3, it suffices to note that, for any two resolutions of singularities $\nu_1\colon X_1\to S$ and $\nu_2\colon X_2\to S$, there exist $X$ and two sequences of blowups of points $\widetilde{\nu}_1\colon X\to X_1$ and $\widetilde{\nu}_2\colon X\to X_2$ such that $\nu_i\circ\widetilde{\nu}_i\colon X\to S$, $i=1,2$, are resolutions of singularities. § 2. Proofs of Proposition 1 and Theorem 12.1. Proof of Proposition 1Consider a graph $\Gamma\in \mathcal G$. Let
$$
\begin{equation*}
\{ v_{1,0},\dots, v_{m,0}\}\cup \{ v_{{m+1},1},\dots, v_{n,1}\}
\end{equation*}
\notag
$$
be the vertex set and let $(k_j,g_j)$ be the bi-weights of $v_{j,0}$ for $j\leqslant m$. For each vertex $v_{j,o}$, $j\leqslant m$, with bi-weight $(k_j,g_j)$, denote by $C_j$ the zero section (and the base) of a complex line bundle $\operatorname{pr}_j\colon L_j:=L_{C_j,D}\to C_j$ associated with a divisor $D$ of degree $k_j$ on a projective curve $C_j$ of genus $g_j$. We have $(C_j^2)_{L_j}=k_j$. Next, let us choose $\operatorname{v}_{v_{j,0}}$ fibres of the projection $\operatorname{pr}_j$, denote them by $F_{j,o_j(1)},\dots, F_{j,o_j(\operatorname{v}_j)}$, and choose a Hermitian metric $(ds_j)^2$ in $L_j$ which is good with respect to the curve $C_j\cup \bigl(\bigcup_{i=1}^{\operatorname{v}_j}F_{j,o_j(i)}\bigr)$. Let $\varepsilon _j$ be a positive number such that the sets
$$
\begin{equation*}
U_{j,\varepsilon}= \{ p\in L_j\mid \operatorname{dist}_{(ds_j)^2}(C_j, p)< \varepsilon\leqslant\varepsilon_j\}
\end{equation*}
\notag
$$
are tubular neighbourhoods of $C_j$. By definition of a good metric, for each $i=1,\dots, \operatorname{v}_j$, there is a neighbourhood $V_{j,o_j(i)}\subset U_{j,\varepsilon}$ of the point $p_{j,o_j(i)}=C_j\cap F_{o_j(i)}$ such that (i) there are local coordinates $z_1,z_2$ in $V_{j,o_j(i)}$ such that
$$
\begin{equation*}
V_{j,o_j(i)}\simeq \{ (z_{j,1},z_{j,2})\in\mathbb C^2\mid |z_{j,1}|< \varepsilon,\, |z_{j,2}|<\varepsilon\};
\end{equation*}
\notag
$$
(ii) $z_{j,1}=0$ is the equation of $C_j\cap V_{j,o_j(i)}$ and $z_{j,2}=0$ is the equation of $F_{o_j(i)}\cap V_{j,o_j(i)}$; (iii) the metric $(ds_j)^2$ in $V_{j,o_j(i)}$ is given by $(ds_j)^2=dz_{j,1}\,d\overline z_{j,1}+dz_{j,2}\,d\overline z_{j,2}$. We set $\varepsilon_0=\min_{1\leqslant j\leqslant m} \varepsilon_j$ and define $C_{o_j(i)}=F_{o_j(i)}\cap V_{j,o_j(i)}\subset U_{j,\varepsilon_0}$ if $o_j(i) > m$. If, for $1\leqslant j_1\neq j_2\leqslant m$, there are $i_1$ and $i_2$ such that $o_{j_1}(i_2)=j_2$ and $o_{j_2}(i_1)=j_1$, then we glue $U_{j_1,\varepsilon_0}$ with $U_{j_2,\varepsilon_0}$ identifying $V_{j_1,j_2}\subset U_{j_1,\varepsilon_0}$ with $V_{j_2,j_1}\subset U_{j_2,\varepsilon_0}$ with the help of the bi-holomorphic isomorphism of $V_{j_1,j_2}$ and $V_{j_2,j_1}$ given by $z_{j_1,1}\leftrightarrow z_{j_2,2}$, $z_{j_1,2}\leftrightarrow z_{j_2,1}$. As a result of all possible such glues, we obtain the tubular neighbourhood $\widetilde U_{\varepsilon_0}=\bigcup_{j=1}^mU_{j,\varepsilon_0}$ of the curve $\widetilde C=\bigcup_{j=1}^nC_j$. Obviously, the partially bi-weighted dual graph $\Gamma(\widetilde C)$ of the curve $\widetilde C\subset \widetilde U_{\varepsilon_0}$ and the partially bi-weighted graph $\Gamma\in\mathcal G$ are isomorphic as partially bi-weighted graphs. 2.2. Proof of Theorem 1In the notation of § 1.2, for each point
