Completeness of the homeomorphism relation of locally connected continua

TL;DR

This paper proves that classifying locally connected continua up to homeomorphism is as complex as possible within orbit equivalence relations, answering a question by Chang and Gao.

Contribution

It establishes that the homeomorphism relation of locally connected continua is a complete orbit equivalence relation, a significant result in topological classification.

Findings

Homeomorphism relation is a complete orbit equivalence relation.

Answers a previously open question by Chang and Gao.

Advances understanding of topological classification complexity.

Abstract

In this paper we prove that the homeomorphism relation of locally connected continua is a complete orbit equivalence relation. This answers a question posed by Chang and Gao.