The complexity of the homeomorphism relation between compact metric spaces

TL;DR

This paper establishes the precise complexity of classifying compact metric spaces up to homeomorphism, showing it matches the most complex orbit equivalence relations in descriptive set theory.

Contribution

It proves that the homeomorphism relation on compact metric spaces is Borel bi-reducible with the complete orbit equivalence relation of Polish group actions, linking topology and descriptive set theory.

Findings

Homeomorphism relation is Borel bi-reducible with complete orbit equivalence relation.

Results extend to isomorphism of separable commutative C*-algebras.

Findings clarify the complexity of classification problems in topology and operator algebras.

Abstract

We determine the exact complexity of classifying compact metric spaces up to homeomorphism. More precisely, the homeomorphism relation on compact metric spaces is Borel bi-reducible with the complete orbit equivalence relation of Polish group actions. Consequently, the same holds for the isomorphism relation between separable commutative C*-algebras and the isometry relation between C(K)-spaces.