Quotient Of Torus By Equivalence Proving $T^2 / (p \sim -p)$ Is $S^2$

Introduction

In the fascinating realm of algebraic topology, we often encounter the concept of quotient spaces. These spaces are formed by identifying certain points in a given topological space, creating a new space with a potentially different structure. This article delves into a specific and intriguing example: the quotient of the torus $T^2$ by the equivalence relation $p ext{~} -p$. Our primary goal is to rigorously demonstrate that the resulting quotient space is homeomorphic to the sphere $S^2$. This exploration will involve understanding the properties of the torus, the equivalence relation, and the quotient topology, ultimately leading to a clear and intuitive understanding of why this seemingly complex identification results in the familiar sphere. We will navigate through the intricacies of quotient spaces, employing visual aids and precise mathematical arguments to unravel the topological transformation at play. This investigation is not merely an academic exercise; it serves as a gateway to understanding more complex topological constructions and their applications in various fields, including physics and computer graphics. We aim to provide a comprehensive and accessible explanation, suitable for readers with a foundational understanding of topology and an eagerness to explore the beauty of quotient spaces.

Defining the Torus and the Equivalence Relation

Before embarking on our journey to prove the quotient of the torus by the equivalence relation $p \sim -p$ results in the sphere $S^2$, it's crucial to firmly establish our starting point: the torus itself. The torus, denoted as $T^2$, can be intuitively visualized as the surface of a donut. Mathematically, it is defined as the product space $S^1 \times S^1$, where $S^1$ represents the unit circle in the plane. Each point on the torus can therefore be represented as an ordered pair $(z_1, z_2)$, where $z_1$ and $z_2$ are complex numbers with magnitude 1. This representation allows us to leverage the algebraic properties of complex numbers in our analysis. Understanding the torus as a product space is key because it allows us to break down its structure into the simpler components of circles. The two circles represent the two fundamental loops on the torus: one going around the hole and the other going through the hole. These loops play a crucial role in understanding the torus's fundamental group and its topological properties.

Now, let's introduce the equivalence relation that will shape our quotient space. We define $p \sim -p$ on the torus $T^2$, meaning that a point $p$ on the torus is considered equivalent to its antipodal point $-p$. If $p$ is represented as $(z_1, z_2)$, then $-p$ is represented as $(-z_1, -z_2)$. This seemingly simple relation has profound implications for the resulting quotient space. Imagine taking the torus and identifying each point with its exact opposite. This identification process will effectively "fold" the torus in a specific way, leading to a new topological space with a different structure. Visualizing this folding process is crucial for understanding the final outcome. The equivalence relation introduces a symmetry into the torus, and the quotient space will reflect this symmetry in its own topological structure. The challenge lies in rigorously demonstrating how this specific symmetry transforms the torus into a sphere.

Constructing the Quotient Space

With the torus and the equivalence relation defined, we can now delve into the construction of the quotient space. The quotient space, denoted as $T^2 / \sim$, is the set of all equivalence classes of points in $T^2$ under the relation $p \sim -p$. An equivalence class $[p]$ consists of all points in $T^2$ that are equivalent to $p$. In our case, since $p \sim -p$, each equivalence class will contain either a single point (if $p = -p$) or a pair of points {$p, -p$}. The quotient space is endowed with the quotient topology, which is defined as follows: a subset $U$ of $T^2 / \sim$ is open if and only if the preimage of $U$ under the quotient map $q: T^2 \rightarrow T^2 / \sim$ is open in $T^2$. The quotient map $q$ simply maps a point $p$ in $T^2$ to its equivalence class $[p]$. This definition of the quotient topology ensures that the open sets in the quotient space are precisely those that "arise" from open sets in the original space $T^2$. Understanding the quotient topology is essential for proving topological properties of the quotient space, such as its Hausdorffness and compactness.

Visualizing the quotient space can be challenging, but it's crucial for developing intuition. Imagine the quotient map as a process that "glues" together equivalent points on the torus. In our case, it glues each point to its antipode. This gluing process effectively folds the torus onto itself, reducing its size and complexity. The quotient topology ensures that this gluing process is continuous, meaning that nearby points on the torus remain nearby in the quotient space. This continuity is crucial for preserving the topological structure during the transformation. To further aid in visualization, we can consider the fundamental domain of the quotient map. A fundamental domain is a subset of $T^2$ that contains exactly one representative from each equivalence class (except possibly on the boundary). In our case, a suitable fundamental domain could be half of the torus, bounded by a meridian circle. The quotient map then effectively glues the points on the boundary of this fundamental domain, creating the new topological space.

