Section 9.2 Topology
¶In section Section 7.4, it was pointed out that the Klein disk, and hence the real projective plain, can be obtained by identifying antipodal points on the sphere. But what does the result look like? We digress briefly to discuss the peculiar topological structure of the real projective plane.
Begin by considering (a piece of) the Euclidean plane, which we represent by a rectangle, as shown in the first image in Figure 9.2.1. Now identify two sides, as shown in the second image, where the arrows indicate the identification. What do we get? All we've done is to roll up the plane; the result is a cylinder.
What happens if we identify the same two sides, but in the opposite orientation, as shown in the third image in Figure 9.2.1? You may already know the answer: the result is a Möbius strip.
Activity 9.2.1. Cutting up a Möbius strip.
A Möbius strip can be made by cutting paper into strips and taping them together, with a half-twist before adding the final piece of tape. What happens if you cut a Möbius strip in half (lengthwise)? Try it! What if you give the paper two half-twists before taping it? What happens if you cut a Möbius strip in thirds?
What happens if we identify both pairs of opposite sides? For instance, what if we take the cylinder and identify the remaining sides, as shown in the first image in Figure 9.2.2? That construction amounts to thinking of the cylinder as a hose, wrapping the hose into a circle, and joining the ends. The result is a donut shape known as a torus.
However, this torus is not very much like a donut, which is clearly a curved surface. Our torus is flat! A better analogy would be with the “wraparound” feature of some video games, in which crossing the edge of the screen on one side returns you to the other side. A donut obviously exists in three dimensions. Our flat torus can be thought of as the product of two circles, each one in two dimensions, and therefore lives in four dimensions!
Again, we can change the orientation when we identify opposite sides. One way to do that is shown in the second image in Figure 9.2.2, which represents a Klein bottle. Like our flat torus, a Klein bottle can not be built in three dimensions, but requires four.
Finally, if we change both orientations, we obtain the model shown in the last image in Figure 9.2.2 – which is just the real projective plane! Imagine rounding off the corners; all we've done is identify antipodal points, resulting in a square version of the Klein disk. The projective plane can not even be built in four dimensions, but requires five.
