Some notes on Pontryagin duality of abelian topological groups
Linus Kramer, Karl Heinrich Hofmann
TL;DR
The paper investigates Pontryagin duality for abelian topological groups beyond local compactness, focusing on abelian pro-Lie groups and the role of $k$-groups. It presents the Leptin–Noble–Banaszczyk exponent-2 example $E$ to show that the duality map $\eta_E$ can be bijective yet discontinuous, with a dense but incomplete dual $\widehat{E}$ and a discrete bidual, illustrating duality pathologies. It then formalizes $k$-groups via the $k$-ification functor $G\mapsto kG$, establishing a coreflective subcategory of topological groups and showing that products of $k$-groups remain $k$-groups, while providing counterexamples like $E$. In the abelian pro-Lie setting, the paper proves that $\eta_G$ is bijective and, under suitable conditions, its inverse is continuous, and it develops a framework in which $k$-group structure interacts with duality through the functors $k(-)$ and $\widehat{(-)}$, culminating in open questions about when $\widehat{\widehat{G}}$ inherits the $k$-group property for abelian pro-Lie $G$. The work clarifies the boundaries of Pontryagin duality beyond locally compact groups and highlights a categorical, $k$-theoretic approach to salvage duality in broader classes of abelian topological groups.
Abstract
We consider several questions related to Pontryagin duality in the category of abelian pro-Lie groups.