Comparison of Topologies on Homotopy Groups with Subgroup Topology Viewpoint

TL;DR

This paper reviews and compares different topologies on homotopy groups of topological spaces, focusing on subgroup topology, to enhance understanding and utility in algebraic topology.

Contribution

It introduces a comparison framework for various topologies on homotopy groups using subgroup topology, providing new insights into their relationships.

Findings

Comparison of known topologies on homotopy groups

Results on topologized homotopy groups using subgroup topology

Enhanced understanding of algebraic topology structures

Abstract

By introducing various topologies on the homotopy groups of a topological space, some researchers make these well known notions in algebraic topology more useful and powerful. In this paper, first we recall and review some known topologies on homotopy groups. Then by reviewing some famous subgroups of homotopy groups and using the concept of subgroup topology, we intend to compare these topologies in order to present some results on topologized homotopy groups.