Computable topological abelian groups

TL;DR

This paper investigates the computability aspects of Pontryagin - van Kampen duality for certain topological abelian groups, providing new insights into their algorithmic structure and applications.

Contribution

It establishes the computability of dualization for compact and locally compact totally disconnected Polish abelian groups, with applications to Polish space presentations and solenoid groups.

Findings

Dualization is computable for specific classes of groups

Provides arithmetical characterization of solenoid groups

Answers questions on presentations of Polish spaces

Abstract

We study the algorithmic content of Pontryagin - van Kampen duality. We prove that the dualization is computable in the important cases of compact and locally compact totally disconnected Polish abelian groups. The applications of our main results include solutions to questions of Kihara and Ng about presentations of connected Polish spaces, and an unexpected arithmetical characterisation of direct products of solenoid groups among all Polish groups.