Multiplication is discontinuous in the Hawaiian earring group (with the quotient topology)

TL;DR

This paper demonstrates that the multiplication operation in the fundamental group of the Hawaiian earring, equipped with the quotient topology, is discontinuous, providing a counterexample to a common assumption in topological group theory.

Contribution

It introduces a specific example where the fundamental group with the quotient topology is not a topological group, answering a longstanding open question negatively.

Findings

The quotient map q is not a quotient map when considering the product topology.

The fundamental group of the Hawaiian earring is not a topological group under the quotient topology.

The result settles the question of whether all such fundamental groups are topological groups, providing a negative answer.

Abstract

The natural quotient map q from the space of based loops in the Hawaiian earring onto the fundamental group provides a new example of a quotient map such that q x q fails to be a quotient map. This also settles in the negative the question of whether the fundamental group (with the quotient topology) of a compact metric space is always a topological group with the standard operations.