On Subgroup Topologies on Fundamental Groups

TL;DR

This paper introduces new topologies on the fundamental group of a space to classify coverings and analyze their properties, including when the fundamental group forms a topological group.

Contribution

It develops and studies subgroup topologies on fundamental groups, providing a framework for classifying coverings and understanding their topological structures.

Findings

Identifies conditions under which the fundamental group becomes a topological group.

Provides examples comparing different topologies on fundamental groups.

Classifies coverings using topological properties of the fundamental group.

Abstract

It is important to classify covering subgroups of the fundamental group of a topological space using their topological properties in the topologized fundamental group. In this paper, we introduce and study some topologies on the fundamental group and use them to classify coverings, semicoverings, and generalized coverings of a topological space. To do this, we use the concept of subgroup topology on a group and discuss their properties. In particular, we explore which of these topologies make the fundamental group a topological group. Moreover, we provide some examples of topological spaces to compare topologies of fundamental groups.