The Fundamental Group Of A Torus - Tufts University

The Fundamental Group of a TorusPejmon ShariatiTufts University, Medford, MassachusettsAbstract. This paper aims to illustrate the process of visualizing andconstructing fundamental groups and how they are related to previousalgebraic structures we have studied earlier this semester. We will startby studying the very definition of a fundamental group and analyzingseveral examples. From there we will introduce the torus and how toconstruct the fundamental group of this structure.Keywords: Torus · Fundamental Group.11.1IntroductionBasic TopologyWe start by introducing some new concepts in topology that are crucial in understanding 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 suchthat α(a) α(b) x0Definition 3. A homotopy of two loops α, β on (X, x0 ) is a continuous mapF : [0, 1] [a, b] 7 X with F (0, t) α(t) for all t [a, b] and F (1, t) β(t) forall t [a, b].We start by explaining why the fundamental group of a line, plane, closed/opendisc are all trivial, i.e. there is only one loop, the loop at the base point. Asmentioned before we define two loops to be the same if we can “wiggle” one toresemble the other. We will start by demonstrating how the fundamental groupof the closed disc is trivial. Depicted below is our closed disc with the basepointx0 in black with the blue and red loops:

2P. ShariatiThe blue “loop” is the trivial loop which is just our basepoint. We claimthat 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 thisdescription but it is more analogous to a “contraction” or a “deformation” if thathelps paint a clearer image of the visual process. This same process can be applyto any arbitrary loop in our closed disc which confirms that the fundamentalgroup of a closed disc is just the trivial loop. Apart from a visual argument wecan provide a more rigorous explanation based on the definition of two elementsin the fundamental group being equal. Two elements (loops) of the fundamentalgroup 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, andthe real number line, but obviously with some slight modifications.2The Definition of a Fundamental GroupDefinition 4. The fundamental group of a space X relative to the basepoint x0is 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 twopoints (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 forall points x X, π1 (X, x). Really all this is saying is a path-connected space Xis 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, theline segment joining x to y is also contained in X. All convex sets are simplyconnected and since any closed/open disc, as well as Rn , a plane, and thereal 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 niceto deal with, but now we will move onto a space that is not convex that willeventually lead us to the fundamental group of a torus, the circle.2.1The circleWe now want to analyze the fundamental group of a circle defined as S 1 {(x, y) R2 x2 y 2 1}. This is different from the closed disc example sincea circle implies we are only dealing with the border. Let us start by analyzingthe circle depicted below with the basepoint x0 in purple and the red and blueloops:

The Fundamental Group of a Torus3The “surface” we are considering is solely the border. The blue loop is drawnon the inside because of the lack of space and the fact that the image is onedimensional. Based on our description of two loops being equivalent above weknow that despite the blue loop looking significant different from the red loopthey are actually the same since we can deform one to look like the other. Letus 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 thegreen loop we can never get it to resemble the red or blue loop. This is furtherdemonstrated by the fact that by the intermediate value theorem the green loopgl must satisfy the following conditions: gl(0) 1, gl(1) 1, and gl(t) 1for some t such that 0 t 1. The red and blue loops do not satisfy theseconditions 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 wecan distinguish loops based on how many times they go around the basepointx0 which in our case is 1 since we want to analyze the circle in the complexcoordinate system. If we assign directions to these loops with counterclockwiserepresenting non-negative integers and clockwise as negative integers then we cansurely observe how the fundamental group of a circle is isomorphic to the groupZ under addition. The operation of the fundamental group would be just tracingone loop after the other. Note that we have not proven this conjecture because

4P. Shariatiit is just a precursor to the main topic of this paper which is the fundamentalgroup of a torus.2.2The TorusWe now want to claim that the fundamental group of a Torus which is defined asT 2 S 1 S 1 is just the direct product of the fundamental group of a circle withitself. A broader general claim we now want to make is that the fundamentalgroup of the torus is isomorphic to Z Z. Although we will not prove thisrigorously we can provide a sound argument as to why this is true. Let us thinkabout the torus and how it relates to the plane R2 . We can represent the planeas a lattice with each square being a unit square so for example the first squareto the right of the origin has coordinates (0, 0), (0, 1), (1, 0), (1, 1). We claim thatany loop in T 2 with basepoint x0 can be represented as a straight line segmentbetween (0, 0) which is the image of our basepoint x0 to any (p, q) Z Z. Letus depict the torus below with the two loops a, bBased on this picture we can define the fundamental group of the torus π1 (T 2 )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 wiggledor deformed to resemble the straight line segment. Depending on the location of

The Fundamental Group of a Torus5the basepoint of the torus our line segment on the plane may lie somewhere elsebut we can still guarantee our claim if we apply the homotopy lifting propertywhich essentially claims that regardless of our starting point on the plane we canfiguratively speaking, lift the line segment to start at the origin. In this sense wecan guarantee that the elements of the fundamental group of a torus will neverexceed that of Z2 . A little more work is required to show that the fundamentalgroup of a torus is isomorphic to Z2 , but it is clear that our visual argumentshould convince the reader that this isomorphism is valid.References1. Margalit, Dan. Office Hours with a Geometric Group Theorist. Princeton University Press, 2017.2. https://web.stanford.edu/ aaronlan/assets/fundamental-group.pdf3. groups-of-common-spaces4. https://www.math.arizona.edu/ \sim glickenstein/math534 1011/fundgrp1-2.pdf5. http://pi.math.cornell.edu/ \sim hatcher/AT/AT.pdf

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. De nition 1. A path is a continuous map f: I7!X. De nition 2. A loop in a pointed space (X;x 0) is a path : [a;b] 7!Xsuch