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Unit 4

Topology's Twists and Turns

4.1 Introduction

"Abstractness, sometimes hurled as a reproach at mathematics, is its chief glory and its surest title to practical usefulness. It is also the source of such beauty as may spring from mathematics"

-Eric Temple Bell

Does the universe go on forever? If not, what happens when we get to the edge? What are the possible shapes that our space can take? What makes these possible shapes different from each other? These questions are fundamental to the mathematical study of topology. Topology, originally known as analysis situs—roughly, "geometry of position", seeks to describe what is fundamental about shape in general.

Subway MapTo envision what we mean, imagine a subway map. A subway map shows the connections between stops and which train lines transfer to others, but it does not give any indication of the geography of the ground. Neither does it accurately portray the distance between stops. It basically shows you only how many stops are in between others and which connections you must make in order to get to your destination stop. This emphasis on connections at the expense of relatively superficial characteristics, such as distance, is the key idea behind topology.

Pretend that you are in an unfamiliar city and, unfortunately, you are without a subway map. You know which stop you wish to get to, but without a map, you are hopelessly lost as to how to get there. You ask a kind-looking stranger for help, and she tells you to get on the blue line towards Flatsburgh, go three stops, then transfer to the red line towards Square City, and get off at the fifth stop. You can follow these directions and get to your desired destination without ever having to look at a map.

In following these directions, you are experiencing the subway system firsthand; your mental image of your journey would not necessarily be that of a map, but rather that of your first-person perspective. This is an important perspective known as an "intrinsic" view. In topology, this correlates to the study of a surface or spatial shape from the perspective of someone who is in it. Looking at a map of the subway system, on the other hand, is an example of taking an "extrinsic" view, because you can see the system from the point of view of an outside observer. In this unit we will look at topology from both the intrinsic and extrinsic views.

Now that we understand how we will be looking at things, we can ask, "what are these things that we wish to study?" In short, they are topological spaces, such as graphs and manifolds. Our understanding of the full meaning of this term, "manifold," will develop over the course of the unit, but for now we can think of manifolds as surfaces that, when viewed up close, appear to be flat. Our system of subway tunnels could be thought of as a 1-manifold, as it is essentially a system in which one can go only forward or backward. A 2-manifold is the surface of something like a sphere. A 3-manifold is like our universe and can be thought of analogously to the 2-manifold being a surface. This may not be intuitive; one of our goals in this unit is to develop a better understanding of the concept of 3-manifolds.

This shows the subway in 3-D. This shows the subway in 3-D.

Topology, the study of position without regard to distance, is an area of mathematics that deals with highly abstract, idealized notions of shape, connectedness, and other properties. It is a true exercise for the mind, and as such is best appreciated for its intellectual and aesthetic value. Although most topology is studied for its own sake, some ideas can be applied to problems in the real world. Configuration space, for example, is a way to view all possible physical arrangements of a system, such as the equipment on a factory’s manufacturing floor, as a topological space. This can aid in high-level design processes.

In this unit we will look at what is essential about shapes from both the intrinsic and extrinsic views. Examining concepts such as connectedness, embedding, and orientability, we will see how surfaces are classified and learn a bit about the recent classification of 3-manifolds. Finally, we will see how concepts such as the Euler characteristic apply to the manufacturing floor, and we will close with an exploration of what our universe might be like on the largest of scales.

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Next: 4.2 What is Essential about Shape?

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