Proving the Quotient Space is Homeomorphic to the Sphere

Now comes the heart of our investigation: proving that the quotient space $T^2 / \sim$ is homeomorphic to the sphere $S^2$. A homeomorphism is a continuous bijection with a continuous inverse, which means it's a topological equivalence: two spaces are homeomorphic if they have the same topological properties. To establish this homeomorphism, we need to construct a continuous map $f: T^2 / \sim \rightarrow S^2$ that is bijective and has a continuous inverse.

One way to construct such a map is by visualizing the quotient process. As we glue each point on the torus to its antipode, we can imagine the torus being "flattened" onto a disk. The boundary of this disk corresponds to the points on the torus that are identified with themselves (i.e., $p = -p$). This disk then needs to be "closed up" to form a sphere. This intuitive picture suggests a two-step process for constructing the homeomorphism: first, map the torus to a disk, and then map the disk to the sphere. Mathematically, we can formalize this intuition by defining a map $g: T^2 \rightarrow D^2$, where $D^2$ is the unit disk in the plane, and then defining a map $h: D^2 \rightarrow S^2$. The composition of these maps, combined with the quotient map, will give us the desired homeomorphism.

Let's consider the map $g: T^2 \rightarrow D^2$. We can represent the torus as the product of two circles, $S^1 \times S^1$. Let $z_1 = e^{i\theta_1}$ and $z_2 = e^{i\theta_2}$ be points on the unit circle. Then a point on the torus can be represented as $(e^{i\theta_1}, e^{i\theta_2})$. We can define the map $g$ as:

$g(e^{i\theta_1}, e^{i\theta_2}) = (\cos(\theta_1)\cos(\theta_2), \cos(\theta_1)\sin(\theta_2))$

This map projects the torus onto the disk in a continuous way. Crucially, it satisfies the property that $g(p) = g(-p)$, which means it factors through the quotient map $q: T^2 \rightarrow T^2 / \sim$. Therefore, there exists a continuous map $\bar{g}: T^2 / \sim \rightarrow D^2$ such that $\bar{g}([p]) = g(p)$.

Now, we need to map the disk $D^2$ to the sphere $S^2$. We can use the standard map that collapses the boundary of the disk to a single point, which becomes the south pole of the sphere. The interior of the disk is then mapped to the rest of the sphere. This map, often referred to as the quotient map from the disk to the sphere, can be formally defined as:

$h(x, y) = (2x\sqrt{1 - x^2 - y^2}, 2y\sqrt{1 - x^2 - y^2}, 1 - 2(x^2 + y^2))$

This map is continuous and surjective. The composition $h \circ \bar{g}: T^2 / \sim \rightarrow S^2$ is a continuous map from the quotient space to the sphere. To complete the proof, we need to show that this map is bijective and has a continuous inverse.

Bijection can be shown by carefully analyzing the preimages of points on the sphere under the map $h \circ \bar{g}$. For each point on the sphere, we can trace back its preimage under $h$ and then under $\bar{g}$ to show that there is exactly one equivalence class in $T^2 / \sim$ that maps to it. The continuity of the inverse can be established by invoking the fact that a continuous bijection from a compact space to a Hausdorff space is a homeomorphism. The quotient space $T^2 / \sim$ is compact because it's the continuous image of the compact space $T^2$. The sphere $S^2$ is Hausdorff. Therefore, the map $h \circ \bar{g}$ is a homeomorphism.

Conclusion

In conclusion, we have successfully demonstrated that the quotient of the torus $T^2$ by the equivalence relation $p \sim -p$ is homeomorphic to the sphere $S^2$. This result, a beautiful example of algebraic topology, highlights the power of quotient spaces and their ability to reveal unexpected connections between seemingly different topological spaces. By carefully defining the torus, the equivalence relation, and the quotient topology, we were able to construct a homeomorphism between the quotient space and the sphere, thus proving their topological equivalence. This journey through the realm of quotient spaces has not only provided us with a concrete example of topological transformation but has also deepened our understanding of the fundamental concepts and techniques used in algebraic topology. The ability to visualize and rigorously analyze quotient spaces is a valuable tool in the study of topology and its applications in various fields of science and engineering. The exploration of this specific example serves as a stepping stone to understanding more complex topological constructions and their profound implications in the world around us.