Fundamental group of non-singular locus of Lauricella's $F_C$

TL;DR

This paper determines the fundamental group of the non-singular locus of Lauricella's hypergeometric functions by providing explicit generators and relations, advancing understanding of its topological structure.

Contribution

It introduces a specific set of generators and relations for the fundamental group, offering a detailed algebraic description of the non-singular locus.

Findings

Explicit generators for the fundamental group

Relations among generators established

Enhanced understanding of the topology of Lauricella's functions

Abstract

In this paper, we give a set of generators and relations of the fundamental group of the non-singular locus of Lauricella's hypergeometric functions

Equations

$$F_C^{(n)}(a,b;c_1,\dots ,c_n;z_1,\dots ,z_n)$$

$$=$$

$$(\text{const.})\int \prod _{k=1}^n{t}_k^{-c_k}\cdot (1-\sum _{k=1}^n{t}_k{)}^{\sum c_k-a-n}\cdot (1-\sum _{k=1}^n\frac{z_k}{t_k}{)}^{-b}d{t}_1\cdots d{t}_n.$$

$$\{((t_i{)}_i,\lambda )\in ({\mathbb{C}}^{\times }{)}^{n+1}\mid \lambda (1-\sum _{k=1}^n{t}_k)+(1-\sum _{k=1}^n\frac{z_i}{t_k})=0\}$$

$$\{\lambda (1+\sum _{i\in I}{t}_i)+\sum _{j\in J}\frac{a_j}{t_j}=0\},$$

$$\{\lambda (\sum _{i\in I}{t}_i)+1+\sum _{j\in J}\frac{a_j}{t_j}=0\}$$

$$\{\begin{array}{l}1-{\sum }_{k=1}^n{t}_k=0,\\ \lambda -\frac{z_i}{t_i^2}=0,\\ \lambda (1-{\sum }_{k=1}^n{t}_k)+(1-{\sum }_{k=1}^n\frac{z_i}{t_k})=0.\end{array}$$

$$t_i\mu ={ϵ}_i{x}_i,\mu -\sum _{i=1}^n{ϵ}_i{x}_i=\mu (1-\sum _{i=1}^n{t}_i)=0.$$

$$0$$

$$=(1+\sum _{i=1}^n{ϵ}_i{x}_i)(1-\sum _{i=1}^n{ϵ}_i{x}_i).$$

$${\mathbb{C}}^n=\{(x_1,\dots ,x_n)\}\ni (x_i{)}_i\mapsto (x_i^2{)}_i=(z_i{)}_i\in {\mathbb{C}}^n=\{(z_1,\dots ,z_n)\}.$$

$$Y_n=\{(x_i{)}_i\mid \prod _{k=1}^n{x}_k\prod _{{ϵ}_i\in \{-1,1\}}(1-\sum _{i=1}^n{ϵ}_i{x}_i)\ne 0\}.$$

$$[{\Gamma }_i,{\Gamma }_j]=1,(1\le i,j\le n),({\Gamma }_0{\Gamma }_i{)}^2=({\Gamma }_i{\Gamma }_0{)}^2,(1\le i\le n),$$

$$[M(I{)}^{-1}{\Gamma }_{0}M(I),M(J{)}^{-1}{\Gamma }_{0}M(J)]=1$$

$$Y=Y(\mathcal{H})={\mathbb{C}}^n-{\cup }_{i\in I}{H}_i.$$

$$\overline{{\sigma }_1}\supset \overline{{\sigma }_2}\supset \overline{\sigma }.$$

$${\pi }_1(X_2)\to {\pi }_1(Y).$$

$$H_{ϵ}:{ϵ}_1{x}_1+\cdots +{ϵ}_n{x}_n=1.$$

$$L_i:x_i=0,(i=1,\dots ,n).$$

$$H_{ϵ,{ϵ}^{\prime }}=H_{ϵ}\cap H_{{ϵ}^{\prime }}.$$

$$H_{ϵ,i}=H_{ϵ}\cap L_i.$$

$$g^{(i)}({ϵ}_1,\dots ,{ϵ}_n)=({ϵ}_1,\dots ,\stackrel{i}{-{ϵ}_i},\dots ,{ϵ}_n).$$

