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

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.