On a question of Slaman and Steel

We consider an old question of Slaman and Steel: whether Turing equivalence is an increasing union of Borel equivalence relations none of which contain a uniformly computable infinite sequence. We show this question is deeply connected to problems surrounding Martin’s conjecture and in the theory of countable Borel equivalence relations. In particular, if Slaman and Steel’s question has a positive answer, it implies there is a universal countable Borel equivalence relation which is not uniformly universal, that there is a ( ≡ T , ≡ m ) (≡ _T,≡ _m) -invariant function which is not uniformly invariant on any pointed perfect set, and which is everywhere ≤ m ≤ _m -incomparable with both the Turing jump j j and its complement.