$$H_{i,j}=L_i\cap L_j.$$

$$g:{\mathbb{C}}^n\to {\mathbb{C}}^n:(x_1,\dots ,x_n)\mapsto (g_1{x}_1,\dots ,g_n{x}_n)$$

$${\mathbb{R}}_{>0}=\{x\in \mathbb{R}\mid x>0\},{\mathbb{R}}_{\ge 0}=\{x\in \mathbb{R}\mid x\ge 0\}.$$

$${\pi }_1(\overline{X_2})\to {\pi }_1({\overline{X_2}}^{(s)},s)$$

$$a=c,b=d,ab=ba.$$

$$\partial {\tilde{D}}_{{\sigma }_1,\tau }:a=d,$$

$$\partial {\tilde{D}}_{{\sigma }_2,\tau }:ba=dc,$$

$$\partial {\tilde{D}}_{{\sigma }_3,\tau }:cba=dcb.$$

$$d=a,c=a^{-1}ba,(ab{)}^2=(ba{)}^2.$$

$${\sigma }_0=\{(x_i)\in {\mathbb{R}}^n\mid x_i>0(0\le i\le n),\sum x_i<1\},$$

$${\tau }_0=\{(x_i)\in {\mathbb{R}}^n\mid x_i>0(0\le i\le n),\sum x_i=1\},$$

Full text

1. 1 Introduction and motivation

2. 2 Recall of a result of Salvetti

3. 3 $F_{C}$-hyperplane arrangement

4. 3.1 The arrangement $\mathcal{H}_{n}$

5. 3.2 Group action

6. 3.3 Cell complex for the quotient space

7. 3.4 Relations for type 1 and type 2

8. 3.4.1 Type 1 relation

9. 3.4.2 Type 2 relation

10. 3.5 Definition of $\gamma_{i}$ and their relations

11. 4 Fundamental relation

12. 4.1 Main theorem

13. 4.1.1 The definition of $\varphi$

14. 4.1.2 The definition of $\psi$

15. 4.2 Simplification

1. Introduction and motivation

The Lauricella hypergeometric function $F_{C}^{(n)}$ of $n$ variable defined by

[TABLE]

and has the following integral expression ([G]):

[TABLE]

Using Caylay technique [GKZ], the function $F_{C}^{(n)}$ is locally holomorphic on$(z_{i})\in(\mathbb{C}^{\times})^{n}$ if the toric hypersurface

[TABLE]

is non-degenerate for Newton polyhadra. For non-degneracy condition, see [T].

Since the non-degeneracy condition for a proper Newton polyhedra is equal to the smoothness of the varieties

[TABLE]

for $I,J\subset\{1,\dots,n\}$ and $I\cap J=\emptyset$. Therefore the non-degeneracy condition is equivalent to the smoothness to the open face. Using Jacobian criterion, the singular locus is defined by

[TABLE]

By setting $\mu^{2}=\lambda,x_{i}^{2}=z_{i}$, and using the first and the second equations, $\mu$ is obtained by

[TABLE]

Here $\epsilon_{i}\in\{-1,1\}$. Again, using the first and the second equations, the third equation is equal to

[TABLE]

Therefore under the $\mu_{2}^{n}$-covering map,

[TABLE]

the pull back $Y_{n}$ of $\overline{Y_{n}}$ is given by

[TABLE]

See also [HT]. Therefore $\overline{Y_{n}}\subset\{(z_{1},\dots,z_{n})\}$ is isomorphic to $Y_{n}/\mu_{2}^{n}$.

In the study of monodromy of hypergeometric function of type $F_{C}$, it is a basic problem to give an expression of the fundamental group of $\overline{Y_{n}}$. The generator and relations of the fundamental group for $n=2$ and $n=3$ is determined in [GK]. We prove the following presentation of the fundamental group which is conjectured in [GK].

Theorem 1.1** **(Main Theorem, see Theorem 4.1 and Proposition

4.2).

The fundamental group of $\overline{Y_{n}}$ is generated by elements $\Gamma_{0},\Gamma_{1},\dots,\Gamma_{n}$ with the relations

[TABLE]

and

[TABLE]

for all subsets $I$ and $J$ of $\{1,\dots,n\}$ satisfying $I\cap J=\emptyset,I\neq\emptyset,J\neq\emptyset$ and $\#I+\#J\leq n-1$. Here we set $M(I)=\prod_{i\in I}\Gamma_{i}$.

