Archipelago groups

TL;DR

This paper investigates the fundamental groups of classical archipelago spaces, revealing their properties and classifying archipelago groups built from countable groups, with conjectures on their isomorphisms.

Contribution

It characterizes the fundamental group of archipelago spaces and classifies archipelago groups formed from countable groups, establishing isomorphism results and conjectures.

Findings

The fundamental group is locally free and not indicable.

Archipelago groups from countable groups are isomorphic to either () or (_2).

For large cardinalities, (G_i) is not isomorphic to these cases.

Abstract

The classical archipelago is a non-contractible subset of $\mathbb{R}^3$ which is homeomorphic to a disk except at one non-manifold point. Its fundamental group, $\mathcal{A}$, is the quotient of the topologist's product of $\mathbb Z$, the fundamental group of the shrinking wedge of countably many copies of the circle (the Hawaiian earring), modulo the corresponding free product. We show $\mathcal{A}$ is locally free, not indicable, and has the rationals both as a subgroup and a quotient group. Replacing $\mathbb Z$ with arbitrary groups yields the notion of archipelago groups. Surprisingly, every archipelago of countable groups is isomorphic to either $\mathcal{A}(\mathbb Z)$ or $\mathcal{A}(\mathbb Z_2)$, the cases where the archipelago is built from circles or projective planes respectively. We conjecture that these two groups are isomorphic and prove that for large enough cardinalities of $G_i$, $\mathcal{A}(G_i)$ is not isomorphic to either.