Uniform Martin's conjecture, locally

Abstract

We show that part I of uniform Martin's conjecture follows from a local phenomenon, namely that if a non-constant Turing invariant function goes from the Turing degree $\boldsymbol x$ to the Turing degree $\boldsymbol y$, then $\boldsymbol x \le_T \boldsymbol y$. Besides improving our knowledge about part I of uniform Martin's conjecture (which turns out to be equivalent to Turing determinacy), the discovery of such local phenomenon also leads to new results that did not look strictly related to Martin's conjecture before. In particular, we get that computable reducibility $\le_c$ on equivalence relations on $\mathbb N$ has a very complicated structure, as $\le_T$ is Borel reducible to it. We conclude raising the question "Is part II of uniform Martin's conjecture implied by local phenomena, too?" and briefly indicating a possible direction.