Turing Degrees in Polish Spaces and Decomposability of Borel Functions

In this article we give a partial answer to an important open problem in descriptive set theory, the Decomposability Conjecture on Borel functions from an analytic subset of a Polish space into a separable metrizable space. Our techniques employ deep results from effective descriptive set theory and recursion theory. In fact it is essential to prove that several prominent results in recursion theory are extended in the setting of Polish spaces. These results include the Shore-Slaman Join Theorem, and as a by-product, we also give both positive and negative results on the Martin Conjecture on degree preserving Borel functions between Polish spaces. Finally we give 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.