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Visually-friendly manifolds with arbitrary finite fundamental group

Luca Tanganelli Castrillón

Abstract

We exhibit a family of metrizable manifolds such that any finite group appears as the fundamental group of one of them. These spaces are especially interesting as they can be easily visualized, as opposed to classical examples of spaces with arbitrary fundamental group.

Visually-friendly manifolds with arbitrary finite fundamental group

Abstract

We exhibit a family of metrizable manifolds such that any finite group appears as the fundamental group of one of them. These spaces are especially interesting as they can be easily visualized, as opposed to classical examples of spaces with arbitrary fundamental group.

Paper Structure

This paper contains 3 sections, 7 theorems, 14 equations, 1 figure.

Table of Contents

  1. Metrizing the quotient
  2. The fundamental group of the quotient
  3. One-for-all

Key Result

Proposition 1

Let $G$ be a group acting isometrically and closedly on a metric space $(X,d)$. Then $X/G$ inherits a natural metric $\overline d$ given by and $\overline d$ induces the quotient topology on $X/G$.

Figures (1)

  • Figure 1: Two examples of loop based at $\overline{(A,B,C)}\in X^3/\Sigma_3$. The first one is null-homotopic. The second isn't because its lift to $X^3$ "permutes $A$ and $B$".

Theorems & Definitions (10)

  • Proposition 1
  • Example 1
  • Proposition 2
  • Proposition 3
  • Lemma 1
  • Definition 1
  • Lemma 2
  • Proposition 4
  • Proposition 5
  • Example 2