A Problem Related To Coverings And Lens Spaces

Introduction

In the realm of algebraic topology, covering spaces and lens spaces are fundamental concepts that have been extensively studied. Covering spaces are topological spaces that are locally homeomorphic to a given space, while lens spaces are a type of topological space that can be constructed from a 3-sphere by identifying certain points. In this article, we will delve into a problem related to coverings and lens spaces, specifically focusing on the action of the group $\mathbb{Z}_6$ on the 3-sphere $S^3$.

The Problem

The problem we are trying to solve is as follows:

Let $\mathbb{Z}_6$ act on $S^3=\{ (z,w)\in \mathbb{C}^2:|z|^2+|w|^2=1\}$ via $(z,w) \rightarrow (\epsilon z,\epsilon w)$, where $\epsilon$ is a primitive 6th root of unity. We need to determine the covering space of $S^3$ with respect to this action.

Understanding the Action

To begin with, let's understand the action of $\mathbb{Z}_6$ on $S^3$. The group $\mathbb{Z}_6$ consists of six elements, which can be represented as $\{1, \omega, \omega^2, \omega^3, \omega^4, \omega^5\}$, where $\omega$ is a primitive 6th root of unity. The action of $\mathbb{Z}_6$ on $S^3$ is defined as follows:

$$\begin{align*} \mathbb{Z}_6 \times S^3 &\rightarrow S^3 \\ (\epsilon, (z,w)) &\mapsto (\epsilon z, \epsilon w) \end{align*}$$

where $\epsilon$ is an element of $\mathbb{Z}_6$ and $(z,w)$ is a point in $S^3$.

The Covering Space

To determine the covering space of $S^3$ with respect to this action, we need to find a space that is locally homeomorphic to $S^3$ and has a group of deck transformations isomorphic to $\mathbb{Z}_6$. Let's consider the space $S^3 \times \mathbb{Z}_6$, which consists of all pairs $(x, \epsilon)$, where $x \in S^3$ and $\epsilon \in \mathbb{Z}_6$. The action of $\mathbb{Z}_6$ on $S^3 \times \mathbb{Z}_6$ is defined as follows:

$$\begin{align*} \mathbb{Z}_6 \times (S^3 \times \mathbb{Z}_6) &\rightarrow S^3 \times \mathbb{Z}_6 \\ (\epsilon, (x, \delta)) &\mapsto (\epsilon x, \epsilon \delta) \end{align*}$$

where $\epsilon$ is an element of $\mathbb{Z}_6$ and $(x, \delta)$ is a point in $S^3 \times \mathbb{Z}_6$.

The Covering Map

To show that $S^3 \times \mathbb{Z}_6$ is a covering space of $S^3$, we need to define a covering map $\pi: S^3 \times \mathbb{Z}_6 \rightarrow S^3$. Let's define $\pi$ as follows:

$$\begin{align*} \pi: S^3 \times \mathbb{Z}_6 &\rightarrow S^3 \\ (x, \epsilon) &\mapsto x \end{align*}$$

where $x \in S^3$ and $\epsilon \in \mathbb{Z}_6$.

Properties of the Covering Map

To show that $\pi$ is a covering map, we need to verify that it satisfies the following properties:

1. Local homeomorphism: For each point $x \in S^3$, there exists a neighborhood $U$ of $x$ such that $\pi^{-1}(U)$ is a disjoint union of open sets, each of which is homeomorphic to $U$.
2. Group of deck transformations: The group of deck transformations of $\pi$ is isomorphic to $\mathbb{Z}_6$.

Local Homeomorphism

To show that $\pi$ is a local homeomorphism, we need to find a neighborhood $U$ of each point $x \in S^3$ such that $\pi^{-1}(U)$ is a disjoint union of open sets, each of which is homeomorphic to $U$. Let's consider the neighborhood $U = \{x \in S^3: |x| < 1\}$, which is an open ball in $S^3$. Then, $\pi^{-1}(U)$ is a disjoint union of open sets, each of which is homeomorphic to $U$. Specifically, we have:

$$\begin{align*} \pi^{-1}(U) &= \{x \in S^3: |x| < 1\} \times \mathbb{Z}_6 \\ &= \bigcup_{\epsilon \in \mathbb{Z}_6} \{x \in S^3: |x| < 1, \pi(x) = x\} \times \{\epsilon\} \end{align*}$$

where each set $\{x \in S^3: |x| < 1, \pi(x) = x\} \times \{\epsilon\}$ is homeomorphic to $U$.

Group of Deck Transformations

To show that the group of deck transformations of $\pi$ is isomorphic to $\mathbb{Z}_6$, we need to find a homomorphism $\phi: \mathbb{Z}_6 \rightarrow \text{Homeo}(S^3 \times \mathbb{Z}_6)$ such that $\phi(\epsilon)$ is a deck transformation of $\pi$ for each $\epsilon \in \mathbb{Z}_6$. Let's define $\phi$ as follows:

$$\begin{align*} \phi: \mathbb{Z}_6 &\rightarrow \text{Homeo}(S^3 \times \mathbb{Z}_6) \\ \epsilon &\mapsto \phi_\epsilon \end{align*}$$

where $\phi_\epsilon$ is the homeomorphism defined by:

$$\begin{align*} \phi_\epsilon: S^3 \times \mathbb{Z}_6 &\rightarrow S^3 \times \mathbb{Z}_6 \\ (x, \delta) &\mapsto (\epsilon x, \epsilon \delta) \end{align*}$$

Then, $\phi$ is a homomorphism, and $\phi(\epsilon)$ is a deck transformation of $\pi$ for each $\epsilon \in \mathbb{Z}_6$. Specifically, we have:

$$\begin{align*} \phi(\epsilon \delta) &= \phi_{\epsilon \delta} \\ &= \phi_\epsilon \circ \phi_\delta \\ &= \phi_\epsilon \circ \phi_\delta \circ \phi_\epsilon^{-1} \circ \phi_\epsilon \\ &= \phi_{\epsilon \delta} \circ \phi_\epsilon \end{align*}$$

where $\phi_\epsilon^{-1}$ is the inverse homeomorphism of $\phi_\epsilon$.

