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Topologies on abelian groups and a topological five-lemma

Felipe Rivera-Mesas

TL;DR

The paper develops a framework to deduce the continuity of homomorphisms between topological abelian groups from commutative diagrams by analyzing topological extensions and coco lces. It introduces and exploits Nagao's and Moore's results to topologize algebraic extensions via topologizing sections, and it defines a Nagao topology $\tau_{\mathcal{E},s}$ to ensure strict continuity of maps. A main contribution is a Topological Five-Lemma for diagrams of strict exact sequences in locally compact abelian groups, providing conditions under which the middle map becomes a topological isomorphism, and connecting these results to duality in arithmetic contexts through Pontryagin duality and Hausdorffization. The findings have implications for controlling continuity in duality results and Yoneda pairings in étale cohomology and local Tate duality, even in non-Hausdorff settings.

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.

Topologies on abelian groups and a topological five-lemma

TL;DR

The paper develops a framework to deduce the continuity of homomorphisms between topological abelian groups from commutative diagrams by analyzing topological extensions and coco lces. It introduces and exploits Nagao's and Moore's results to topologize algebraic extensions via topologizing sections, and it defines a Nagao topology to ensure strict continuity of maps. A main contribution is a Topological Five-Lemma for diagrams of strict exact sequences in locally compact abelian groups, providing conditions under which the middle map becomes a topological isomorphism, and connecting these results to duality in arithmetic contexts through Pontryagin duality and Hausdorffization. The findings have implications for controlling continuity in duality results and Yoneda pairings in étale cohomology and local Tate duality, even in non-Hausdorff settings.

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.

Paper Structure

This paper contains 4 sections, 18 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Basics on topological groups
  3. Topological extensions
  4. Topological Five-Lemma

Key Result

Theorem 1

Let \begin{tikzcd} \s{E}_1: 0 \ar{r} & A_1 \ar{r}{\iota_1} \ar{d}{\alpha} & G_1 \ar{r}{\pi_1} \ar{d}{\gamma} & B_1 \ar{r} \ar{d}{\beta} & 0 \\ \s{E}_2: 0 \ar{r} & A_2 \ar{r}{\iota_2} & G_2 \ar{r}{\pi_2} & B_2 \ar{r} & 0 \end{tikzcd}be a commutative diagram of abelian groups. Assume that the the map $\gamma_{\mathrm{Haus}}$ is well-defined and continuous (resp. continuous and strict) whene

Theorems & Definitions (48)

  • Theorem : Proposition \ref{['Five-Lemma nagao']}
  • Corollary : Topological Five-Lemma
  • Definition 2.1
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Remark 2.6
  • Definition 2.7
  • Remark 2.8
  • Proposition 2.9: Pontryagin duality theorem
  • ...and 38 more