The complexity of homeomorphism relations on some classes of compacta

TL;DR

This paper investigates the complexity of homeomorphism relations on various classes of compact spaces, establishing their reducibility and classifiability within the Borel hierarchy and orbit equivalence relations.

Contribution

It demonstrates that homeomorphism relations on certain compacta are Borel reducible to universal orbit equivalence, and classifiable by countable structures, advancing understanding of their descriptive complexity.

Findings

Homeomorphism relation between compact spaces reduces to that between absolute retracts.

Homeomorphism relation of absolute retracts is Borel bireducible with the universal orbit equivalence.

Homeomorphism relation between regular continua is classifiable by countable structures.

Abstract

We prove that the homeomorphism relation between compact spaces can be continuously reduced to the homeomorphism equivalence relation between absolute retracts which strengthens and simplifies recent results of Chang and Gao, and Cie\'sla. It follows then that the homeomorphism relation of absolute retracts is Borel bireducible with the universal orbit equivalence relation. We also prove that the homeomorphism relation between regular continua is classifiable by countable structures and hence it is Borel bireducible with the universal orbit equivalence relation of the permutation group on a countable set. On the other hand we prove that the homeomorphism relation between rim-finite metrizable compacta is not classifiable by countable structures.