$$
\begin{equation*}
\begin{aligned} \, p_{j_1,j_2} &=C_{j_1}\cap C_{j_2}\subset V_{p_{j_1,j_2}}\simeq \mathbb B_2 \\ &=\bigl\{ (z_1,z_2)\in \mathbb C^2\bigm| \sqrt{|z_1|^2+|z_2|^2}<2\bigr\}, \qquad 1\leqslant j_1,j_2\leqslant m, \end{aligned}
\end{equation*}
\notag
$$
consider in $V_{p_{j_1,j_2}}\cap \partial U_{\varepsilon}$ the subset
$$
\begin{equation*}
\partial U_{j_1,j_2,\varepsilon}\simeq \{ (z_1,z_2)\mid |z_1|= \varepsilon,\, \varepsilon\leqslant |z_2|\leqslant 1\}\cup \{ (z_1,z_2)\mid \varepsilon \leqslant |z_1|\leqslant 1,\, |z_2|=\varepsilon\}.
\end{equation*}
\notag
$$
Let $M_{j_1,j_2,\varepsilon}\,{=}\,\{ (|z_1|,|z_2|)\in \mathbb R^2\mid (z_1,z_2)\in\partial U_{j_1,j_2,\varepsilon}\} \subset \mathbb R^2$ be the set of modules of coordinates of points in $\partial U_{j_1,j_2,\varepsilon}$. For $\varepsilon_1<\varepsilon_2<\varepsilon_0$, define the mapping $\rho_{\varepsilon_1,\varepsilon_2}\colon \partial U_{j_1,j_2,\varepsilon_2} \to \partial U_{j_1,j_2,\varepsilon_1}$,
$$
\begin{equation*}
\rho_{\varepsilon_1,\varepsilon_2}\colon (z_1,z_2)\in \partial U_{j_1,j_2,\varepsilon_2}\mapsto \bigl(\rho_{\varepsilon_1,\varepsilon_2}(z_1), \rho_{\varepsilon_1,\varepsilon_2}(z_2)\bigr)\in \partial U_{j_1,j_2,\varepsilon_1},
\end{equation*}
\notag
$$
as follows: $\operatorname{arg}(\rho_{\varepsilon_1,\varepsilon_2}(z_i))= \operatorname{arg}(z_i)$ for $i=1,2$ and the induced by $\rho_{\varepsilon_1,\varepsilon_2}$ mapping $|\rho_{\varepsilon_1,\varepsilon_2}|\colon M_{j_1,j_2,\varepsilon_2}\to M_{j_1,j_2,\varepsilon_1}$ is the projection of $M_{j_1,j_2,\varepsilon_2}$ from the point $(1,1)\in\mathbb R^2$ to $M_{j_1,j_2,\varepsilon_1}$ (see Figure 3). It is easy to check that $\rho_{\varepsilon_1,\varepsilon_2}$ is given by
$$
\begin{equation}
\begin{alignedat}{3} \rho_{\varepsilon_1,\varepsilon_2}(z_1,z_2) &= \biggl(\frac{(1-\varepsilon_1)(|z_1|-1)+ (1-\varepsilon_2)}{(1-\varepsilon_2)|z_1|}z_1, \, \frac{\varepsilon_1}{\varepsilon_2}z_2\biggr) &\quad &\text{if } |z_2|=\varepsilon_2, \\ \rho_{\varepsilon_1,\varepsilon_2}(z_1,z_2) &= \biggl(\frac{\varepsilon_1}{\varepsilon_2}z_1, \, \frac{(1-\varepsilon_1)(|z_2|-1)+(1-\varepsilon_2)} {(1-\varepsilon_2)|z_2|}z_2\biggr) &\quad &\text{if } |z_1|=\varepsilon_2. \end{alignedat}
\end{equation}
\tag{17}
$$
