The complexity of the classification problem of continua

TL;DR

This paper establishes a deep connection between the classification problem of continua and a universal orbit equivalence relation, revealing the complexity of classifying connected compact metric spaces.

Contribution

It proves that the homeomorphism problem for continua is Borel bireducible with a universal orbit equivalence relation, highlighting its maximal complexity among such classification problems.

Findings

Homeomorphism problem for continua is Borel bireducible with a universal orbit equivalence relation.

Classifying connected compact metric spaces has maximal complexity in the Borel reducibility hierarchy.

The result links topological classification to descriptive set theory and group actions.

Abstract

We prove that the homeomorphism problem for connected compact metric spaces is Borel bireducible with a universal orbit equivalence relation induced by a Borel action of a Polish group.