Conclusion

In this article, we have shown that the space $S^3 \times \mathbb{Z}_6$ is a covering space of $S^3$ with respect to the action of $\mathbb{Z}_6$ on $S^3$. We have also shown that the group of deck transformations of the covering map $\pi: S^3 \times \mathbb{Z}_6 \rightarrow S^3$ is isomorphic to $\mathbb{Z}_6$. This result has important implications for the study of covering spaces and lens spaces in algebraic topology.

References

• [1] L. Maxim, "Lecture Notes on Algebraic Topology"
• [2] M. M. Postnikov, "Lectures on Algebraic Topology"
• [3] J. P. May, "A Concise Course in Algebraic Topology"

Future Work

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Introduction

In our previous article, we discussed a problem related to coverings and lens spaces, specifically focusing on the action of the group $\mathbb{Z}_6$ on the 3-sphere $S^3$. We showed that the space $S^3 \times \mathbb{Z}_6$ is a covering space of $S^3$ with respect to this action, and that the group of deck transformations of the covering map $\pi: S^3 \times \mathbb{Z}_6 \rightarrow S^3$ is isomorphic to $\mathbb{Z}_6$. In this article, we will answer some of the most frequently asked questions about this problem.

Q: What is the significance of the group $\mathbb{Z}_6$ in this problem?

A: The group $\mathbb{Z}_6$ plays a crucial role in this problem because it acts on the 3-sphere $S^3$ via the map $(z,w) \rightarrow (\epsilon z, \epsilon w)$, where $\epsilon$ is a primitive 6th root of unity. This action gives rise to a covering space of $S^3$, which is the space $S^3 \times \mathbb{Z}_6$.

Q: What is the relationship between the covering space $S^3 \times \mathbb{Z}_6$ and the lens space $L(3,1)$?

A: The covering space $S^3 \times \mathbb{Z}_6$ is closely related to the lens space $L(3,1)$. In fact, the lens space $L(3,1)$ can be obtained from the covering space $S^3 \times \mathbb{Z}_6$ by identifying certain points. Specifically, the lens space $L(3,1)$ is the quotient space of the covering space $S^3 \times \mathbb{Z}_6$ under the action of the group $\mathbb{Z}_6$.

Q: What are the homotopy groups of the covering space $S^3 \times \mathbb{Z}_6$?

A: The homotopy groups of the covering space $S^3 \times \mathbb{Z}_6$ are closely related to the homotopy groups of the 3-sphere $S^3$. Specifically, the homotopy groups of the covering space $S^3 \times \mathbb{Z}_6$ are isomorphic to the homotopy groups of the 3-sphere $S^3$.

Q: What are the applications of this result to other areas of mathematics?

A: This result has important implications for the study of covering spaces and lens spaces in algebraic topology. It also has applications to other areas of mathematics, such as geometry and topology. Specifically, this result can be used to study the properties of other covering spaces and lens spaces, and to investigate the relationships between these spaces.

Q: What are some of the open problems related to this result?

A: There are several open problems related to this result. One of the most important open problems is to investigate the properties of the covering space $S^3 \times \mathbb{Z}_6$ in more detail. Specifically, it would be interesting to study the homotopy groups of the covering space $S^3 \times \mathbb{Z}_6$ and to investigate the relationship between the covering space and the lens space $L(3,1)$.

Q: What are some of the future directions for research in this area?

A: There are several future directions for research in this area. One of the most promising areas of research is to investigate the properties of other covering spaces and lens spaces. Specifically, it would be interesting to study the homotopy groups of other covering spaces and lens spaces, and to investigate the relationships between these spaces. Another area of research is to investigate the applications of this result to other areas of mathematics, such as geometry and topology.

Conclusion

In this article, we have answered some of the most frequently asked questions about the problem related to coverings and lens spaces. We have shown that the space $S^3 \times \mathbb{Z}_6$ is a covering space of $S^3$ with respect to the action of $\mathbb{Z}_6$ on $S^3$, and that the group of deck transformations of the covering map $\pi: S^3 \times \mathbb{Z}_6 \rightarrow S^3$ is isomorphic to $\mathbb{Z}_6$. We have also discussed some of the open problems related to this result and some of the future directions for research in this area.

References

• [1] L. Maxim, "Lecture Notes on Algebraic Topology"
• [2] M. M. Postnikov, "Lectures on Algebraic Topology"
• [3] J. P. May, "A Concise Course in Algebraic Topology"

Future Work

In future work, we plan to investigate the properties of the covering space $S^3 \times \mathbb{Z}_6$ in more detail. Specifically, we plan to study the homotopy groups of the covering space $S^3 \times \mathbb{Z}_6$ and to investigate the relationship between the covering space and the lens space $L(3,1)$. We also plan to explore the applications of this result to other areas of mathematics, such as geometry and topology.