For the proof of this theorem, we use a cell complex constructed by Salvetti [S], which is homotopic to the complement of a hyperplane arrangement in $\mathbb{C}^{N}$ and stable under a group action. The author is grateful for discussions with Y. Goto and K. Matsumoto in “Workshop on Special Varieties in Tambara, 2017”, in Tambara International Seminar House.

2. Recall of a result of Salvetti

We recall a construction of $2$-skeleton of a cell complex which is homotopic to the complement of real hyperplane arrangement. A finite set $\mathcal{H}=\{H_{i}\}_{i\in I}$ of complex hyperplanes in $\mathbb{C}^{n}$ is called a hyperplane arrangement. In this paper, we are interested in the topological space

[TABLE]

A hyperplane arrangement is called a real hyperplane arrangement if the defining equations of $H_{i}$ is defined over $\mathbb{R}$ for all $i\in I$. For a real hyperplane arrangement $\mathcal{H}$, we set $H_{i,\mathbb{R}}=H_{i}\cap\mathbb{R}^{n}$. The set $\{H_{i,\mathbb{R}}\}_{i\in I}$ is denoted by $\mathcal{H}_{R}$. A subset of $\mathbb{R}^{n}$ which can be obtained by the intersection of finite number of $H_{i,\mathbb{R}}$’s is simply called a linear subset of $\mathcal{H}_{\mathbb{R}}$. As a special case, the total space $\mathbb{R}^{n}$ is an $n$-dimensional linear subset. Let $L$ be an $i$-dimensional linear subset of $\mathcal{H}_{\mathbb{R}}$. A connected component of the complement of the union of proper linear subsets of $L$ in $L$ is called an $i$-chamber of $\mathcal{H}_{\mathbb{R}}$ and the set of $i$-chamber is denoted by $\operatorname{Ch}_{i}=\operatorname{Ch}_{i}(\mathcal{H}_{\mathbb{R}})$. Each $i$-chamber is a convex set.

We define the dual cell complex of $\mathcal{H}_{\mathbb{R}}$ as follows. For each $i$-dimensional chamber $\sigma$, we choose a vertex $v_{\sigma}$ in the interior of $\sigma$. The set of [math]-cell of the dual cell complex is given by $D_{\sigma}=v_{\sigma}$, where $\sigma$ is an $n$-chamber.

Let $\tau$ be an $(n-1)$-chamber. Then there exist exactly two $n$-chambers $\tau_{1}$ and $\tau_{2}$ such that $\overline{\tau_{i}}\supset\overline{\tau}$ for $i=1,2$. Here $\overline{\tau}$ is the closure of $\tau$ in $\mathbb{R}^{n}$. We consider 1-cell $D_{\tau}$ by considering the union of segments $\Delta(v_{\tau_{1}},v_{\tau})$ and $\Delta(v_{\tau_{2}},v_{\tau})$. We continue this procedure to define $2$-cell $D_{\sigma}$ attached to $(n-2)$-dimensional chamber as follows. If $\sigma_{1},\sigma_{2}$ and $\sigma$ are $n,(n-1)$ and $(n-2)$-chambers, such that

[TABLE]

A sequence $F=F(\sigma_{1},\sigma_{2},\sigma)$ as above is called a (descending) flag of length $3$. The triangle $\Delta(v_{\sigma_{1}},v_{\sigma_{2}},v_{\sigma})$ is called the dual flag $F^{*}$ of $F$. The union of dual flags containing $v_{\sigma}$ is called the $2$-dimensional dual cell $D_{\sigma}$ of $\sigma$.

We recall the construction of the $2$-skeleton $X_{2}$ of the cell complex $X$ after Salvetti [S], which is homotopy equivalent to the space $Y=Y(\mathcal{H})$. The set $C_{0}(X)$ of [math]-cell in $X$ is the set $\{\widetilde{D_{\sigma}}\}_{\sigma\in\operatorname{Ch}_{n}}$ of the copy $\widetilde{D_{\sigma}}$ of $D_{\sigma}$.

