Topologies on abelian groups and a topological five-lemma

TL;DR

This paper develops new topological tools, including a version of the Five-Lemma, to establish the continuity of homomorphisms in topological abelian groups, with applications to duality in arithmetic cohomology.

Contribution

It introduces a topological Five-Lemma and related results to determine homomorphism continuity in topological abelian groups, aiding duality theories in arithmetic contexts.

Findings

Established a topological version of the Five-Lemma.

Provided criteria for the continuity of homomorphisms.

Applied results to duality in cohomology groups.

Abstract

In this article we establish some results that allow to deduce the continuity of homomorphisms of (topological) abelian groups from commutative diagrams. In particular, we present a new topological version of the classical Five-Lemma. These results aim to be applied in duality results between cohomology groups in arithmetical contexts. In such a topological-arithmetical context, Pontryagin duality plays a central role and it becomes necessary to know whether certain homomorphisms are continuous.