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 (G) over cap 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 a > 0((alpha)) there is a connected compact Polish space having proper ath jump degree 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 is an element of 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-low2 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.
展开▼
