The Fundamental Group of a Torus

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 path is a continuous map f : I 7→ 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 with F (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 x0 in black with the blue and red loops: 2 P. Shariati

The blue “loop” is the trivial loop which is just our basepoint. We claim that the red loop is the same as our blue loop, because of the fact that we can “wiggle” the red loop to the base point. “Wiggle” is a common term used in this description but it is more analogous to a “contraction” or a “deformation” if that helps paint a clearer image of the visual process. This same process can be apply to any arbitrary loop in our closed disc which confirms that the fundamental group of a closed disc is just the trivial loop. Apart from a visual argument we can provide a more rigorous explanation based on the definition of two elements in the fundamental group being equal. Two elements (loops) of the fundamental group are equivalent if they are homotopic as defined above earlier in this section. This same argument can be applied to an open disc, as well as Rn, 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 space X relative to the basepoint x0 is defined as π1(X, x0) = {[f]| f is a loop based at x0 }.

Definition 5. A space X is path-connected if 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 is simply-connected if it is path connected and for all points x ∈ X, π1(X, x). Really all this is saying is a path-connected space X is simply-connected if π1(X, x0) is trivial which we defined earlier.

Roughly speaking a space X is convex if for any two points x, y ∈ X, the line segment joining x to y is also contained in X. 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 basepoint x0 in purple and the red and blue loops: The Fundamental Group of a Torus 3

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 Z under addition. The operation of the fundamental group would be just tracing one loop after the other. Note that we have not proven this conjecture because 4 P. Shariati it is just a precursor to the main topic of this paper which is the fundamental group of a torus.

2.2 The Torus

We now want to claim that the fundamental group of a Torus which is defined as T 2 = S1 ×S1 is just the direct product of the fundamental group of a circle with itself. A broader general claim we now want to make is that the fundamental group of the torus is isomorphic to Z × Z. Although we will not prove this rigorously we can provide a sound argument as to why this is true. Let us think about the torus and how it relates to the plane R2. 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 2 any loop in T with basepoint x0 can be represented as a straight line segment between (0, 0) which is the image of our basepoint x0 to any (p, q) ∈ Z × Z. Let us depict the torus below with the two loops a, b

2 Based on this picture we can define the fundamental group of the torus π1(T ) 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 for a + 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 The Fundamental Group of a Torus 5 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 5. http://pi.math.cornell.edu/$\sim$hatcher/AT/AT.pdf