Project's nick-name
Loop Groups
Title
Developing code that determines the loop groups that identify and distinguish one topological object from
another.
Instructor
Dr. Emily Landes, Department of Mathematics
Supervisor
Professor Yoav Moriah, Department of mathematics
Abstract / Background:
Is an inner-tube different than a ball? Is a cake doughnut different than a chocolate truffle? How and in
what context?
Figure 1
For this project we will be working in the world of topology. In topology, intuitively speaking, two objects
(called manifolds) are the “same” if one can be continuously deformed into the other. So, in this context, is
an inner-tube the same as a ball? Can you continuously deform an inner-tube into a ball? Your intuition
may tell you that the inner-tube is NOT the same as the ball since regardless of the deformation the innertube has a hole while ball does not. This is good intuition and we may think of the hole in the inner-tube as
an obstruction.
The first goal of this project is to turn our intuition into something mathematically rigorous. Here is where
loop groups, i.e. fundamental groups, come in. Imagine a closed loop, like a rubber band sitting on the
inner-tube and another sitting on the ball. What happens as you shrink these loops through the space they
are each sitting on?
Figure 2
The loops on the ball all shrink through continuous deformations to a point whereas some loops on the
inner-tube shrink to a point and others cannot shrink at all (compare the green loops with the red and blue
loops). Consider the loops on the inner-tube that do not shrink to a point (red and blue). Notice that all such
loops either go around or go through the hole. So in some sense, the loops that do not shrink on the innertube capture the “hole” obstruction. By considering properties of loops that exist on a given space, we can
actually distinguish different spaces.
What we do is associate to each topological space, a group of loops called the fundamental group. A
general group is a set equipped with some extra structure (for example a rule that describes how to put two
elements together to create another element in the set). The elements of the fundamental group of a
topological space are classes of loops where two loops on a space are equivalent (i.e. in the same class) if
one loop can be continuously deformed through the space into the other loop. Let us again consider the case
of the ball and the inner-tube. On the ball every loop can be continuously deformed into any other. So, for
the ball there is just one element in the fundamental group and this element is the set of loops that all shrink
to a point. On the inner-tube there are many different classes of loops. In addition to loops that shrink to a
point, there are loops that go around the hole (blue in Figure 2) and loops that go through the hole (red in
Figure 2). Are the loops that go through the hole equivalent to the loops that go around the hole? What
about the loops that go around the hole three times (red loop in figure 3)? Are these loops equivalent to
those which go through the hole once (blue loop in figure 3)?
Figure 3
Understanding how to distinguish two manifolds or two topological spaces is a fundamental mathematical
problem whose results have many important scientific applications including identifying and classifying
DNA molecules and improving the reliability of computer-aided-geometric-design systems used in
engineering analysis tasks.
Loop groups or fundamental groups are a key element in distinguishing topological spaces. Although there
are algorithms for computing the fundamental groups of topological spaces, writing programs to do so
quickly and concisely is a challenge. This project provides a rare opportunity to join pure mathematics with
the real world through developing background in topological spaces and writing code to determine the
fundamental groups that distinguish them.
Student mission / Objective:
Develop an understanding of topological spaces and the loop groups (fundamental groups) that distinguish
them.
Review existing code and develop your own program to not only compute fundamental groups but also
reduce these groups to a nice form.
Requirements:
Basic Mathematics
A willingness to draw and make paper models to investigate mathematical ideas
A willingness/desire to learn/do some programing
Questions from the Project Description:
Consider a double inner-tube (shown right). Is this space topologically the same as a regular
inner-tube? Provide support for your answer using classes of equivalent loops. What are the
dimensions of these inner-tubes? What is the dimension of the ball?
Compare the cake doughnut to the chocolate truffle. What dimension are these topological objects?
Describe the elements of the loop groups for each of the cake doughnut and the chocolate truffle. Are these
two topological spaces different or the same? Why?
Consider the cake doughnut and the inner-tube. Are they topologically the same? What is the topological
dimension of each? How are they similar? Does a copy of one object exist inside the other? What are the
loop groups of each topological space? Do you know notice any relationship between the associated loop
groups?
Additional Reading:
I have included two papers. The first is Chapter 1 of Allen Hatcher's book, Algebraic Topology. It gives a
nice intuitive description of the fundamental group and introduces the important concept of homotopy. Read
this chapter an answer the questions below.
In order to understand how this project brings together programing and topology (via the fundamental
group), you can also look at the second paper, Computing Fundamental Group of General 3-Manifold . We
will go over this paper in detail during the project so you do not have to worry about understanding all of
definitions or details on your own.
Questions from Chapter 1 of Hatcher's Algebraic Topology book:
Consider the loop A, called the trefoil. Give an intuitive description of the fundamental group of the
complement of the trefoil in R3. Namely, describe the equivalence classes of loops that describe the space
R3 – A. (Remember that two loops area equivalent if once can be continuously deformed in R3 – A to the
other).
A
We can think of a circle or radius 1 as a path in R2 with the image of the starting point the same as the image
of the end point. Using polar coordinates, the path can be written as the map f: I → R2 defined by
f(x) = Cos(x) + iSin(x). What is the map g: I→ R2 that describes the path of a circle of radius 3 ? Are the
paths f and g homotopic? Give an explicit function F: I × I → R2 that describes the homotopy from f to g
(i.e. F(x,0) = f(x) and F(x,1) = g(x) and F(x,t) = ft(x) are all continuous).