The Fundamental Group of a Torus

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Pejmon Shariati

Tufts University, Medford, Massachusetts

Abstract. This paper aims to illustrate the process of visualizing and constructing fundamental groups and how they are related to previous algebraic structures we have studied earlier this semester. We will start by studying the very definition of a fundamental group and analyzing several examples. From there we will introduce the torus and how to construct the fundamental group of this structure.

Keywords: Torus·Fundamental Group.

1

Introduction

1.1 Basic Topology

We start by introducing some new concepts in topology that are crucial in un-derstanding fundamental groups.

Definition 1. A pathis a continuous map f :I7→X.

Definition 2. A loop in a pointed space (X, x0) is a path α: [a, b]7→X such that α(a) =α(b) =x0

Definition 3. A homotopy of two loops α, β on (X, x0) is a continuous map F : [0,1]×[a, b]7→X withF(0, t) =α(t)for all t∈[a, b]and F(1, t) =β(t)for all t∈[a, b].

We start by explaining why the fundamental group of a line, plane, closed/open disc are all trivial, i.e. there is only one loop, the loop at the base point. As mentioned before we define two loops to be the same if we can “wiggle” one to resemble the other. We will start by demonstrating how the fundamental group of the closed disc is trivial. Depicted below is our closed disc with the basepoint x0in black with the blue and red loops:

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This same argument can be applied to an open disc, as well asR , a plane, and the real number line, but obviously with some slight modifications.

2

The Definition of a Fundamental Group

Definition 4. The fundamental group of a spaceX relative to the basepointx0 is defined as π1(X, x0) ={[f]|f is a loop based atx0 }.

Definition 5. A spaceX ispath-connectedif there is a path joining any two points (i.e., for all x, y ∈ X there is some path f : I 7→ X with f(0) = x, f(1) =y)

Definition 6. A space X issimply-connected if it is path connected and for all points x∈X, π1(X, x). Really all this is saying is a path-connected spaceX is simply-connected ifπ1(X, x0)is trivial which we defined earlier.

Roughly speaking a spaceX is convex if for any two points x, y ∈ X, the line segment joiningxtoy is also contained inX. All convex sets are simply-connected and since any closed/open disc, as well as Rn, a plane, and the real number line are all convex then their fundamental groups will be trivial. However, it is important to note that while convex implies simply connected, the converse is not necessarily true. As you can see convex spaces are very nice to deal with, but now we will move onto a space that is not convex that will eventually lead us to the fundamental group of a torus, the circle.

2.1 The circle

We now want to analyze the fundamental group of a circle defined as S1 =

{(x, y) ∈R2|x2+y2 = 1}. This is different from the closed disc example since a circle implies we are only dealing with the border. Let us start by analyzing the circle depicted below with the basepointx0 in purple and the red and blue loops:

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The “surface” we are considering is solely the border. The blue loop is drawn on the inside because of the lack of space and the fact that the image is one dimensional. Based on our description of two loops being equivalent above we know that despite the blue loop looking significant different from the red loop they are actually the same since we can deform one to look like the other. Let us now analyze a third image with the green loop:

The green loop started at 0 but notice that the endpoint is located at 4π, and went around the circle twice. Notice that no matter how we deform the green loop we can never get it to resemble the red or blue loop. This is further demonstrated by the fact that by the intermediate value theorem the green loop gl must satisfy the following conditions: gl(0) = 1, gl(1) = 1, and gl(t) = 1 for some t such that 0 < t < 1. The red and blue loops do not satisfy these conditions and thus there cannot exist a homotopy, or a continuous mapping, between the green loop and the red/blue loop. We can further conclude that we can distinguish loops based on how many times they go around the basepoint x0 which in our case is 1 since we want to analyze the circle in the complex coordinate system. If we assign directions to these loops with counterclockwise representing non-negative integers and clockwise as negative integers then we can surely observe how the fundamental group of a circle is isomorphic to the group

Zunder addition. The operation of the fundamental group would be just tracing

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about the torus and how it relates to the planeR . We can represent the plane

as a lattice with each square being a unit square so for example the first square to the right of the origin has coordinates (0,0),(0,1),(1,0),(1,1). We claim that any loop inT2 with basepointx0 can be represented as a straight line segment between (0,0) which is the image of our basepointx0 to any (p, q)∈Z×Z. Let

us depict the torus below with the two loopsa, b

Based on this picture we can define the fundamental group of the torusπ1(T2) as{0, a, b, a+b,2a+b, a−b, ...}, which can be generalized to{pa+qb:p, q∈Z}. We can define the line segment fora+b for example to look like:

It is easy to see that any path on the plan from (0,0) to (a, b) can be wiggled or deformed to resemble the straight line segment. Depending on the location of

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the basepoint of the torus our line segment on the plane may lie somewhere else but we can still guarantee our claim if we apply the homotopy lifting property which essentially claims that regardless of our starting point on the plane we can figuratively speaking, lift the line segment to start at the origin. In this sense we can guarantee that the elements of the fundamental group of a torus will never exceed that of Z2. A little more work is required to show that the fundamental

group of a torus is isomorphic to Z2, but it is clear that our visual argument

should convince the reader that this isomorphism is valid.

References

1. Margalit, Dan. Office Hours with a Geometric Group Theorist. Princeton Univer-sity Press, 2017.

2. https://web.stanford.edu/~aaronlan/assets/fundamental-group.pdf

3. http://mathonline.wikidot.com/list-of-fundamental-groups-of-common-spaces

4. https://www.math.arizona.edu/$\sim$glickenstein/math534_1011/

fundgrp1-2.pdf

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