Turing degrees in Polish spaces and decomposability of Borel functions

TL;DR

This paper advances the understanding of the Decomposability Conjecture for Borel functions on Polish spaces by extending recursion theory results and analyzing degree structures, with implications for the Martin Conjecture.

Contribution

It extends key recursion theory results to Polish spaces and provides new insights into the Decomposability and Martin Conjectures in descriptive set theory.

Findings

Partial progress on the Decomposability Conjecture.

Positive and negative results on the Martin Conjecture.

Analysis of the transfinite and computable versions of the conjecture.

Abstract

We give a partial answer to an important open problem in descriptive set theory, the Decomposability Conjecture for Borel functions on an analytic subset of a Polish space to a separable metrizable space. Our techniques employ deep results from effective descriptive set theory and recursion theory. In fact it is essential to extend several prominent results in recursion theory (\eg the Shore-Slaman Join Theorem) to the setting of Polish spaces. As a by-product we give both positive and negative results on the Martin Conjecture on the degree preserving Borel functions between Polish spaces. Additionally we prove results about the transfinite version as well as the computable version of the Decomposability Conjecture, and we explore the idea of applying the technique of turning Borel-measurable functions into continuous ones.