Obviously, for $C_{j_1}\cap C_{j_2}\neq \varnothing$, the mappings $\rho_{\varepsilon_1,\varepsilon_2}\colon \partial U_{j_1,j_2,\varepsilon_2} \to \partial U_{j_1,j_2,\varepsilon_1}$ are homeomorphisms. To extend the homeomorphisms $\rho_{\varepsilon_1,\varepsilon_2}\colon \partial U_{j_1,j_2,\varepsilon_2} \to \partial U_{j_1,j_2,\varepsilon_1}$ to a homeomorphism $\rho_{\varepsilon_1,\varepsilon_2}\colon \partial\widetilde U_{\varepsilon_2} \to \partial \widetilde U_{\varepsilon_1}$, we let $\bigcup_{i=1}^{\operatorname{v}_j}\partial U_{j,o_j(i),\varepsilon}$ denote the union over all $o_j(i)$ such that $\overline o_j(i)=(o_j(i),0)$. Applying Claim 1, it follows from (17) that, for the points $q=(z_1,z_2)\in \partial U_{j_1,j_2,\varepsilon_2}$ such that $|z_1|=1$ and $|z_2|=\varepsilon_2$ (respectively, $|z_1|=\varepsilon_2$ and $|z_2|=1$), the images $\rho_{\varepsilon_1,\varepsilon_2}(q)$ belong to the geodesic lines $\gamma\subset V_{j_1,j_2}$, transversal to $C_{j_2}$ (respectively, $C_{j_1}$) and passing through the points $q$. Therefore, the homeomorphisms $\rho_{\varepsilon_1,\varepsilon_2}\colon \partial U_{j_1,j_2,\varepsilon_2} \to \partial U_{j_1,j_2,\varepsilon_1}$ can be extended continuously to a homeomorphism $\rho_{\varepsilon_1,\varepsilon_2}\colon \partial\widetilde U_{\varepsilon_2} \to \partial \widetilde U_{\varepsilon_1}$ such that its restrictions
$$
\begin{equation*}
\rho_{j,\varepsilon_1,\varepsilon_2}\colon \partial U_{j,\varepsilon_2} \setminus \bigcup_{i=1}^{\operatorname{v}_j}\partial U_{j,o_j(i),\varepsilon_2} \to \partial U_{j,\varepsilon_1}\setminus \bigcup_{i=1}^{\operatorname{v}_j}\partial U_{j,o_j(i),\varepsilon_1}
\end{equation*}
\notag
$$
to $\partial U_{j,\varepsilon_2}\setminus \bigcup_{i=1}^{\operatorname{v}_j}\partial U_{j,o_j(i),\varepsilon_2}$ are defined as follows: the homeomorphisms $\rho_{j,\varepsilon_1,\varepsilon_2}$ send $q\in \partial U_{j,\varepsilon_2}\setminus \bigcup_{i=1}^{\operatorname{v}_j}\partial U_{j,o_j(i),\varepsilon_2}$ to $\rho_{j,\varepsilon_1,\varepsilon_2}(q)=\gamma_q\cap \bigl(\partial U_{j,\varepsilon_1}\setminus \bigcup_{i=1}^{\operatorname{v}_j}\partial U_{j,o_j(i),\varepsilon_1}\bigr)$, where $\gamma_q$ are the geodesic lines transversal to $C_j$ and passing through the points $q$. As a result, we obtain a homeomorphism $\rho_{\varepsilon_1,\varepsilon_2}\colon \partial U_{\varepsilon_2}\simeq \partial \widetilde U_{\varepsilon_2}\to \partial \widetilde U_{\varepsilon_1} \simeq \partial U_{\varepsilon_1}$. For $q\in \widetilde U_{\varepsilon}$, denote by $d(q)=\operatorname{dist}_{(ds)^2}(q,\widetilde C)$ the distance between $q$ and the curve $\widetilde C$. We have $U_{\varepsilon}\setminus C\simeq \widetilde U_{\varepsilon}\setminus\widetilde C$ and it is easy to see that the mapping
$$
\begin{equation*}
\rho_{\varepsilon}\colon U_{\varepsilon}\setminus C\simeq \widetilde U_{\varepsilon}\setminus \widetilde C \to \partial \widetilde U_{\varepsilon}\times (0,\varepsilon)\simeq \partial U_{\varepsilon}\times (0,\varepsilon),
\end{equation*}
\notag
$$
given by $\rho_{\varepsilon}(q)=(\rho^{-1}_{d(q),\varepsilon}(q),d(q))$ for $q\in \widetilde U_{\varepsilon}\setminus \widetilde C$, is an homeomorphism. Therefore, the groups $\pi_1(U_{\varepsilon}\setminus C)$ and $\pi_1(\partial U_{\varepsilon})$ are isomorphic, and now assertions 1) and 2) of Theorem 1 follow from Theorem 3.
