Turing degrees in Polish spaces and decomposability of Borel functions

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 (e.g. 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.

ON A METRIC GENERALIZATION OF THE tt-DEGREES AND EFFECTIVE DIMENSION THEORY

“…In particular, the theory of generalized Turing degrees has achieved great success. The researchers have found a number of unexpected applications of the notion of metric/topological Turing degrees: Day-Miller [6] explained the behavior of Levin's neutral measures in algorithmic randomness theory; Gregoriades-Kihara-Ng [9] gave a partial answer to the open problem on generalizing the Jayne-Rogers theorem in descriptive set theory; Kihara-Pauly [21] proved a result related to descriptive set theory, infinite dimensional topology, and Banach space theory; Andrews-Igusa-Miller-Soskova [1] showed that the PA degrees are first-order definable in the enumeration degrees; Kihara-Ng-Pauly [20] established a classification theory of the enumeration degrees; et cetera.…”

Enumeration Degrees and Topology

“…Here, the partition of X in Definition 10 can even be chosen to be Π 0 2 . Intricate arguments from descriptive set theory [13,33,38] then show that for Polish spaces, σ-homeomorphism agrees with second-level Borel isomorphism. Jayne had explored second-level Borel isomorphism of Polish spaces in 1974 [20] motivated by applications in Banach space theory.…”

On some topics around the Wadge rank $ω_2$

“…These hybrid difference operators seem relevant for studying σ-continuous functions (ω-decomposable functions; see e.g. [12]). As in Propositions 2.2 and 2.3, the classes…”