The set $C_{1}(X)$ of $1$-cell consists of $\widetilde{D}_{\sigma,\tau}$ for $\sigma\in\operatorname{Ch}_{n},\tau\in\operatorname{Ch}_{n-1}$ such that $\overline{\sigma}\supset\tau$. The $n$-chamber lying on the opposite side of $\sigma$ with respect to the $(n-1)$-chamber $\tau$ is denoted by $\rho_{\tau}(\sigma)$. The attaching map $\partial\widetilde{D}_{\sigma,\tau}\to X_{0}$ is given by connecting two points $\sigma$ and $\rho_{\tau}(\sigma)$. The $1$-cell $\widetilde{D}_{\sigma,\tau}$ is called an arrow from $\sigma$ to $\rho_{\tau}(\sigma)$. The composite of several arrows compatible with the directions is called an oriented path.

The set $C_{2}(X)$ of $2$-cell consists of $\widetilde{D}_{\sigma,\tau}$ for $\sigma\in\operatorname{Ch}_{n},\tau\in\operatorname{Ch}_{n-2}$ such that $\overline{\sigma}\supset\overline{\tau}$. The $n$-chamber lying on the opposite side of $\sigma$ with respect to the $(n-2)$-chamber $\tau$ is denoted by $\rho_{\tau}(\sigma)$ and the vertex in $\rho_{\tau}(\sigma)$ is denoted by $\rho_{\tau}(v_{\sigma})$ (see Figure 2).

Then there exist exactly two shortest paths from $v_{\sigma}$ to $\rho_{\tau}(v_{\sigma})$. The attaching map $\partial\widetilde{D}_{\sigma,\tau}\to X_{1}$ is given by bounding the two shortest paths from $v_{\sigma}$ to $\rho_{\tau}(v_{\sigma})$ (see Figure 2).

Proposition 2.1** (Salvetti).**

The natural inclusion $X_{2}\to Y$ induces an isomorphism of fundamental groups

[TABLE]

As a consequence, the fundamental groupoid is generated by $\widetilde{D}_{\tau_{1},\tau_{2}}$ for $\tau_{1}\in\operatorname{Ch}_{n},\tau_{2}\in\operatorname{Ch}_{n-1},\overline{\tau_{1}}\supset\overline{\tau_{2}}$, and the relation is given by $\widetilde{D}_{\sigma_{1},\sigma_{2}}$ for $\sigma_{1}\in\operatorname{Ch}_{n},\sigma_{2}\in\operatorname{Ch}_{n-2},\overline{\sigma_{1}}\supset\overline{\sigma_{2}}$.

3. $F_{C}$-hyperplane arrangement

3.1. The arrangement $\mathcal{H}_{n}$

For an element $\epsilon=(\epsilon_{1},\dots,\epsilon_{n})\in\{-1,1\}^{n}$, we define a hyperplane $H_{\epsilon}$ by

[TABLE]

We define $n$-dimensional $F_{C}$-arrangement $\mathcal{H}_{n}$ by the union of the set of hyperplanes $\{H_{\epsilon}\}$ $(\epsilon\in\{-1,1\}^{n})$ and that of coordinate hyperplanes

[TABLE]

The following proposition is used to classify $(n-2)$-chambers in $\mathcal{H}_{n}$.

Proposition 3.1**.**

1. (1)

Let $\epsilon,\epsilon^{\prime}$ be elements in $\{-1,1\}^{n}$ such that $\#\{i\mid\epsilon_{i}\neq\epsilon^{\prime}_{i}\}\geq 2$ and set

[TABLE]

A hyperplane in $\mathcal{H}$ containing $H_{\epsilon,\epsilon^{\prime}}$ is equal to $H_{\epsilon}$ or $H_{\epsilon^{\prime}}$. 2. (2)

For an element $\epsilon$ in $\{-1,1\}^{n}$ and an integer $i$ with $1\leq i\leq n$, we set

[TABLE]

A hyperplane in $\mathcal{H}$ containing $H_{\epsilon,i}$ is equal to $L_{i}$, $H_{\epsilon}$ or $H_{g^{(i)}(\epsilon)}$. Here

[TABLE] 3. (3)

Let $i,j$ be distinct integers such that $1\leq i,j\leq n$ and set

[TABLE]

A hyperplane in $\mathcal{H}$ containing $H_{i,j}$ is equal to $L_{i}$ or $L_{j}$.