§ 3. Proof of Theorem 43.1. Extended Euclidean algorithmLet $\mathbb Z[u_1,\dots,u_n,\dots]$ be the ring of polynomials in variables $u_1,\dots,u_n,\dots$ with coefficients in $\mathbb Z$, and let $\mathbb Q(u_1,\dots,u_n,\dots)$ be its field of quotients. Consider the rational functions
$$
\begin{equation}
R_n(u_1,\dots, u_n)=u_n-\frac{1}{u_{n-1}-\cfrac{1}{u_{n-2}- \cfrac{1}{\dots-\cfrac{1}{u_1}}}}\in \mathcal R=\mathbb Q(u_1,\dots,u_n,\dots)
\end{equation}
\tag{18}
$$
and $R_0=P_0=1$. By induction, it is easy to check that $R_n(u_1,\dots,u_n)=P_n(u_1,\dots,u_n)/P_{n-1}(u_1,\dots,u_{n-1})$, where the polynomials $P_n(u_1,\dots,u_n)$ are given recursively by
$$
\begin{equation}
\begin{gathered} \, P_0:=1,\qquad P_1:=P_1(u_1)=u_1, \nonumber \\ P_2:=P_2(u_1,u_2)= u_2P_1-P_0=u_1u_2-1, \nonumber \\ P_3:=P_3(u_1,u_2,u_3)=u_3P_2-P_1=u_1u_2u_3-u_1-u_3, \nonumber \\ \dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots \nonumber \\ P_n:=P_n(u_1,\dots,u_n)=u_nP_{n-1}-P_{n-2}. \end{gathered}
\end{equation}
\tag{19}
$$
We set $Q_n:=P_{n-1}(u_2,\dots,u_n)$, $Q_{0}:=0$, and consider the set of $(2\times 2)$-matrices
$$
\begin{equation*}
\mathcal M=\biggl\{ M_m=(-1)^m\begin{pmatrix} -Q_{m-1} & Q_{m} \\ -P_{m-1} & P_m \end{pmatrix} \in \operatorname{Mat}(2,\mathcal R) \biggm| m\in\mathbb N \biggr\}.
\end{equation*}
\notag
$$
We also define
$$
\begin{equation*}
M_1(u_m):=(-1)\begin{pmatrix} 0 & 1 \\ -1 & u_m \end{pmatrix} \in \operatorname{Mat}(2,\mathcal R),\qquad m\in\mathbb N;
\end{equation*}
\notag
$$
in particular, $M_1=M_1(u_1)$. Lemma 4. $\mathcal M=\{ M_m=M_1(u_1)\cdots M_1(u_m)\mid m\in \mathbb N\} \subset \operatorname{SL}(2,\mathcal R)$. Proof. Using induction on $m$ and applying the equality
$$
\begin{equation*}
\begin{aligned} \, M_m M_1(u_{m+1}) &= (-1)^{m+1} \begin{pmatrix} -Q_{m-1} & Q_{m} \\ -P_{m-1} & P_m \end{pmatrix} \begin{pmatrix} 0 & 1 \\ -1 & u_{m+1} \end{pmatrix} \\ &=(-1)^{m+1} \begin{pmatrix} -Q_{m} & -Q_{m-1}+u_{m+1}Q_{m} \\ -P_{m} & -P_{m-1}+u_{m+1}P_m \end{pmatrix} \\ &\!\!\stackrel{(19)}= (-1)^{m+1} \begin{pmatrix} -Q_{m} & Q_{m+1} \\ -P_{m} & P_{m+1} \end{pmatrix}=M_{m+1}, \end{aligned}
\end{equation*}
\notag
$$
we find that $M_m=M_1(u_1)\cdots M_1(u_m)$.
