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Connecting topological dimension theory and recursion theory

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Mathematical, Foundational and Computational Aspects of the Higher Infinite

We introduce the point degree spectrum of a represented spaces as a substructure of the Medvedev degrees, which integrates the notion of Turing degrees, enumeration degrees, continuous degrees, and so on. The point degree spectrum connects descriptive set theory, topological dimension theory and computability theory. Through this new connection, for instance, we construct a family of continuum many infinite dimensional Cantor manifolds possessing Haver’s property C whose Borel structures at an arbitrary finite rank are mutually non-isomorphic, which strengthen various theorems in infinite dimensional topology such as Roman Pol’s solution to Pavel Alexandrov’s old problem.

This talk is part of the Isaac Newton Institute Seminar Series series.

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