On Matsushita $π_1^2$ discrete fundamental groups

TL;DR

The paper develops Seifert–van Kampen-type tools for Matsushita's discrete fundamental group $\pi_1^2(X)$, both at the groupoid level and the group level, enabling computation of these invariants via graph covers that respect 4-cycles. It then proves a universality result: every group $G$ is realizable as $\pi_1^2(X)$ for some graph $X$, using a covering-theoretic construction and a carefully designed simply connected cover with a free $G$-action. The methods extend prior results for related discrete fundamental groups and provide a constructive approach to realize arbitrary groups as graph-based invariants. Overall, the work highlights the richness of $\pi_1^2(X)$ beyond the classical fundamental group of graphs and connects to classifying-space ideas through explicit coverings.

Abstract

The Matsushita fundamental groups of a graph $X$, denoted $π_1^r(X)$, 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 $π_1^2$. 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 $π_1^2(X)$ for some graph $X$. Our construction works equally well for the aforementioned alternate discrete fundamental group, and our second result thus generalizes a theorem of Kapulkin and Mavinkurve which applies only to finitely presented groups $G$.

Figures (1)

• Figure 1: The basic "building block" graph $B$ for $\tilde{X}$

Theorems & Definitions (17)

• Definition 2.1
• Definition 2.2
• Example 2.3
• Example 2.4
• Definition 3.1
• Theorem 3.2
• proof
• Theorem 3.3
• proof
• Corollary 3.4