Question

Man zeige ohne Verwendung der \(j\)-Funktion, dass \(\mathrm{H} / \Gamma\) zur Ebene C topologisch ?quivalent ist. Anleitung. Man studiere die Rand?quivalenzen im Fundamentalbereich.

Short answer

\(\mathrm{H} / \Gamma\) is topologically equivalent to \(\mathbb{C}\) by identifying boundaries in the fundamental domain.

Step-by-step solution

Understanding the Problem

The problem asks us to show that the quotient space \(\mathrm{H} / \Gamma\) is topologically equivalent to the complex plane \(\mathbb{C}\). We are instructed not to use the \(j\)-function and to focus on boundary equivalences in the fundamental domain.

Identify the Fundamental Domain

The fundamental domain for \(\mathrm{H}\) under the action of \(\Gamma = \text{SL}(2, \mathbb{Z})\) is usually given by the set \(\{ z \in \mathrm{H} \mid -1/2 < \text{Re}(z) \leq 1/2, \ |z| \geq 1 \} \). This domain can be visualized as a region in the upper half-plane.

Identify the Boundaries of the Fundamental Domain

The boundaries of the fundamental domain are the vertical lines \(\text{Re}(z) = -1/2\) and \(\text{Re}(z) = 1/2\), and the arc of the unit circle \(|z| = 1\) within the upper half-plane. These boundaries need to be identified under the transformations of \(\Gamma\).

Understand Boundary Identification

In this context, points on opposite sides of the fundamental domain are identified. The vertical boundary lines are identified by the transformation \(T:z\mapsto z+1\) from \(\Gamma\). The arc of the unit circle is self-identified due to the modular transformation \(S:z\mapsto -1/z\).

Show Topological Equivalence

By identifying these boundaries, the fundamental domain can be mapped continuously onto a punctured sphere model (which covers \(\mathbb{C}\)) after accounting for the identifications which effectively create a loop space equivalent to a sphere with one point removed. Thus, \(\mathrm{H}/\Gamma\) is topologically equivalent to \(\mathbb{C}\).

Key concepts

Fundamental Domain

A fundamental domain is a specific region of the mathematical space that essentially contains all the distinct information needed to describe a whole group action. In simple terms, when you have a space where a mathematical group acts, this domain helps by providing a smaller, representative segment that, when copies are arranged appropriately, can cover the entire space.

For the upper half-plane \(H\) under the action of the special linear group \(\Gamma = \text{SL}(2, \mathbb{Z})\), the fundamental domain is crucial. It is often depicted visually as the set of complex numbers \( z \) satisfying: \(-1/2 < \text{Re}(z) \leq 1/2\) and \(|z| \geq 1\).

This description might seem technical at first, but imagine it as a slice of the whole space that, through certain transformations, can constructively fill the entire region of interest. This domain is like a puzzle piece that replicates to form the complete picture.

Upper Half-Plane

The concept of the upper half-plane is simple but powerful. It refers to all complex numbers with a positive imaginary part. Mathematically, if you imagine the complex plane (which has real and imaginary numbers), the upper half-plane is basically the portion of this plane that lies above the real line.

This region is essential in many areas of mathematics, especially in complex analysis and number theory. In context, the upper half-plane \(H\) serves as the domain upon which we examine the group action of \(SL(2, \mathbb{Z})\) and study its effects in the field of modular functions and forms.

Understanding this plane helps us to see how groups of mathematical transformations operate to form new structures and equivalences. Here, all our operations and transformations, such as the modular transformations, take place.

Modular Transformations

Modular transformations are special function transformations related to the theory of modular forms. More specifically, a modular transformation is an action using a matrix from the group SL(2, \(\mathbb{Z}\)), which has the form \([\begin{smallmatrix} a & b \ c & d \end{smallmatrix}])\), where \(a, b, c, d\) are integers, and \(ad-bc=1\).

Such a transformation can be expressed as \(z \mapsto \frac{az+b}{cz+d}\), mapping complex numbers from the upper half-plane to itself. Two elementary types of these transformations frequently used are \(T: z \mapsto z+1\) and \(S: z \mapsto -\frac{1}{z}\).

These transformations manage how different pieces of the fundamental domain are identified and merged, playing a crucial role in making \(\mathrm{H}/\Gamma\) resemble the complex plane topologically. Thus, modular transformations help in understanding symmetrical properties and identifying equivalent points.

Boundary Identification

Boundary identification is the process of recognizing and matching the boundaries of a fundamental domain to create a seamless, larger structure. For the fundamental domain of the upper half-plane, it means connecting points on the domain periphery as dictated by modular transformations.

For instance, using the transformation \(T: z \mapsto z + 1\), the vertical lines \(\text{Re}(z) = -1/2\) and \(\text{Re}(z) = 1/2\) are considered identical. Similarly, the transformational operation \(S: z \mapsto -\frac{1}{z}\) makes the arc of the unit circle \(|z|=1\) self-consistent.

By performing these identifications, the fundamental domain morphs into a new shape, akin to connecting edges of a cut-out pattern to assemble a three-dimensional shape. This process results in a topological structure equivalent to a punctured sphere, establishing the equivalence to the complex plane \(\mathbb{C}\). Hence, boundary identification is a pivotal step in achieving the desired topological equivalence.