Comparison of Topologies on Homotopy Groups with Subgroup Topology Viewpoint

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.