Teacher professional development and classroom resources across the curriculum
Teacher professional development and classroom resources across the curriculum
In our subway example above, we saw that there are two ways to view a manifold. The first, and probably most intuitive, way is to look at the manifold as a whole as it sits in space. This kind of view is called an extrinsic view and it is the kind of view that we get when we, for instance, look at a subway map. Although this view is the most intuitive, it is, in some sense, not the most fundamental way to view a surface or manifold. This is because a single topological object can be represented extrinsically in many different ways. This idea, known as "embedding," will be covered in more detail in the next section, but for now what we care about is that the extrinsic view is in some ways not as fundamental as the intrinsic view.
The intrinsic view, remember, is the view from inside a surface or manifold. For a surface, or 2-manifold, this view can be thought of as what a bug would see if it landed on the surface. For a line, or 1-manifold, this is what a bug would see if it landed on a wire. For a 3-manifold, the intrinsic view is what we see in our daily lives as we look out into outer space. The intrinsic view is a way of viewing a manifold without regard to how it is embedded. This enables us to distinguish between which properties are inherent in the manifold and which properties are the results of the way the manifold is represented.
To get a better sense of the intrinsic perspective, let's consider the donut-shaped torus that was introduced earlier.
Let's think about what a person living on this surface would experience. Let's say that our person is completely two-dimensional, a Flatlander, and she is curious to find out what her world is like. Remember that because this is a manifold, it always appears to her to be a flat plane. She, being naturally curious, sets out to prove it. To do this, she leaves the front of her house and begins walking "south" leaving a trail of blue thread behind her to mark her path.
After traveling for a while without turning, she spots a building in the distance. As she approaches, she recognizes the building as her own house, except now she is facing the back of it. She correctly deduces that her world is not, in fact, an infinite plane but, rather, is a curve that turns back in on itself. This indicates to her that her world could be a closed manifold. A closed manifold does not have to go on forever and yet has no boundary. An open manifold, on the other hand, extends forever in all directions.
As our traveler approaches the backside of her house, she decides to tie the end of the blue string that she is carrying to the end of the string at the front of her house, which marks the beginning of her journey. She surmises that this effectively creates some sort of loop around her world.
Having realized that her world has some sort of global topology, she resolves to discover exactly which kind of "shape" she lives in. To do this, she sets out heading west, this time leaving a trail of red thread to mark her path. After walking for quite a while without turning, she begins to wonder why she hasn't seen her blue thread anywhere. She had thought that she would cross it at some point and that that would imply that her world is some sort of "hyper-circle." Flatlanders know about circles, so our explorer had thought that her world was some sort of two-dimensional analog to the circle, sort of an "inflated circle." We three-dimensional beings call such a structure a "sphere."
To our explorer's surprise, after continuing on for a considerably longer time than the duration of her first journey, she arrives at the east face of her house. Furthermore, she has managed to return to her house without seeing her blue thread. This disturbs her greatly, because with her two–dimensional mindset, she has trouble envisioning the donut surface that we can see as the perfect explanation for what she has experienced.
We can clearly envision the donut surface that makes the traveler's experience possible, but let's try to get a feel for how she sees the situation. We need some sort of device or mechanism for drawing the donut surface from an insider's perspective. To do this, we will represent both a torus and a sphere intrinsically with what are known as box-diagrams, or gluing diagrams. Gluing diagrams are simply flat shapes, squares in this case, that have a set of rules governing what happens when an object crosses one of the sides, or boundaries. We can imagine the boundaries being glued to one another according to the specific markings in the diagram.
When an object crosses a single-arrow line, it returns from the analogous position from the other single-arrow line. The same holds true when the double-arrow boundaries are crossed. With our advantage of seeing in three dimensions, we can easily imagine these box diagrams being curled up with their edges glued together to make the familiar surfaces of a sphere and a torus (with some help from our topologically allowed deformations of course).
To our explorer however, this view makes no sense; she would probably think of her experience like this:
These diagrams represent an intrinsic view of the surfaces of a torus and a sphere. We could perform any topologically allowed operations to either surface in our external view, and these diagrams would not change.
The diagram on the right demonstrates what our explorer expected to happen; it represents a sphere and shows that the two threads would have crossed. Notice that the paths on this diagram are not straight. This is a result of the fundamental difference between the local geometry of a torus and a sphere. The local geometry of a sphere is one of positive curvature, whereas that of a torus is flat. We will explore these ideas of geometry in more depth in unit 8.
Let's return to our earlier example of a single-loop subway system. Remember that we don't have a map, and that it never really feels as if we're turning when we ride it, and that we return to our initial stop after a while. This is our intrinsic experience, but the extrinsic view of our subway does not have to be a large oval, or even a circle, for that matter. It can be any convoluted shape, even crossing over itself, and we would have no idea.
Our final point about intrinsic topology is that it is the only choice we have when it comes to experiencing and attempting to understand a 3-manifold. Our Flatlander from before had no choice but to explore the intrinsic topology of her two-dimensional world. Similarly, we have no choice but to explore the intrinsic topology of our own 3-manifold world. This is a topic to which we will return a bit later, but for now, let's look at some possible ways to think of the intrinsic topology of a 3-manifold.
The above diagram represents a "flat" 3-torus. If we were inside such a manifold, we would find that as we "exited" one face, we would "enter" at the analogous spot on the other face having the same marking. Notice that this is similar to the situation of the flat 2-torus from before, except that in this 3-torus we can travel up or down and experience the same behavior.
If this were the shape of our universe and we decided to carry out the Flatlander's experiment, the first thing we should notice is that we are going to need another color of thread. If we leave out of the front of the box carrying a blue thread, we will find that we eventually return through the back of the box; if we leave out of the side of the box carrying a red thread, we will find that we return through the opposite side of the box, having never crossed the blue thread; and if we depart from the top of the box carrying a green thread, we will find ourselves returning through the bottom of the box, having seen neither the red nor the blue threads! This may seem very strange to us, as our Flatlander's experiment must seem to her. Of course, it's impossible to carry out such an experiment in our universe, so let's consider what a person inside this manifold must see.
This person looks forward and sees his back, looks to his right side and sees his left, and looks up and sees his own feet. This gives the experiment some reference points that we can have some hope of duplicating in our own universe. For instance, we can use telescopes to map the night sky and look for regions that seem to repeat themselves. This, of course, is very complicated, but it is considerably less complicated than traveling to the edge of the universe in all directions.
Now that we have an idea of how manifolds look when we view them intrinsically, let's turn our attention to the different ways these surfaces can be viewed from outside. By gaining an understanding of how 1- and 2-manifolds behave when viewed extrinsically, we'll gain insight into how a 3-manifold, such as our universe, might behave.