The Unipotent Tropical Fundamental Group

TL;DR

This paper introduces the unipotent tropical fundamental group for polyhedral complexes, establishes its computability and properties, and proves a correspondence with the classical unipotent de Rham fundamental group via a new isomorphism.

Contribution

It defines a new tropical fundamental group, proves its computability, and establishes a correspondence with classical fundamental groups in algebraic geometry.

Findings

The unipotent tropical fundamental group is computable and satisfies a Seifert--Van Kampen theorem.

An explicit description for fans using a bar complex is provided.

A main theorem establishes an isomorphism between tropical and classical unipotent fundamental groups.

Abstract

We define the unipotent tropical fundamental group of a polyhedral complex in $\mathbb{R}^n$ as the Tannakian fundamental group of the category of unipotent tropical vector bundles with integrable connection. We show that it is computable in that it satisfies a Seifert--Van Kampen theorem and has a description for fans in terms of a bar complex. We then review an analogous classical object, the unipotent de Rham fundamental group of a sch\"{o}n subvariety of a toric variety. Our main result is a correspondence theorem between classical and tropical unipotent fundamental groups: there is an isomorphism between the unipotent completion of the fundamental group of a generic fiber of a tropically smooth family over a disc and the tropical unipotent fundamental group of the family's tropicalization. This theorem is established using Kato--Nakayama spaces and a descent argument. It requires a slight enlargement of the relevant categories, making use of enriched structures and partial compactifications.