Teacher resources and professional development across the curriculum
Teacher professional development and classroom resources across the curriculum
In 1884, the English mathematician and writer Edwin Abbott wrote a novel in which almost all of the characters are two-dimensional beings. They live on a surface called Flatland because everywhere it seems to be—well, flat. When we refer to a "surface," we generally mean something that appears to be nice and flat when we look at it closely, as the Flatlander does.
However, just because something is flat in a given region does not mean that it is an infinite plane that extends this flatness in all directions forever.
The global structure of our object that appears to be so nice and flat on the local level might be very complicated, having hills, valleys, holes, and strange, reversing regions (we'll come to those a little later).
In topology, there is an important distinction between the local and global view of surfaces. We generally regard a two-dimensional surface as an object that appears locally as a flat plane, regardless of its global behavior. Some common two-dimensional surfaces are the sphere, the torus, and the double torus.
Remember, though, that we are concerned only with what is essential about shape, so there are many surfaces with a variety of looks that are actually the same topologically.
What separates each topological shape from all other types is the numbers of holes. No matter what is done to a shape, as long as it is topologically allowed, the number of holes will remain constant (although, as we shall see, a hole may not always look like a hole). Hence, the number of holes is another topological invariant, just like the Euler number.
In fact, the Euler characteristic is related to the number of holes a surface has. Notice that a sphere, whose Euler characteristic is two, has no holes; a torus, whose Euler characteristic is zero has one hole; and a double torus, whose Euler characteristic is -2, has two holes. An examination of this pattern reveals that for every hole, the Euler characteristic decreases by two. This implies that the relationship is linear and follows this formula: Euler characteristic = something – twice the number of holes. The number of holes is also known as a surface's genus, so we now have a rough idea of how the Euler characteristic of a surface relates to its genus.
The genus of a surface is a feature of its global topology. The local topology, remember, is always that of a flat plane. The fact that the local topology is flat, however, doesn't mean that the geometry has to be. In unit 8, we will discuss different types of geometry in detail. For the moment we are concerned only with the difference between geometry and topology. In geometry, the primary concern is the measurement of things such as lengths and angles. In topology, it is possible to manipulate shapes without tearing or gluing, so these concepts are pretty meaningless.
We have been discussing two-dimensional surfaces up until this point, but there is no reason that our ideas need to be limited to such objects. We can generalize the idea of a surface into that of a manifold. A 2-manifold is an object that has the local topology of a plane, just like a two-dimensional surface. A 1-manifold is an object that has the local topology of a line segment, regardless of how twisted and knotted it is globally. These descriptions reveal the key property of a manifold: in the local view, it looks straight, or flat, and featureless, but when viewed globally, it may present a more-interesting structure.
A 3-manifold is the generalization of a surface in three dimensions. It is an object that has the local topology of what we normally think of as "space." It too, like the 1- and 2-manifolds, can have rather convoluted global topology. It's a bit hard for us to visualize what topologies might be possible on a global scale, because we are stuck inside such a manifold; consequently, we cannot gain an external view as we can with the 1- and 2-manifolds. Nonetheless, there are some things that we can observe to acquire some ideas about the global topology of a 3-manifold. One of these meaningful observations is of what happens as we leave a particular point and head off on a straight–line path. If, after traveling a sufficiently long distance without turning, we find ourselves back where we started, we might have a clue as to the global topology of the 3-manifold we inhabit.
Perhaps this is a bit hard to visualize, so let's return to a subway example. Let's pretend that this subway system is very large, but very simple, consisting of a single, large oval. It is so large, in fact, that at any given moment, it feels as if we are traveling in a straight line. Furthermore, let's assume that our movement along the track is restricted to only forward or backward motion. Basically, we are treating this subway as a 1-manifold. If we were newcomers to the subway system, and we didn't have a map, we might be able to deduce the global topology of this system by observing the sequence of stops.
If we board the train at stop A, stay on the train for a long time, and eventually find ourselves at stop A again, we could safely assume that we are traveling in some kind of loop, even though it doesn't feel as if we're turning anywhere. This experience in 1-dimension gives us some idea of what it is like to be inside a manifold. At any given point or moment, it seems like a straight line, flat plane, or normal space, but as we attain a greater perspective on the system's structure, we find that it is not as simple as a line, plane, or space that extends forever in all directions. As we shall see in the next section, we can go a step further and actually use this interior, or intrinsic, view to understand topology in a completely different light.
Next: 4.4 Intrinsic Topology