# Uniform Martin's conjecture, locally

## TL;DR

The paper links part I of uniform Martin's conjecture to a local Turing degree phenomenon, revealing new complexity results and raising questions about part II's implications.

## Contribution

It shows that part I of the conjecture follows from a local property of Turing invariants and explores the complexity of computable reducibility on equivalence relations.

## Key findings

- Part I of uniform Martin's conjecture is equivalent to Turing determinacy.
- Computable reducibility on equivalence relations is highly complex, with Turing reducibility embedded.
- The local phenomenon leads to new insights beyond the original conjecture.

## 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.