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.

中文翻译：

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波兰空间中的图灵度和 Borel 函数的可分解性

我们对描述性集合论中的一个重要的开放性问题给出了部分答案，即波雷尔函数在波兰空间的分析子集上的可分解性猜想到可分离的可度量空间。我们的技术采用了有效的描述性集合理论和递归理论的深刻结果。事实上，有必要将递归理论中的几个突出结果（例如 Shore-Slaman 连接定理）扩展到波兰空间的设置。作为一个副产品，我们对 Martin 猜想给出了关于波兰空间之间保度 Borel 函数的正反两方面的结果。此外，我们证明了关于可分解性猜想的超限版本和可计算版本的结果。