3.2. Group action

On the space $Y$, the group $\mu_{2}^{n}=\{1,-1\}^{n}$ acts by

[TABLE]

for $g=(g_{1},\dots,g_{n})\in\mu_{2}^{n}$. The group $\mu_{2}^{n}$ acts on the sets $\operatorname{Ch}_{i}$. We can choose the set of vertex $\{v_{\sigma}\}_{\sigma\in\operatorname{Ch}_{i}}$ so that they are stable under the action of $\mu_{2}^{n}$.

Lemma 3.2**.**

On the topological space $X_{2}$, the action of the group $\mu_{2}^{n}$ on $X_{2}$ is cell-wise and fixed point free.

Proof.

The group acts on $\operatorname{Ch}_{n}$ freely. Therefore it acts freely on the set of $0,1$ and $2$-cells. ∎

3.3. Cell complex for the quotient space

We consider topological space $\overline{X_{2}}=X_{2}/\mu_{2}^{n}$. Then $\overline{X_{2}}$ is a cell complex. We have the following proposition.

Proposition 3.3**.**

The natural map $\pi_{1}(\overline{X_{2}})\to\pi_{1}(Y/\mu_{2}^{n})$ is an isomorphism.

We describe the cell complex $\overline{X_{2}}$ in this subsection. We set

[TABLE]

The subset of $i$-chambers in $\operatorname{Ch}_{i}$ contained in $\mathbb{R}_{\geq 0}$ is denoted by $\overline{\operatorname{Ch}}_{i}$.

The set $C_{0}(\overline{X})$ of [math]-cells in $\overline{X}$ is identified with $\{\widetilde{D}_{\sigma}\mid\sigma\in\overline{\operatorname{Ch}}_{n}\}$. There are the following two kinds of $1$-cells in $\overline{X}$: The image of $\widetilde{D}_{\sigma,\tau}$, $(\sigma\in\overline{\operatorname{Ch}}_{n},\tau\in\overline{\operatorname{Ch}}_{n-1})$ such that

1. (1)

(type 1, non-closed one cell) $\tau\subset H_{\epsilon}$, $(\epsilon\in\{-1,1\}^{n})$. 2. (2)

(type 2, closed one cell) $\tau\subset L_{i}$, $(1\leq i\leq n)$.

There are three kinds of $2$-cells in $\overline{X_{2}}$: The image of $\widetilde{D}_{\sigma,\tau}$, $(\sigma\in\overline{\operatorname{Ch}}_{n},\tau\in\overline{\operatorname{Ch}}_{n-2})$ such that

1. (1)

(type 1, interior disc) $\tau\subset H_{\epsilon}\cap H_{\epsilon^{\prime}}$, 2. (2)

(type 2, boundary disc) $\tau\subset H_{\epsilon}\cap L_{i}$, 3. (3)

(type 3, coordinate disc) $\tau\subset L_{i}\cap L_{j}$.

Definition 3.4**.**

*Let $\sigma$ be an element in $\overline{\operatorname{Ch}}_{n}$. we define height $h(v_{\sigma})$ of $v_{\sigma}=\widetilde{D}_{\sigma}$ by the number of hyperplanes of the form $H_{\epsilon}$ $(\epsilon\in\{-1,1\}^{n})$ separating $\mathbb{0}$ and $v_{\sigma}$. The number $h(v_{\sigma})$ is also denoted by $h(\sigma)$. *

Proposition 3.5**.**

1. (1)

A interior disc (type 1) is attached to four $1$-cells and contains four [math]-cells. The shape of height is as follows. 2. (2)

A boundary disc (type 2) is attached to six $1$-cells and contains three [math]-cells. 3. (3)

A coordinate disc (type 3) is attached to two $1$-cells and contains one [math]-cells.

