Relative twisted homology and cohomology groups associated with Lauricella's $F_D$

TL;DR

This work extends the twisted (co)homology framework to Lauricella FD by introducing relative twisted groups H_1(T,D;L) and H^1(T,D;L) that remain well-behaved for integral exponents. It proves a canonical isomorphism between H_1(T,D;L) and the local solution space Sol_x(a,b,c), enabling a robust, parameter-agnostic study of FD through intersection pairings. The paper develops dual relative structures, twisted period relations, and a Gauss–Manin connection that yields a Pfaffian system for FD, and analyzes the monodromy representation, including reducibility and triviality criteria in the integral-exponent regime. These results provide a versatile toolkit for connection problems, parameter-difference equations, and further hypergeometric systems. Potential applications include explicit period matrices, monodromy computations, and a deeper understanding of FD’s parameter space.

Abstract

We introduce relative twisted homology and cohomology groups associated with Euler type integrals of solutions to Lauricella's system $F_D(a,b,c)$ of hypergeometric differential equations. We define an intersection form between relative twisted homology groups and that between relative twisted cohomology groups, and show their compatibility. We prove that the relative twisted homology group is canonically isomorphic to the space of local solutions to $F_D(a,b,c)$ for any parameters $a,b,c$. Through this isomorphism, we study $\cF_D(a,b,c)$ by the relative twisted homology and cohomology groups and the intersection forms without any conditions on $a,b,c$.

Figures (2)

• Figure 1: Chains and relative chains
• Figure 2: Regions with respect to a twisted cycle $\gamma^u$

Theorems & Definitions (27)

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