We have $M_m\in \operatorname{SL}(2,\mathcal R)$, since $\det M_1(u_n)=1$. Lemma 4 is proved. Denote by $\overline k^{\,m}=(k_1,\dots,k_m)$, $k_j\in\mathbb Z$, the elements of $\mathbb Z^m$. The matrices $M_m\in \mathcal M$ define the mappings $\mu_m \colon \mathbb Z^m\to \operatorname{SL}(2,\mathbb Z)$ given by
$$
\begin{equation*}
\begin{aligned} \, &\mu_m \colon \overline k^{\,m}=(k_1,\dots,k_m) \\ &\qquad\qquad\mapsto M_{\overline k^{\,m}}=(-1)^m \begin{pmatrix} -Q_{m-1}(k_2,\dots, k_{m-1}) & Q_{m}(k_2,\dots, k_m) \\ -P_{m-1}(k_1,\dots,k_{m-1}) & P_m(k_1,\dots, k_m) \end{pmatrix}. \end{aligned}
\end{equation*}
\notag
$$
Proposition 5. $ \bigcup_{n=1}^{\infty}\mu_{4n}(\mathbb Z^{4n})= \bigcup_{n=1}^{\infty}\mu_n(\mathbb Z^{n})=\operatorname{SL}(2,\mathbb Z). $ Proof. It is well-known that $\operatorname{SL}(2,\mathbb Z)$ is generated by the matrices
$$
\begin{equation*}
T_{1,2}(m)=\begin{pmatrix} 1 & m \\ 0 & 1 \end{pmatrix},\qquad T_{2,1}(m)=\begin{pmatrix} 1 & 0 \\ m & 1 \end{pmatrix},\qquad m\in\mathbb Z,
\end{equation*}
\notag
$$
that is, any matrix $A\in \operatorname{SL}(2,\mathbb Z)$ is a product $A=T_1(m_1)\cdots T_n(m_n)$ for some $n\in \mathbb N$, were each $T_i(m_i)$ is either $T_{1,2}(m_i)$, or $T_{2,1}(m_i)$ and if $T_i(m_i)=T_{1,2}(m_i)$, then $T_{i+1}(m_{i+1})=T_{2,1}(m_{i+1})$ and if $T_i(m_i)=T_{2,1}(m_i)$, then $T_{i+1}(m_{i+1})=T_{1,2}(m_{i+1})$. On the other hand, by Lemma 4, it is easy to check that
$$
\begin{equation*}
\begin{aligned} \, M_{(0,0,0,-k_4)} &=M_1(0)^3M_1(k_4)=T_{1,2}(k_4)\in \mu_4(\mathbb Z^4), \\ M_{(k_1,0,0,0)} &=M_1(k_1)M_1(0)^3=T_{2,1}(k_1)\in \mu_4(\mathbb Z^4). \end{aligned}
\end{equation*}
\notag
$$
Therefore, by Lemma 4, the matrix $A=T_1(m_1)\cdots T_n(m_n)\in \mu_{4n}(\mathbb Z^{4n})$. This proves Proposition 5. 3.2. Proof of assertion 1) of Theorem 4Consider a partially bi-weighted graph $\Gamma_{m,2}\in\mathcal Ch_2^0$ depicted in Figure 4 in which $(k_j,0)$ are bi-weights of vertices $v_{j,0}$. By Theorem 2, the fundamental group $\pi_1^w(\Gamma_{m,2})$ has the presentation
$$
\begin{equation}
\begin{aligned} \, \pi_1^w(\Gamma_{m,2}) &=\langle x_{1,0},\dots,x_{m,0},x_{m+1,1},x_{m+2,1} \mid x_{1,0}^{k_1}x_{m+1,1}x_{2,0}=x_{2,0}^{k_2}x_{1,0}x_{3,0} \nonumber \\ &\qquad\qquad =\dots=x_{m-1,0}^{k_{m-1}}x_{m-2,0}x_{m,0}= x_{m,0}^{k_m}x_{m-1,0}x_{m+2,1}=1, \nonumber \\ &\qquad\qquad [x_{1,0},x_{m+1,1}]=[x_{1,0},x_{2,0}]=[x_{2,0},x_{3,0}] \nonumber \\ &\qquad\qquad=\dots=[x_{m-1,0},x_{m,0}]=[x_{m,0},x_{m+2,1}]=1\rangle. \end{aligned}
\end{equation}
\tag{20}
$$
By (20),