We define spanning complex which is a slight generalization of spanning tree. A $1$-cell $\widetilde{D}_{\sigma,\tau}$ is called a spanning $1$-cell if it is type 1 and $h(\sigma)+1=h(\rho_{\tau}(\sigma))$, i.e. $\rho_{\tau}(\sigma)$ is farer from the origin than $\sigma$. A $2$-cell $\widetilde{D}_{\sigma,\tau}$ is called a spanning $2$-cell if it is type 1 and $h(\sigma)$ is the smallest among vertices contained in $D_{\tau}$. The union of spanning $1$ and $2$-cells forms a sub cell complex $\mathcal{S}$ of $\overline{X_{2}}$. The complex $\mathcal{S}$ is called the spanning complex of $\overline{X_{2}}$.

Lemma 3.6**.**

The spanning complex $\mathcal{S}$ is simply connected.

Proof.

It is identified with a $2$-skeleton of the dual cell complex of $\mathbb{R}^{n}_{>0}$ which is simply connected. ∎

We define $\overline{X_{2}}^{(s)}$ by obtaining contracting a subset $\mathcal{S}\subset\overline{X_{2}}$ to a point $s$. By the above proposition, we have

Proposition 3.7**.**

The natural map

[TABLE]

is an isomorphism.

Definition 3.8**.**

A $1$-cell $\widetilde{D}_{\sigma,\tau}$ in $\overline{X_{2}}$ is called a generator if it is

1. (1)

type 1 and not spanning, or 2. (2)

type 2.

A generator defines a closed path in $\overline{X_{2}}^{(s)}$. Then the set of generator generates the group $\pi_{1}(\overline{X_{2}}^{(s)})$.

3.4. Relations for type 1 and type 2

3.4.1. Type 1 relation

First, we consider a type 1 $2$-cells $\widetilde{D}_{\sigma,\tau}$ in $\overline{X}$ with $\tau\subset H_{\epsilon}\cap H_{\epsilon^{\prime}}$.

The arrows $\overline{a},\overline{b},\overline{c}$ and $\overline{d}$ are spanning $1$-cells and $a,b,c$ and $d$ define elements in $\pi_{1}(\overline{X_{2}}^{(s)},s)$. Their relations are given as

[TABLE]

3.4.2. Type 2 relation

Next we consider a type 2 $2$-cells $\widetilde{D}_{\sigma,\tau}$ in $\overline{X}$ as in Figure 4.

The arrows $\overline{b}$ and $\overline{c}$ are spanning $1$-cells and reduced to one point in $\overline{X_{2}}^{(s)}$. We consider a $(n-2)$-chamber $\tau$ contained in $L_{i}$. The relations beginning from $v_{1}=v_{\sigma_{1}},v_{2}=v_{\sigma_{2}}$ and $v_{3}=v_{\sigma_{3}}$ are the following:

[TABLE]

We can easily check the following proposition.

Proposition 3.9**.**

The above relations are equivalent to

[TABLE]

We consider the above situation and set $\epsilon=(\epsilon_{1},\dots,\epsilon_{n})$ and $\epsilon^{\prime}=(\epsilon^{\prime}_{1},\dots,\epsilon^{\prime}_{n})$. Here $\epsilon^{\prime}=g^{(i)}(\epsilon)$ where $g^{(i)}$ is defined as (3.1). By the definition of height, $v_{1}$ is the closest vertex from the origin. Therefore we have $\epsilon_{j}=\epsilon^{\prime}_{j}$ if $j\neq i$ and $\epsilon_{i}=1$ and $\epsilon^{\prime}_{i}=-1$.

3.5. Definition of $\gamma_{i}$ and their relations

In this subsection, we define $\gamma_{i}$ and study their relations.

Definition 3.10**.**

We define $\gamma_{i}=\widetilde{D}_{\sigma_{0},\tau_{i}}$ for $i=0,1,\dots,n$ where

[TABLE]

Actually $\sigma_{0}$ is a chamber since if $x=(x_{1},\dots,x_{n})\in\sigma_{0}$ and $\epsilon\neq(1,\dots,1)$ then

[TABLE]

and the point $x$ is not contained in $H_{\epsilon,\mathbb{R}}$.

