Degree spectra of homeomorphism types of compact Polish spaces
Abstract
A Polish space is not always homeomorphic to a computably presented Polish space. In this article, we examine degrees of non-computability of presenting homeomorphic copies of compact Polish spaces. We show that there exists a -computable low compact Polish space which is not homeomorphic to a computable one, and that, for any natural number , there exists a Polish space such that exactly the high-degrees are required to present the homeomorphism type of . We also show that no compact Polish space has a least presentation with respect to Turing reducibility. The first version of this article appeared in April 2020. A major update was made in September 2023, with improved proofs and results. This is the final version from January 2024, with more results on \v{C}ech homology groups.
Keywords
Cite
@article{arxiv.2004.06872,
title = {Degree spectra of homeomorphism types of compact Polish spaces},
author = {Mathieu Hoyrup and Takayuki Kihara and Victor Selivanov},
journal= {arXiv preprint arXiv:2004.06872},
year = {2024}
}