On the Complexity of the Uniform Homeomorphism Relation between Separable Banach Spaces

Recently, there has been a growing interest in understanding the complexity of common analytic equivalence relations between separable Banach spaces via the notion of Borel reducibility in descriptive set theory (see [Bos] [FG] [FLR] [FR1] [FR2] [Me]). In general, the notion of Borel reducibility yields a hierarchy (though not linear) among equivalence relations in terms of their relative complexity. The most important relations between separable Banach spaces include the isometry, the isomorphism, the equivalence of bases, and in nonlinear theory, Lipschitz and uniform homeomorphisms. The exact complexity of the first four relations has been completely determined by recent work in the field. Using earlier work of Weaver [W], Melleray [Me] proved that the isometry between separable Banach spaces is a universal orbit equivalence relation. Rosendal [Ro] studied the equivalence of bases and showed that it is a complete Kσ equivalence relation. Using the work of Argyros and Dodos [AD] on amalgamations of Banach spaces, Ferenczi, Louveau, and Rosendal [FLR] recently showed that the isomorphism, the (complemented) biembeddability and the Lipschitz equivalence between separable Banach spaces, as well as the permutative equivalence of Schauder bases, are complete analytic equivalence relations. The Borel reducibility among these equivalence relations, as well as some other equivalence relations we will be dealing with in this paper, is illustrated in Figure 1 (see Section 2 below for the definitions of the equivalence relations). Note that in particular the complete analytic equivalence relation E Σ 1 1 is the most complex one in the Borel reducibility hierarchy of all analytic equivalence relations.