By previous subsection, we have

[TABLE]

Proposition 3.11**.**

1. (1)

Under the notation of figure 1, we have $a=\gamma_{i}$ in $\pi_{1}(\overline{X_{2}}^{(s)})$. 2. (2)

Let $\tau_{1}$ and $\tau_{2}$ be two elements in $\overline{\operatorname{Ch}}_{n-1}$ contained a common hyperplane $H_{\epsilon,\mathbb{R}}$. Suppose that $1$-cells $\widetilde{D}_{\sigma_{1},\tau_{1}}$ and $\widetilde{D}_{\sigma_{2},\tau_{2}}$ are generators. Then the paths obtained by them are homotopic to each other. These paths defines a common element in $\pi_{1}(\overline{X_{2}}^{(s)},s)$ which is denoted by $\gamma_{\epsilon}$. 3. (3)

For $\epsilon\in\{-1,1\}^{n},\epsilon\neq(-1,\dots,-1)$, we set

[TABLE]

Then we have

[TABLE]

Proof.

(1) We use the first relation (3.2) iteratively and have the statement. (2) This follows from the relations obtained by a type 1 $2$-cells. (3) Let $\epsilon$ be an element in $\{-1,1\}^{n}$ and $\epsilon\neq(-1,\dots,-1)$. We set $S(\epsilon)=\{i_{1},\dots,i_{k}\}$. We consider a chain $\epsilon^{(0)},\dots,\epsilon^{(k)}\in\{-1,1\}^{n}$ defined by

[TABLE]

Then $\epsilon^{(k)}=\epsilon$.

Lemma 3.12**.**

$H_{\epsilon^{(j)}}\cap\mathbb{R}^{n}_{>0}\neq\emptyset$, $(j=0,\dots,k)$ and

[TABLE]

Proof of Lemma 3.12.

By descending induction, it is enough to prove the lemma for $j=k$. We set $\epsilon^{(k)}=(\epsilon_{1},\dots,\epsilon_{n})$. First statement holds since $\epsilon_{j}=1$ for some $j$. We prove the second statement. Since $\epsilon_{i_{k}}=-1$, there exsists $j\neq i_{k}$ such that $\epsilon_{j}=1$. Therefore the equation

[TABLE]

has a solution satisfying $x_{l}>0$ for $l\neq i_{j}$. ∎

By applying the second relation in (3.2) iteratively, we have statement (3). ∎

Using Proposition 3.11 (3) and the relation of type 1, we have the following theorem.

Theorem 3.13**.**

We have

[TABLE]

for $\epsilon,\epsilon^{\prime}\in\{-1,1\}^{n}$ and $H_{\epsilon,\mathbb{R}}\cap H_{\epsilon^{\prime},\mathbb{R}}\cap R_{>0}^{n}\neq\emptyset.$

4. Fundamental relation

4.1. Main theorem

In this section, we prove the following theorem.

Theorem 4.1**.**

The relations (3.3) and (3.5) are fundamental relations for $\pi_{1}(\overline{X_{2}}^{(s)},s)$ with generators $\gamma_{0}$ and $\gamma_{i}$ $(1\leq i\leq n)$.

We define $G$ as a group generated by $\Gamma_{0}$ and $\Gamma_{i}$ $(1\leq i\leq n)$ with the relations

[TABLE]

and

[TABLE]

for $H_{\epsilon,\mathbb{R}}\cap H_{\epsilon^{\prime},\mathbb{R}}\cap R_{>0}^{n}\neq\emptyset.$ Here we set $M_{\epsilon}=\prod_{i\in S(\epsilon)}\Gamma_{i}$. We define group homomorphisms

[TABLE]

which are inverse to each other.

4.1.1. The definition of $\varphi$

We define $\varphi$ by $\varphi(\Gamma_{i})=\gamma_{i}$ for $i=0,1,\dots,n$. We check that fundamental relations of $G$ are satisfied in $\pi_{1}(\overline{X_{2}}^{(s)},s)$. The relation (4.1) is satisfied by the definition of $\varphi$. By the definition of $\varphi$, we have

[TABLE]

Thus the relation (4.2) is satisfied by Theorem 3.13.

