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Comparison of Topologies on Homotopy Groups with Subgroup Topology Viewpoint

Naghme Shahami, Behrooz Mashayekhy

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.

Comparison of Topologies on Homotopy Groups with Subgroup Topology Viewpoint

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.
Paper Structure (4 sections, 4 theorems, 17 equations)

This paper contains 4 sections, 4 theorems, 17 equations.

Table of Contents

  1. Introduction and Motivation
  2. Some Subgroups of Homotopy Groups
  3. Some Topologies on Homotopy Groups
  4. Comparison of Topologies on Homotopy Groups

Key Result

Theorem 4.1

$(i)$ If $H=\pi^{sp}(X,x_0)$, then $\pi_n^{Span}(X,x_0) \preccurlyeq \pi_n^H(X,x_0)$. $(ii)$ If $\pi_n^{Span}(X,x_0)=\pi_n^K(X,x_0)$ for a subgroup $K$ of $\pi_n(X,x_0)$, then $K=\pi^{sp}(X,x_0)$.

Theorems & Definitions (12)

  • Remark 2.1
  • Theorem 4.1
  • proof
  • Remark 4.2
  • Remark 4.3
  • Definition 4.4
  • Theorem 4.5
  • proof
  • Corollary 4.6
  • Corollary 4.7
  • ...and 2 more