New degree spectra of Polish spaces

The main result is as follows: Fix an arbitrary prime $q$. A $q$-divisible torsion-free (discrete, countable) abelian group $G$ has a $\Delta^0_2$-presentation if, and only if, its connected Pontryagin–van Kampen Polish dual $\widehat{G}$ admits a computable complete metrization (in which we do not require the operations to be computable). We use this jump-inversion/duality theorem to transfer the results on the degree spectra of torsion-free abelian groups to the results about the degree spectra of Polish spaces up to homeomorphism. For instance, it follows that for every computable ordinal $\alpha>1$ and each $=\mathbf{a} > 0^{(\alpha)}$ there is a connected compact Polish space having proper $\alpha^{th}$ jump degree $\mathbf{a}$ (up to homeomorphism). Also, for every computable ordinal $\beta$ of the form $1+\delta + 2n +1$, where $\delta$ is zero or is a limit ordinal and $n \in \omega$, there is a connected Polish space having an $X$-computable copy if and only if $X$ is $non$-$low_{\beta}$. In particular, there is a connected Polish space having exactly the $non$-$low_{2}$ complete metrizations. The case when $\beta=2$ is an unexpected consequence of the main result of the author's M.Sc. Thesis written under the supervision of Sergey S. Goncharov.