4.1.2. The definition of $\psi$

The group $\pi_{1}(\overline{X_{2}}^{(s)},s)$ is generated by type 1 non-spanning arrow $\gamma_{\epsilon,\tau}$ and type 2 generators $\gamma_{i,\tau}$ with the relation of type 1, type 2 and type 3 relations. We set

[TABLE]

Type 1 and type 3 relation are satisfied by the fundamental relations of $G$. The first relations of (3.2) is easy to check. The second relation is obtained by the relation between $\epsilon$ and $g^{(i)}(\epsilon)$. We check the third relation of (3.2) by using $\psi(a)=\Gamma_{i},\psi(b)=M_{\epsilon}^{-1}\Gamma_{0}M_{\epsilon}$. Since $\Gamma_{i}$ and $M_{\epsilon}$ are commutative in $G$, we have

[TABLE]

and

[TABLE]

Thus we have the equality $\psi(abab)=\psi(baba)$. Thus the homomorphism $\psi$ is well defined.

Proof of Theorem 4.1.

By the definition of $\varphi$ and $\psi$, we see that the homomorphisms $\psi$ and $\varphi$ are inverse to each other. ∎

4.2. Simplification

We modify the relation of (4.2) and get the simpler form cited in [GK]. By Theorem 4.1 and the following proposition, we get Main Theorem 1.1.

Proposition 4.2**.**

For a subset $I$ of $\{1,\dots,n\}$ we set $M(I)=\prod_{i\in I}\Gamma_{i}$. Under the relation (4.1), the relation (4.2) for $H_{\epsilon,\mathbb{R}}\cap H_{\epsilon^{\prime},\mathbb{R}}\cap R_{>0}^{n}\neq\emptyset$ is equivalent to the following set of relations.

[TABLE]

for all $I,J$ satisfying $I\cap J=\emptyset,I\neq\emptyset,J\neq\emptyset$ and $\#I+\#J\leq n-1$.

Proof.

Throughout this proof we assume the commutativity of $\Gamma_{1},\dots,\Gamma_{n}$. First we assume the condition (4.3) and prove the relation (4.2). We set $K=S(\epsilon)\cap S(\epsilon^{\prime})$. By the definition of $M_{\epsilon}$ and the commutativity of $\Gamma_{i}$, the condition (4.2) can be rewrite as

[TABLE]

where $M_{\epsilon}^{*}=\prod_{i\in S(\epsilon)-K}\Gamma_{i}$ and $M_{\epsilon^{\prime}}^{*}=\prod_{i\in S(\epsilon^{\prime})-K}\Gamma_{i}$. This is one of the conditions in (4.3) by setting $I=S(\epsilon)-K$ and $J=S(\epsilon^{\prime})-K$. We check that $I$ and $J$ satisfies the required conditions. The condition $I\cap J=\emptyset$ is clear. If $M_{\epsilon}^{*}=\emptyset$, then $S(\epsilon)\subset S(\epsilon^{\prime})$ and this contradicts to the condition $H_{\epsilon,\mathbb{R}}\cap H_{\epsilon^{\prime},\mathbb{R}}\cap\mathbb{R}_{>0}^{n}\neq\emptyset$. If $\#I+\#J=n$, then $\epsilon^{\prime}=-\epsilon$. This also contradicts to the condition for $\epsilon$ and $\epsilon^{\prime}$ since $H_{\epsilon,\mathbb{R}}\cap H_{-\epsilon,\mathbb{R}}=\emptyset$.

Next we assume the condition (4.2) and prove the relation (4.3). Let $I$ and $J$ be subsets in $\{1,\dots,n\}$ satisfying the condition of (4.3). We define $\epsilon=(\epsilon_{1},\cdots,\epsilon_{n})$ and $\epsilon^{\prime}=(\epsilon^{\prime}_{1},\cdots,\epsilon^{\prime}_{n})$ by

[TABLE]

Then the relation (4.4) becomes the relation (4.3). We check the condition $H_{\epsilon,\mathbb{R}}\cap H_{\epsilon^{\prime},\mathbb{R}}\cap\mathbb{R}^{n}_{>0}\neq\emptyset$. We set $K=\{1,\dots,n\}-(I\cup J)$. Then we have $K\neq\emptyset$. The system of equations

[TABLE]

is equivalent to

[TABLE]

Thus it has a solution $x=(x_{i})\in\mathbb{R}_{>0}^{n}$. ∎