On the complexity of the uniform homeomorphism relation between separable Banach spaces

TL;DR

This paper investigates the complexity of the uniform homeomorphism relation among separable Banach spaces, establishing its position within the Borel reducibility hierarchy and exploring its behavior on specific subclasses.

Contribution

It proves that the complete $K_{\sigma}$ equivalence relation reduces to uniform homeomorphism and characterizes the local equivalence relation's complexity.

Findings

Uniform homeomorphism relation is as complex as the complete $K_{\sigma}$ relation.

The local equivalence relation is bireducible with $K_{\sigma}$.

Constructed a class of Banach spaces where the isomorphism relation encodes the equality of countable sets of reals.

Abstract

We consider the problem of determining the complexity of the uniform homeomorphism relation between separable Banach spaces in the Borel reducibility hierarchy of analytic equivalence relations. We prove that the complete $K_{\sigma}$ equivalence relation is Borel reducible to the uniform homeomorphism relation, and we also determine the possible complexities of the relation when restricted to some small classes of Banach spaces. Moreover, we determine the exact complexity of the local equivalence relation between Banach spaces, namely that it is bireducible with $K_{\sigma}$. Finally, we construct a class of mutually uniformly homeomorphic Banach spaces such that the equality relation of countable sets of real numbers is Borel reducible to the isomorphism relation on the class.