Teacher resources and professional development across the curriculum
Teacher professional development and classroom resources across the curriculum
To get an idea of how a topologist views the world, let's look at a famous problem considered by many to be the inspiration for the birth of topology. In the mid-1700s residents of the city of Königsberg, Prussia (now called Kaliningrad, Russia), tried to find a route that traversed each of the city's seven bridges exactly once.
Leonhard Euler, an influential Swiss mathematician who was living in Königsberg at the time, took an interest in this problem. His solution provided the basis not only for the study of topology, but also for graph theory, a topic that we will take up in another unit.
Euler recognized that the distances overland and the lengths of the bridges had no bearing whatsoever on the issue of the possible existence of a path that traversed each bridge only once. He was able to condense, or simplify, the map of Königsberg much in the same way that we simplify a city's geography when creating a subway map. His drawing looked like this:
Gone were any geographical or man-made features such as the river, streets, buildings, parks, etc. Euler reduced the entire arrangement to a diagram of edges and nodes (points), in which the distances between points and the angles between edges were not at all important. In fact, from a topological viewpoint, all of the following diagrams would be equivalent to the above drawing.
Euler's graph of the Königsberg bridges and the different versions shown here have the same fundamental connections. No matter how we stretch or bend the graph, the connections remain the same, or invariant. It is as though the edges and nodes are made of rubber, and we are allowed to do anything we want to them as long as we don't cut or glue the rubber. For this reason, topology is often known as "rubber sheet geometry." We haven't seen the "sheet" part of this yet, but it is coming up very soon when we extend our discussion from graphs to surfaces and manifolds.
Let's take a closer look at the connections shared by the graphs above. In all of these drawings there is one node of degree 5 (i.e., a point at which five edges meet), and there are three nodes of degree 3. Now, as it turns out, the degrees of the edges of this graph determine whether or not the sought-after path exists. In our case, which, remember, is analogous to Euler's Königsberg bridges problem, no path exists because there are more than two nodes with an odd degree. We will examine this idea in more depth in another unit; what is important to the development of topology is that the geography of the city was simplified to this representational collection of edges and nodes.
Euler found another property of graphs that remains invariant under stretching and bending. He noticed that graphs in the plane have not only nodes and edges, but also faces. A face is basically the area defined by an associated set of edges and nodes. Faces are topologically the same as disks.
If one takes the number of vertices, subtracts the number of edges, and adds the number of faces, including the face that surrounds the graph, the result is two. This formula holds true for any graph that we can draw on a piece of paper—or a piece of rubber. No matter how much we stretch, twist, or bend a graph, this number will always be two. This number is known as the Euler characteristic, or Euler number.
We must be careful here and note that all of the graphs we have considered so far are flat; that is, they exist on a flat plane, which is only one possible type of surface. A sphere is a different type of surface, as is a torus, or donut shape. As you might imagine, graphs on such surfaces as these do not "behave" the same as graphs on a flat plane.
The above examination of basic graphs has prepared us to think about topological surfaces. This is the "sheet" part of "rubber sheet geometry." In our study of topology, we will be concerned with many different types of surfaces. What's fascinating is that the Euler characteristic is specific to the type of surface upon which a graph is drawn. We can use it to help us determine what kind of a surface we have.
For example, let's look at the surfaces of a sphere and a torus.
Notice that the graph shown on the sphere, corresponding to a horizontal "equator" and a vertical "equator," has 6 vertices, 12 edges, and 8 faces. A configuration such as this, in which the surface is broken up into cells that completely cover it, is called a cell division.. A cell can be thought of as a face, because both are topologically equivalent to a disk. Plugging the known values into Euler's equation, we see that it does indeed yield a result of two as its Euler number. How about for a torus?
Notice that the cell division of a torus shows that it has one node, two edges, and one face (if we unwrap it). This gives us an Euler characteristic of zero. The Euler characteristic is an incredibly powerful concept, and we will see its usefulness demonstrated at several points in our discussion. For now, all we need to remember is that the Euler characteristic is an invariant of the surface with which we are working. That is, we can stretch, twist, or bend a surface as much as we want and the Euler characteristic of graphs on the surface will not change. In other words, the Euler characteristic is considered topologically invariant. The objects studied in topology are malleable, and their true, basic nature can sometimes be obscured by contortions and deformities, so it is quite helpful to have some measures, such as the Euler characteristic, that we can use to identify what kind of things we are dealing with.
In topology, two shapes or surfaces are considered the same if we can continuously deform one into the other. Cutting and pasting are forbidden, but we can bend and squeeze all we want. Consider the example of the "linked chain" you can form by interlocking the index finger and thumb of each hand.
In our normal way of thinking, there would be no way for a person whose hands and fingers are in this position to "unlock" or separate the "chain links" without parting the index finger and thumb of one hand. In the world of topology, however, it's possible to become unlinked without "breaking" either link if the person is sufficiently flexible! Objects in topology that can be transformed into one another are called homeomorphic.
For the remainder of this unit, we will be concerned primarily with surfaces and their generalized cousins, manifolds. We will envision twisting and bending these objects according to the ideas presented in this section in order to learn what fundamental properties they have. Before we do that, however, it would make sense to focus for a moment on what exactly we mean when we speak of surfaces and manifolds.