How to 'See' the 4th Dimension with Topology

In dimension four, all of your intuition starts to just completely go away. And in fact it's worse than going away because you still have the intuition and it's wrong. It's really bad to guess what's true in dimension four. Mathematicians often specifically study four-dimensional topology because there's a lot of behavior in dimension four that's very different from lower and high dimensional settings. It's also just the lowest dimension that we don't understand very well yet. So easy to make a mistake. Every time I finish a paper I have like three days of like, but is it true? We have to ignore our intuition and proceed directly from axioms and definitions to theorems. So what is four-dimensional topology? Topology is the study of abstract spaces. These spaces can have wild behavior. So we tend to focus on category of spaces that are slightly more motivated by the physical world. These are called manifolds, which are really symmetric spaces. A manifold is any space which, if you zoom in close enough, just looks exactly like flat what's called Euclidean space. Normal space that you learned about in school, a line, a plane, three-dimensional space, and so on. When you zoom in close enough you can't tell the difference between any two manifolds, but when you look at the global structure, zoom out, look at the whole manifold all at once, they can have wildly different properties. A one-dimensional manifold or a one manifold would be a manifold where at every point I can zoom in and it looks just like a line. One example would be a circle. A circle isn't a line, but if I zoom in really close it looks like a line. Two-dimensional manifolds are much easier to imagine. Their surfaces, things like the boundary of a ball is called a sphere. The sphere, like the surface of the earth, it's a two-dimensional manifold because even though the earth is not flat, it's very difficult to tell that. If you zoom in really close, you only see a very small portion of the earth. It looks exactly like a flat plane. Like let's say you're an ant and you're sitting on a basketball. At any point I can go forward-backward, I can go left-right, and that's it. You can't tell that that surface is not just flat and goes on for forever. In contrast, a three-dimensional manifold is more like the space we live in now. We're all living inside of a three-dimensional manifold. At least I can only see a very small portion of the universe, and it's very hard to tell if it's just real space that goes on for forever. But that does mean that it is a three-dimensional manifold, even if it's hard for me to know the global structure of it. And so it's three-dimensional because at any instant, there are three different perpendicular directions that I can move in. I can go forward-backward, I can go left-right, and now I can also go up-down. So that's different from two-dimensional space. A four-dimensional manifold is like our three-dimensional space except there's a new perpendicular direction. So in a four-dimensional manifold, starting at any point, you can go forward-backward, you can go left-right, you can go up-down, and now there's a new dimension that I don't have a good name for, something like forward and backward in time. And it's totally perpendicular to space. It doesn't change your position in space. Imagine you're in a movie, and this is time zero. The movie hasn't started yet. You have this three-dimensional space around you. And now you have your remote control, and you're going to press play on the movie, and you can see in the bottom of the screen the time of the movie, the run time. So right now it's time zero, but time is going to start increasing. And so if you pause at any moment, you still have a three-dimensional setting, but now that time value has changed. This is actually a four-dimensional space because time is itself a different direction. So a four-dimensional manifold is any object that locally, if you zoom in close enough, looks just like standard four-dimensional Euclidean space. Four-dimensional topology is a study of these four-dimensional manifolds. There are a lot of them, and we develop new techniques for how do we study these objects. What are the connections between four-dimensional manifolds and physics, or partial differential equations, or complex geometry? So as an example, let's say you conduct a scientific experiment, and you have a data set, and you measure four different variables. Then your data lives inside of a four-dimensional space because you have these four numbers that can all be different from each other. It doesn't have to mean a physical space. In my office, I very often draw pictures of four-dimensional manifolds, or I do that by drawing a sequence of three-dimensional pictures because that's, I think, just logically the best way to do it. But it's important to remember that even though in our drawing or our imagination, it seems like this fourth direction is special, the time direction, we have to remember that topologically, all of the directions are the same. To a mathematician, there's complete symmetry. Let's look at one specific example of a four-dimensional manifold, an interesting one that isn't just standard Euclidean space. In order to get there, we need to start off by understanding some lower-dimensional manifolds so that we can go up by analogy. So first, the very lowest dimensional manifold I'm going to consider is a circle. One way that you could construct a circle is to start out with, in fact, a line segment, which doesn't look like a circle, but I'm going to add the rule that if I am walking along this line segment and I go off the left edge, then I need to come back on the right edge and vice versa. So even though these two points look like they're different points, I'm just declaring that actually these are the same point now. If I do that, you can imagine taking a piece of string, gluing the ends together, you get a circle. So this is the same thing as a circle. All right, I'm going to use this idea to construct a two and a three manifold very explicitly, and then we're going to get a four manifold by analogy. So here's our two manifold, a surface. So this is a picture of a torus. This