The Griffiths double cone group is isomorphic to the triple

TL;DR

This paper proves that the fundamental groups of Griffiths double cone and triple cone spaces are isomorphic, extending to κ-fold cones, and confirms a conjecture relating the Griffiths double cone to the harmonic archipelago, with non-constructive isomorphisms.

Contribution

It establishes the isomorphism of fundamental groups between Griffiths double cone, triple cone, and κ-fold cones, and proves a conjecture linking the double cone to the harmonic archipelago.

Findings

Fundamental groups of Griffiths double cone and triple cone are isomorphic.

Isomorphisms for κ-fold cones are non-constructive.

Confirmed conjecture relating Griffiths double cone to harmonic archipelago.

Abstract

It is shown that the fundamental group of the Griffiths double cone space is isomorphic to that of the triple cone. More generally if $\kappa$ is a cardinal such that $2 \leq \kappa \leq 2^{\aleph_0}$ then the $\kappa$-fold cone has the same fundamental group as the double cone. The isomorphisms produced are non-constructive, and no isomorphism between the fundamental group of the $2$- and of the $\kappa$-fold cones, with $2 < \kappa$, can be realized via continuous mappings. We also prove a conjecture of James W. Cannon and Gregory R. Conner which states that the fundamental group of the Griffiths double cone space is isomorphic to that of the harmonic archipelago.