  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.
