Comparison of Topologies on Fundamental Groups with Subgroup Topology Viewpoint

TL;DR

This paper reviews various topologies on the fundamental group in algebraic topology, compares them using subgroup topology, and presents results to enhance understanding of topologized fundamental groups.

Contribution

It introduces a comparison framework for different topologies on the fundamental group using subgroup topology, providing new insights and results.

Findings

Comparison of multiple topologies on fundamental groups

New results on properties of topologized fundamental groups

Framework for analyzing topologies via subgroup topology

Abstract

In order to make the fundamental group, one of the most well known invariants in algebraic topology, more useful and powerful some researchers have introduced and studied various topologies on the fundamental group from the beginning of the 21st century onwards. In this paper by reviewing these topologies, using the concept of subgroup topology, we are going to compare these topologies in order to present some results on topologized fundamental groups.