On a question of Slaman and Steel

Abstract: 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 also in 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 which is not uniformly universal, and that there is a $(\equiv_T,\equiv_m)$-invariant function which is not uniformly invariant on any pointed perfect set.