The fundamental group of the harmonic archipelago

TL;DR

This paper investigates the fundamental group of the harmonic archipelago, revealing its structure as a quotient of the Hawaiian earring group, and demonstrating its uncountability and topological properties.

Contribution

It provides a detailed description of the fundamental group of the harmonic archipelago, including its kernel and topological features, advancing understanding of complex topological spaces.

Findings

pi1(HA) is a quotient of the Hawaiian earring group

Both pi1(HA) and its kernel are uncountable

pi1(HA) has the indiscrete topology

Abstract

The harmonic archipelago HA is obtained by attaching a large pinched annulus to every pair of consecutive loops of the Hawaiian earring. We clarify the fundamental group pi1(HA) as a quotient of the Hawaiian earring group, provide a precise description of the kernel, show that both pi1(HA) and the kernel are uncountable, and that pi1(HA) has the indiscrete topology.