The Ext group in the categories of topological abelian groups and topological vector spaces

TL;DR

This paper investigates the properties of Ext groups in categories of topological abelian groups and vector spaces, focusing on their behavior under various constructions and applications to known results and three space properties.

Contribution

It provides new insights into the behavior of Ext groups in these categories, especially regarding products, coproducts, and subgroup structures, and applies these to broader contexts.

Findings

Ext groups behave predictably under products and coproducts.

Conditions identified for when topological vector spaces form a three space property.

Applications to locally compact abelian groups and vector space extensions.

Abstract

This paper is devoted to the study of the group ${\rm Ext} (G,H)$ of all extensions of topological abelian groups $0\to H\to X\to G\to 0$ and the group ${\rm Ext}_{TVS} (Z,Y)$ of all extensions of topological vector spaces $0\to Y\to X\to Z\to 0$. We focus on their behaviour under taking products, countable coproducts, dense subgroups and open subgroups. Finally, we apply the obtained properties to formulate in a more general setting some known results in the category of locally compact abelian groups and to determine conditions in which being a topological vector space is a three space property.