Complexity of homeomorphism relations on some classes of compacta

This is a joint work with Benjamin Vejnar (https://arxiv.org/pdf/1808.08760.pdf).
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 Cieśla. It follows 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 is Borel bireducible with the universal orbit equivalence relation of the permutation group on a countable set. On the other hand we observe that the homeomorphism relation between rim-finite metrizable compacta is not classifiable by countable structures.

The next seminar meeting (Ali Enayat talk) is scheduled for November 7th.