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Covering Maps with respect to Topologies on the Fundamental Group

Naghme Shahami, Behrooz Mashayekhy

Abstract

In this paper, using the classical covering theory, we introduce a generalization of covering maps of a space $X$ with respect to a topology $τ$ on the fundamental group of $X$. We show that the famous notions, covering, semicovering, generalized covering and fibration maps are of special cases of this new notion $π_1^τ$-covering map. Moreover, among presenting some properties for this new notion, we compare $π_1^τ$-covering maps of a space $X$ for several famous topologies on the fundamental group of $X$.

Covering Maps with respect to Topologies on the Fundamental Group

Abstract

In this paper, using the classical covering theory, we introduce a generalization of covering maps of a space with respect to a topology on the fundamental group of . We show that the famous notions, covering, semicovering, generalized covering and fibration maps are of special cases of this new notion -covering map. Moreover, among presenting some properties for this new notion, we compare -covering maps of a space for several famous topologies on the fundamental group of .
Paper Structure (4 sections, 7 theorems, 10 equations)

This paper contains 4 sections, 7 theorems, 10 equations.

Table of Contents

  1. Introduction and Motivation
  2. Some Topologies on Fundamental Groups
  3. Some Results on $\pi_1^{\tau}$-Covering Maps
  4. Comparison of $\pi_1^{\tau}$-Covering Maps with respect to Topologies on the Fundamental Group

Key Result

Theorem 3.2

If $p_1: \widetilde{X}_1 \to X$ has pl, $p_2: \widetilde{X}_2 \to X$ has upl and $f: \widetilde{X}_1 \to \widetilde{X}_2$ with $p_2 \circ f = p_1$. Then $f$ has pl property.

Theorems & Definitions (17)

  • Definition 1.1
  • Remark 1.2
  • Definition 3.1
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • Theorem 3.5
  • ...and 7 more