[Paper Review] The fundamental group of the complement of the singular locus of Lauricella's $F_C$

Yoshiaki Gotō, Jyoichi Kaneko|arXiv (Cornell University)|Oct 26, 2017

Nonlinear Waves and Solitons1 references3 citations

TL;DR

This paper presents a complete presentation of the fundamental group of the complement of the singular locus of Lauricella’s hypergeometric function $F_C$ in three variables, using the van Kampen-Zariski theorem and Reidemeister-Schreier method. It identifies three new relations—(R_ij), (R_k), and (R_IJ)—that fully generate the fundamental group for $n=3$, and extends this to a $2^3$-covering space, yielding an explicit 11-generator, 27-relation presentation for the fundamental group of the covering space.

ABSTRACT

We study the fundamental group of the complement of the singular locus of Lauricella's hypergeometric function $F_C$ of $n$ variables. The singular locus consists of $n$ hyperplanes and a hypersurface of degree $2^{n-1}$ in the complex $n$-space. We derive some relations that holds for general $n\geq 3$. We give an explicit presentation of the fundamental groupin the three-dimensional case. We also consider a presentation of the fundamental group of $2^3$-covering of this space. In the version 2, we omit some of the calculations. For all the calculations, refer to the version 1 (arXiv:1710.09594v1) of this article.

Motivation & Objective

• To determine a complete set of defining relations for the fundamental group $\pi_1(X^{(3)})$ of the complement of the singular locus of Lauricella’s $F_C$ in three variables.
• To extend the presentation to the $2^3$-covering space $\tilde{X}^{(3)}$, which becomes a complement of hyperplanes.
• To identify and verify new relations—beyond the known (R_ij) and (R_k)—that govern the group structure for $n \geq 3$, particularly in the three-dimensional case.
• To provide a systematic computation of monodromy relations via plane cuts and pencil of lines, reducing them to a minimal set of generators and relations.

Proposed method

• Use of the van Kampen-Zariskki theorem to compute $\pi_1(X^{(3)})$ by decomposing the space into open sets and analyzing monodromy relations from loops around singular divisors.
• Definition of base loops $\gamma_0, \gamma_1, \gamma_2, \gamma_3$ around the coordinate hyperplanes and the singular hypersurface $S^{(3)}$, with explicit path constructions.
• Introduction of a new class of relations (R_IJ) involving conjugates of $\gamma_0$ by products of $\gamma_i$, valid for disjoint index sets $I, J$ with $p+q \leq n-1$.
• Application of the Reidemeister-Schreier method to compute the fundamental group of the $2^3$-covering space $\tilde{X}^{(3)}$, which resolves the singular hypersurface into a hyperplane arrangement.
• Identification of 11 generators $\lambda_1, \lambda_2, \lambda_3, \lambda_0, \lambda_0^{(i)}, \lambda_0^{(ij)}, \lambda_0^{(123)}$ corresponding to loops in the covering space.
• Verification of 27 defining relations, including commutativity among $\lambda_i$, among $\lambda_0^{(i)}$, and mixed commutativity between $\lambda_0^{(ij)}$ and conjugated $\lambda_i$, as well as relations involving $\lambda_0^{(123)}$.

Experimental results

Research questions

• RQ1 What additional relations, beyond the known (R_ij) and (R_k), govern the fundamental group $\pi_1(X^{(n)})$ for $n \geq 3$?
• RQ2 Can the fundamental group $\pi_1(X^{(3)})$ be completely presented using only the relations (R_ij), (R_k), and the new (R_IJ) relations?
• RQ3 How does the fundamental group of the $2^3$-covering space $\tilde{X}^{(3)}$ relate to the original group $\pi_1(X^{(3)})$?
• RQ4 What is the structure of the fundamental group of the covering space $\tilde{X}^{(3)}$, and can it be presented explicitly with generators and relations?

Key findings

• The fundamental group $\pi_1(X^{(3)})$ is completely presented by the generators $\gamma_0, \gamma_1, \gamma_2, \gamma_3$ and the relations (R_ij), (R_k), and (R_IJ), with no further relations needed.
• The new relation (R_IJ) holds for all disjoint index sets $I, J \subset \{1,2,3\}$ with $|I|, |J| \geq 1$ and $|I| + |J| \leq 2$, and is essential for the full presentation in $n=3$.
• The $2^3$-covering space $\tilde{X}^{(3)}$ is the complement of a hyperplane arrangement, and its fundamental group $\pi_1(\tilde{X}^{(3)})$ has a presentation with 11 generators and 27 defining relations.
• The 27 relations in $\pi_1(\tilde{X}^{(3)})$ include commutativity among $\lambda_i$, among $\lambda_0^{(i)}$, among $\lambda_0^{(ij)}$, and mixed commutativity involving conjugates and triple intersections.
• The relations (5.58)–(5.60) are derived from geometric intersections of hyperplanes in the covering space, corresponding to lines in the plane cuts of $X^{(3)}$, and are essential for consistency.
• The authors verify that all monodromy relations reduce to the presented set, confirming the completeness of the group presentation for $\tilde{X}^{(3)}$.