Uniform Martin’s conjecture, locally

We show that part I of the uniform Martin’s conjecture follows from a local phenomenon, namely that every non-constant uniformly Turing invariant $f:[x]_{\equiv _T}\to [y]_{\equiv _T}$ satisfies $x\le _T y$. Besides improving our knowledge about part I of the 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 by raising the question: Is part II of the uniform Martin’s conjecture implied by local phenomena, too? and briefly indicating possible directions.