On Matsushita $\pi_1^2$ discrete fundamental groups

TL;DR

This paper studies Matsushita's discrete fundamental groups of graphs, proving a Seifert-van Kampen theorem and showing any group can be realized as such a fundamental group, expanding understanding of discrete topological invariants.

Contribution

It establishes a Seifert-van Kampen theorem for $ ext{pi}_1^2$ and demonstrates that any group can be realized as a $ ext{pi}_1^2$ of some graph, providing new tools and constructions.

Findings

Proved a Seifert-van Kampen-type theorem for $ ext{pi}_1^2$.

Showed any group can be realized as $ ext{pi}_1^2$ of a graph.

Provided alternative proofs for existing theorems using these constructions.

Abstract

The Matsushita fundamental groups of a graph $X$, denoted $\pi_1^r(X)$, are certain discrete versions of the fundamental group for topological spaces. For $r=2$, these groups have a nice combinatorial description, due to Sankar. In this paper we prove two results about $\pi_1^2$. First, we prove a Seifert-van Kampen-type theorem. Similar results have previously been obtained by Barcelo, et al. (and strengthened by Kapulkin and Mavinkurve) for a different notion of discrete fundamental group. Second, we prove that an arbitrary group $G$ can be realized as $\pi_1^2(X)$ for some graph $X$. Our construction works equally well for the aforementioned alternate discrete fundamental group $A_1(X)$, and our second result thus also provides an entirely different method of proof for a theorem of Kapulkin and Mavinkurve.