is the surface of a donut. It's not the inside of the donut, it's just like the frosting on the outside, so it's two dimensional. And one way that I could construct the torus is I'm going to start out with the analog of a line segment. In dimension two, that would be something like a square. It seems different from a torus, but what I could do is exactly the same idea here. I'm going to declare if I go off the left edge, then I'm going to come back on the right edge and vice versa. So even though these two points look like they are different points, I'm going to declare that now these are the same point. So imagine taking a piece of paper and gluing the whole left edge to the whole right edge, and if you did that, you would get a tube. Well, that takes care of some of the boundary, but the square has more edges than the line segment did. So I'm going to do the same thing and say, oh, if I go off the top edge, I'm going to come back on the bottom and vice versa. So these two edges are actually the same edge, which means here I'm going to take these two edges and I'm going to fold my tube so that these get glued together as well. And if I take this tube and I fold it, I actually get a torus. So you can see that this is the same object that we started out with. So maybe you can see where this is going. I'm going to repeat this idea. The three manifold is the three torus, and I'm not going to attempt to draw a picture because this is an object that does not fit inside three-dimensional space. We cannot imagine the whole thing at once like we can with the torus and the circle. That's why it's important that we did these by analogy. First, I would take the analog of the line segment or the square, but in three dimensions, which is a solid box, and I'm going to add the same rules. I'm going to say if I go off the top, then I come back to a point on the bottom. So even though these squares look like they are different squares, I just declare that actually all those points on the top are the same as the points on the bottom. It's like video game logic. If you run off the screen, you come back. Okay, I'm going to do the same thing for the left and right faces, which are opposite each other. Just declare that actually those are the same. But the front and back faces of the cube, which are opposite of each other, I'm going to declare that those are also the same. So it's totally analogous to what I did here. If I do that, I'll still have a three-dimensional object, because it was always three-dimensional, but now I won't have any edge, because I got rid of all the edges by gluing them together. So this is a closed three-dimensional manifold called the three-torus. So the first one that's hard to visualize, but we did all of this careful work by analogy, is the four-torus. So I can take a four-dimensional box, and we know how to draw those, because we talked about movies. I'm going to take what seems like a three-dimensional box, and add my time dimension. So this is the box at time t equals zero, and I have the box at time t equals one, and let's say t equals two. And maybe I'll stop there. Maybe I'll just have time go from zero to two. Okay, and now I'm going to glue all of the opposite faces together to get a nice closed manifold. If I'm at a point in this four-dimensional object, and I run off the top, which could happen at any time, then I need to come back at the corresponding point on the bottom. So I'm going to glue all the tops to all the bottoms. And I'm also going to glue all of the rights to all of the lefts. And then I'm also going to glue all of the fronts to all of the backs. And the last thing I have to say is, well, what about this time direction? If I'm a person living in this four-dimensional space, and I go all the way up to the last time, t equals to two, and try to keep going into the future, I don't run off the space. Instead, I come back at time t equals zero. So I just come back on the opposite side of time, just like I did for the space directions. Okay, and that rule gives me a four-dimensional manifold called the four torus. Four-dimensional topology is a special area of topology because there are techniques that work in high dimensions that fail, specifically in dimension four and below, and certain coincidences that hold in dimensions one to three that stop holding when you move to dimension four and above. For example, with four-dimensional manifolds, this is the first setting where our notions of continuous and smooth equivalence start to differ. The difference between smooth and continuous equivalence is easiest to imagine in really low dimensions, like think about a circle. A circle is smoothly equivalent to an oval. We can stretch the circle, just grab the sides and pull it, so it's not a symmetric anymore, it's an oval. But at no point do I ever stop the circle from being a nice smooth curve. It just gets longer. But there's another way of turning a circle into an oval. It takes more steps, but I could first pinch the circle and make corners and turn it into a square. I don't have to break the circle to do that. I just fold it up, make it a square, and then I stretch the square out to be a rectangle and smooth the corners back out, pat them with my hands, and turn it into an oval. But it turns out that when we go into higher dimensions, at least four, sometimes there are ways of turning one manifold into another where I have to add corners and I can't just avoid them. And so these spaces might be continuously equivalent but not necessarily smooth. There are lots of four-dimensional objects, infinite families of four-dimensional objects that are continuously but not smoothly equivalent. So that's the first really different behavior, one of the main aspects that people study, these two notions of equivalence. The big open questions, the things that I think are really interesting are giving new examples of weird behavior. Does there exist a mathematical object with this property? And if the answer is no, then that's great, and I would like to understand that explicitly. But if the answer is yes, something I often do is try to create the specific example, like just a literal picture of it. I really like that direction of four-dimensional topology.