  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
$0'$-computable low$_3$ compact Polish space which is not homeomorphic to a
computable one, and that, for any natural number $n\geq 2$, there exists a
Polish space $X_n$ such that exactly the high$_{n}$-degrees are required to
present the homeomorphism type of $X_n$. 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.
