Homeomorphisms of the Pseudoarc

TL;DR

This paper develops a new approach using graph relations to analyze homeomorphisms of the pseudoarc, providing simplified proofs of key properties and confirming a conjecture about its automorphism group.

Contribution

It introduces a novel Fraïssé theoretic method to study pseudoarc homeomorphisms, recovering classical results and establishing new properties of its automorphism group.

Findings

Reproved Bing's results on pseudoarc uniqueness and homogeneity

Proved the automorphism group of the pseudoarc has a dense conjugacy class

Validated a conjecture of Kwiatkowska regarding the pseudoarc

Abstract

We construct homeomorphisms of compacta from relations between finite graphs representing their open covers. Applied to the pseudoarc, this yields simple Fra\"iss\'e theoretic proofs of several important results, both old and new. Specifically, we recover Bing's classic results on the uniqueness and homogeneity of the pseudoarc. We also show that the autohomeomorphism group of the pseudoarc has a dense conjugacy class, thus confirming a conjecture of Kwiatkowska.