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The Griffiths double cone group is isomorphic to the triple

Samuel M. Corson

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 $κ$ is a cardinal such that $2 \leq κ\leq 2^{\aleph_0}$ then the $κ$-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 $κ$-fold cones, with $2 < κ$, 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.

The Griffiths double cone group is isomorphic to the triple

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 is a cardinal such that then the -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 - and of the -fold cones, with , 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.

Paper Structure

This paper contains 16 sections, 56 theorems, 41 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The cone groups
  3. Theorem \ref{['bigisomorphism']}
  4. P-chunks
  5. Close Subsets
  6. Coherent coi triples
  7. Extensions of coherent collections
  8. $\omega$-type concatenations
  9. $\mathbb{Q}$-type concatenations
  10. Arbitrary extensions
  11. Theorem \ref{['bigisomorphism2']}
  12. Background for $\mathbb{H}\mathbb{A}$
  13. Some basic extension results
  14. $\omega$-type concatenations.
  15. $\mathbb{Q}$-type concatenations.
  16. ...and 1 more sections

Key Result

Theorem A

If $\kappa$ is a cardinal such that $2 \leq \kappa \leq 2^{\aleph_0}$ then $\pi_1(\mathbb{G}\mathbb{S}_2) \simeq \pi_1(\mathbb{G}\mathbb{S}_{\kappa})$.

Figures (2)

  • Figure 1: The Griffiths double cone $\mathbb{G}\mathbb{S}_2$
  • Figure 2: The harmonic archipelago $\mathbb{H}\mathbb{A}$

Theorems & Definitions (105)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • ...and 95 more