$$
\begin{equation*}
x_{2,0}=x_{m+1,1}^{-1}x_{1,0}^{-k_1}\quad\text{and}\quad [x_{2,0},x_{m+1,1}]=1,
\end{equation*}
\notag
$$
since $[x_{1,0},x_{m+1,1}]=1$. Next,
$$
\begin{equation*}
x_{3,0}=x_{1,0}^{-1}x_{2,0}^{-k_2}=x_{m+1,1}^{k_2}x_{1,0}^{k_1,k_2-1}\quad \text{and}\quad [x_{1,0},x_{3,0}]=[x_{m+1,1},x_{3,0}]=1,
\end{equation*}
\notag
$$
etc., and so
$$
\begin{equation}
x_{j,0}=x_{m+1,1}^{(-1)^{j-1}Q_{j-1}}x_{1,0}^{(-1)^{j-1}P_{j-1}},\qquad j=2,\dots,m,
\end{equation}
\tag{21}
$$
$$
\begin{equation}
x_{m+2,1}=x_{m+1,1}^{(-1)^{m}Q_{m}}x_{1,0}^{(-1)^{m}P_m},
\end{equation}
\tag{22}
$$
where $P_j=P_j(k_1,\dots,k_j)$ and $Q_{j}=P_{j-1}(k_2,\dots,k_j)$. It follows from (21) and (22) that $\pi_1^w(\Gamma_{m,2})= \langle x_{1,0},x_{m+1,1}\mid [x_{1,0},x_{m+1,1}]=1\rangle$ is a free commutative group of rank $2$, and $\Pi_w(\mathcal Ch^0_2)=\{ \mathbb Z\times\mathbb Z\}$. Let $\Gamma_{m,1}\in\mathcal Ch_1^0$ be a partially bi-weighted graph depicted in Figure 5. To obtain a presentation of the group $\pi_1^w(\Gamma_{m,1})$, it suffices to add the relation $x_{m+2,1}=1$ to presentation (20). As a result,
$$
\begin{equation}
\pi_1^w(\Gamma_{m,1})= \bigl\langle x_{1,0},x_{m+1,1}\bigm| [x_{1,0},x_{m+1,1}]= x_{m+1,1}^{Q_{m}}x_{1,0}^{P_m}=1\bigr\rangle.
\end{equation}
\tag{23}
$$
Therefore, $\pi_1^w(\Gamma_{m,1})=\mathbb Z$, since $Q_m$ and $P_m$ are coprime by Lemma 4. Let $\Gamma_{m,0}\in\mathcal Ch_0^0$ be a partially bi-weighted graph depicted in Figure 6. To obtain a presentation of the group $\pi_1^w(\Gamma_{m,}0)$, it suffices to add the relation $x_{m+1,1}=x_{m+2,1}=1$ to presentation (20). Hence
$$
\begin{equation*}
\pi_1^w(\Gamma_{m,0})=\bigl\langle x_{1,0}\bigm| x_{1,0}^{P_m}= 1\bigr\rangle \simeq \mathbb Z/|P_m|\mathbb Z
\end{equation*}
\notag
$$
is a cyclic group. If $m=1$, then $P_1=k_1$, and hence, $\Pi_w(\mathcal Ch^0_0)=\{ \mathbb Z/n\mathbb Z\mid n\geqslant 0\}$. 3.3. Proof of assertion 2) of Theorem 4Let $\Gamma_{m}\in\mathcal L$ be a partially bi-weighted loop whose vertex set $V(\Gamma_m)$ is $\{v_{1,0},\dots,v_{m,0}\}$ ($m\geqslant 3$ by condition (G4)). To give a presentation of the group $\pi_1^w(\Gamma_m)$ we need to choose a set of edges $\widetilde E$ and consider a tree $\Gamma_{m,\widetilde E}$ of $\Gamma_m$. We set $\widetilde E=\{ (v_{1,0},v_{m,0})\}$. The vertex set of the tree $\Gamma_{m,\widetilde E}$ is $\{ v_{1,0},\dots,v_{m,0}\}\cup \{v_{1,1},v_{m,1}\}$, and $\Gamma_{m,\widetilde E}$ coincides with the graph $\Gamma_{m,2}$, depicted in Figure 4, where $v_{1,1}=v_{m+2,1}$ and $v_{m,1}=v_{m+1,1}$. Therefore, to obtain a presentation of $\pi_1^w(\Gamma_m)$ it suffices to add an additional generator $y$ to the set of generators in presentation (20) and add two defining relations $y^{-1}x_{1,0}y=x_{m+2,1}$ and $y^{-1}x_{m+1,1}y=x_{m,0}$ to the set of defining relations in presentation (20). It follows from (21) and (22) that $\pi_1^w(\Gamma_m)$ has the presentation
$$
\begin{equation*}
\begin{aligned} \, \pi_1^w(\Gamma_m) &= \bigl\langle x_{1,0}, x_{m+1,1}, y\bigm| [x_{1,0},x_{m+1,1}]=1, \\ &\qquad\qquad y^{-1}x_{1,0}y =x_{1,0}^{(-1)^{m}P_m}x_{m+1,1}^{(-1)^{m}Q_{m}}, \\ &\qquad\qquad y^{-1}x_{m+1,1}y= x_{1,0}^{(-1)^{m-1}P_{m-1}}x_{m+1,1}^{(-1)^{m-1}Q_{m-1}}\bigr\rangle \end{aligned}
\end{equation*}
\notag
$$
or, in additive form,
$$
\begin{equation*}
\pi_1^w(\Gamma_m)\simeq\mathbb Z^2\ltimes_{M^t_{\overline k^{\,m}}}\mathbb F_1 =\bigl\langle (z_1,z_2),t\bigm| (z_1,z_2)\in \mathbb Z^2,\, t^{-1}(z_1,z_2)t=(z_1,z_2)M^{\tau}_{\overline k^{\,m}}\bigr\rangle
\end{equation*}
\notag
$$
is a semidirect product of $\mathbb Z^2$ and $\mathbb F_1\simeq \mathbb Z$, where (by Proposition 5)
$$
\begin{equation*}
M^{\tau}_{\overline k^{\,m}}= (-1)^m \begin{pmatrix} P_m & Q_{m} \\ -P_{m-1} & -Q_{m-1} \end{pmatrix}
\end{equation*}
\notag
$$
is an arbitrary element of the group $\operatorname{SL}(2,\mathbb Z)$ acting on $\mathbb Z^2$ from the right. 3.4. Proof of assertion 3) of Theorem 4Consider the presentation in Definition 1 of the fundamental group $\pi_1^w(\Gamma)$ of $\Gamma\in\mathcal G$. Let $N$ be the normal closure in $\pi_1^w(\Gamma)$ of the subgroup generated by
$$
\begin{equation*}
\! x_{1,0},\ \dots,\ x_{m,0},\ x_{m+1,1},\ \dots,\ x_{m+k,1},\ x_{j(E_1,1),1},\ x_{j(E_1,2),1},\ \dots,\ x_{j(E_r,1),1},\ x_{j(E_r,2),1}.\!
\end{equation*}
\notag
$$
Then the quotient $\pi_1^w(\Gamma)/N$ has the presentation
$$
\begin{equation*}
\begin{aligned} \, \pi_1^w(\Gamma)/N &=\biggl\langle y_1,\dots, y_r, \mu_{j,i},\lambda_{j,i},\ 1\leqslant j\leqslant m,\, 1\leqslant i\leqslant g_j\biggm| \\ &\qquad\prod_{i=1}^{g_{j}}[\mu_{j,i},\lambda_{j,i}]=1,\ 1\leqslant j \leqslant m,\, 1\leqslant g_j \biggr\rangle, \end{aligned}
\end{equation*}
\notag
$$
that is, $\pi_1^w(\Gamma)/N \simeq \bigl(\prod^{n_1}\mathcal R_1*\cdots* \prod^{n_{i_0}}\mathcal R_{i_0}\bigr)*\mathbb F_r$. Acknowledgement. The author is grateful to the referee for valuable comments.
Citation in format AMSBIB
\Bibitem{Kul23} |
|
|||||||||||||||||||||||||||||||||||||||||||||||||
|
Contact us: |
Terms of Use
|
Registration to the website |
Logotypes |
|