Harmonic Galois theory for finite graphs

TL;DR

This paper introduces a harmonic Galois theory for finite graphs, classifying harmonic branched covers via fundamental group homomorphisms and inertia structures, with applications to embedding problems and fiber realization.

Contribution

It develops a new harmonic Galois theory for finite graphs, linking covers to fundamental groups and inertia, and applies it to solve embedding problems and fiber realization issues.

Findings

Finite embedding problems for graphs have proper solutions.

A Grunwald-Wang type theorem for graph covers is established.

Harmonic Galois theory provides a classification framework for graph covers.

Abstract

This paper develops a harmonic Galois theory for finite graphs, thereby classifying harmonic branched $G$-covers of a fixed base $X$ in terms of homomorphisms from a suitable fundamental group of $X$ together with $G$-inertia structures on $X$. As applications, we show that finite embedding problems for graphs have proper solutions and prove a Grunwald-Wang type result stating that an arbitrary collection of fibers may be realized by a global cover.