Real & Complex Projective Space (Part 1)
The idea of “quotienting” out by a subspace or subgroup is a useful way of simplifying an object of study. Although “quotient objects” may appear to be an unnecessary complication, they turn out to be the natural context in which to consider many problems. Today, we will begin to discuss a key example in topology, called “projective spaces.” These spaces are important in algebraic geometry, algebraic topology, differential geometry, and quantum physics.
We begin with real projective spaces, which are easier to understand than their complex cousins, although ultimately less useful. Let our space be and consider the following equivalence relation . Say that if for . Intuitively, considering as a vector space, we are stipulating that all co-linear points be equivalent. Then define . This is, quite simply, the topological space of lines through the origin in .
In order to understand , we want to look at a transversal of its equivalence classes (lines). By this we mean a topological subspace of that intersects each equivalence class (line) in exactly one point. Do this, we take one hemisphere of , the -dimensional sphere. By choosing a hemisphere (and only one half of the rim), we obtain a shape that intersects every line going through the origin in exactly one place.
To understand this better, we consider explicit examples.
is often called the real projective line. It can be thought of as the real number line with and , with the two infinities identified, as in the following image:
As the picture suggest, is topologically equivalent to the circle. Since the circle is compact, so is . We say that is the one-point compactification of the real line.
To see how this arises from our original construction of real projective space, take and quotient out by . We now obtain the space of all lines through the origin in the real plane. As we suggested earlier, we can intersect each line once with the upper hemisphere of a circle, choosing either the left or right edge of the rim.
If we want to, we can also include the other edge of the room, and identify it with its antipode (although this is no longer a transversal). Looking carefully at it, we see a line with endpoints adjoined. This is topologically equivalent to the circle, as well as to with infinities identified.
If we didn’t mind intersecting each line twice, we could instead use the entire circle (. In that sense, we could think of the real projective line as a circle with endpoints identified.
Moving on to , the real projective plane, we start by considering the space of lines through the origin in . As before we consider a transversal: the bottom hemisphere of a sphere (with half of the rim). This can be seen in the following image.
Additionally, as mentioned earlier, we can think of the real projective plane as a sphere with antipodal points identified. In general, it is easy to see a pattern here: we can think of real projective -space as a the -sphere with antipodal points identified. This, together with our formal construction of will have to serve as the bulk of our intuition about , and one can go on to prove that the space has all the nice topological properties we expect of it (I’ll save such considerations for our discussion of complex projective space).
Unfortunately, the real projective plane cannot be embedded into , but it can be embedded in by the following map .
It shouldn’t be too surprising that this map restricts to a map on the sphere , and since it is constant precisely on antipodal points (check it yourself!) it extends to a map . Luckily, this embedding can be projected back down to as the Roman Surface.
Because I want to leave the more general discussion of projective spaces for the complex case, and because I can’t really provide good intuition for higher dimensional real projective spaces, I’ll end with a different way of defining . I leave it to you to ponder why it is equivalent to our construction.
Our approach will be diagrammatic, and (as in some of the previous images), the diagram is borrowed from Wikipedia.
The construction is as follows. We have a square piece of paper, and we need to identify the edges by folding them on top of each other. Doing this in different ways yields various interesting surfaces. For example, if we fold the top and bottom of the paper together, we get a cylinder. Folding the ends together then gives us a torus.
Let’s return to our diagram. In the middle picture, we see that the top and bottom are the same color, so we need to fold them together. However, the arrows are in the opposite direction, so we fold with a twist. No surprise here: we get the Möbius strip.
Now, if we fold the sides of a Möbius strip into a cylinder (without intersections — this can’t be done in ), we get the Klein bottle. This is demonstrated in the last panel of the diagram.
Looking at the first panel of the diagram, we see that if you fold together the ends of Möbius strip with a twist, you get the real projective plane. However, what happens if you only take the following slice of the real projective plane?
This is identical to the diagram for the Möbius strip, which makes sense. Think about what all of this means in the context of the real projective plane being the space of lines through the origin in , or the sphere with antipodal points identified.
Thanks to: dan131m , for pointing out mistakes in the article.