manifolds

There are too many shapes to keep track of, so topologists focus only on the important ones. Manifolds. They sound complex but they’re really not—you actually live on a manifold. Circles, lines, planes, spheres: manifolds are the smooth, simple, uniform shapes that seem to always play a leading role when we’re working with physical spaces in math and science.

They’re so simple, you’d think we would have found them all by now. We haven’t. Topologists are so embarrassed about this, they put out a million-dollar bounty to encourage people to look harder. This is the biggest unsolved question in topology, entertaining and frustrating experts in the field for over a century:

Or, a bit more accurately:

The goal isn’t to literally count them up, but to find them all, name them, and classify them into different species. We’re compiling a field guide of all possible manifolds.

So what exactly is a manifold? The rule for qualifying as a manifold is pretty strict, and most shapes don’t make the cut.

New Rule

A shape is called a “manifold” if it has no special points: no end-points, no crossing-points, no edge-points, no branching-points. It has to be the same everywhere.

This immediately rules out all those infinite families of shapes from last chapter. Anything with hatch marks or asterisks or anything like that won’t count as a manifold. That means the “how many” question might actually have an answer now: There might be an exact, finite number of manifolds. We’ll have to see.

This definition also isn’t limited to flat, wireframe-style shapes like the ones we’ve been working with. You can have manifolds made out of sheetlike material, or doughlike material. The universe we live in is probably a three-dimensional manifold, unless you think there’s a physical boundary where it just stops, or it crosses over itself somehow.

But let’s stick to the wireframe-style shapes for now, the kind you can make out of string or paper clips. In topology we call these shapes one-dimensional, even though the page they sit on is two-dimensional. It’s the material of the shape that matters.

So what manifolds can you make out of string? There aren’t that many options. Most string-shapes you can come up with have special points.

The twists and curls and corners are fine, since those can be smoothed out. The real problem is the end-points. How do you eliminate end-points?

There are only two string-manifolds. If you don’t know what they are, you can take a second now to stare off into space and think about it before turning the page.

The circle (aka S-one) and the infinite line (named R-one) are the only manifolds in the first dimension. To avoid end-points, you either have to loop back around or just go on and on forever. And don’t forget: Because all the shapes in topology are stretchy, this also covers any closed-loop shape and any goes-on-forever shape. It doesn’t have to be literally a circle or a straight line.

That’s it for dimension one. Not bad! You can see we’ve narrowed our search down a lot. The original “how many shapes” question was too big and broad, but this one seems manageable, at least so far. Ready to move up a dimension?

In dimension two, we’re looking for manifolds made out of sheetlike material. Remember, it’s the material that matters! Most of these shapes are what you’d normally consider three-dimensional, but they’re made out of two-dimensional material, and that’s where it counts.

So: What manifolds can be made out of sheet-material? We’re looking for something that’s sheetlike everywhere, with no edges or cliffs where the sheet just stops. Remember how I said you live on a manifold? The surface of the Earth is a sphere, which is a two-dimensional manifold.

With stretching and squeezing, “sphere” includes any closed surface: cube, cone, cylinder, all the hits. But be careful with your terminology! In math, “sphere” only refers to the hollow surface-shape, whereas a “ball” is filled in. A ball is three-dimensional (made out of dough material), so let’s forget about it for now.

This general sphere shape is called S-two, which makes sense because it’s like the leveled-up version of the circle, S-one. We can use the same strategy to find our next sheet-manifold. The equivalent of an infinite line, one dimension up: an infinite plane.

This one’s called R-two, and it includes any infinite surface that divides space into two infinite regions.

You know how some people think the Earth is flat? That makes sense, topologically speaking. A manifold has no special points, so every point looks identical to every other point, if you drop into Street View. There might be some curvature, but if you’re tiny enough you won’t notice. If you lived on any sheet-manifold at all, it would look (locally) like you lived on a flat plane.

And there are more sheet-manifolds than just these two. More dimensions means more freedom of movement. There are new manifolds you can build with two-dimensional material that have no equivalent string-shape.

A hollow donut is a manifold. You can tell it’s a new manifold because of that hole in the middle—no matter how you stretch and squeeze, you can’t get rid of it. But it’s a very curious kind of hole: There’s no hard edge to it. It’s not like you cut a hole out of a piece of paper, leaving a rim of special points. This donut hole is subtler than that. You can only see it from the outside. If you lived on the surface of a donut-shaped planet, you’d never notice from looking around that there was a hole. It would look, locally, just like if you lived on a sphere or a flat plane.

This new manifold is called a torus, or T-two, and it includes anything that has this type of smooth hole through it.

We’re still not done with sheet-manifolds. You can also make a double torus:

Which of course means you can make a triple torus, quadruple torus, and so on. There’s an infinite family of tori, which is the plural of torus.

Okay, so there’s not an exact, finite number of manifolds. That’s fine, we don’t need to literally count up the manifolds in order to find them all. What we’re doing is classifying manifolds. We’re looking for a list of all possible manifolds, and it’s okay if that list has some infinite families in it. A lot of times in abstract math, things just turn out to be infinite, so that’s the best you can hope to do.

Believe it or not, we’re still not done with the second dimension. There’s still more you can build out of sheet-material.

There’s just a little issue here. The next sheet-manifold I want to tell you about is very strange. I’ll tell you what it’s called: It’s the “real projective plane.” But I can’t show you what it looks like. I don’t know what it looks like. No one knows what it looks like, because it doesn’t exist in our universe and never can.

Here’s why: It needs a minimum of four dimensions to exist. Regardless of material, each shape has a minimum dimension that it can actually exist in. A plane can fit in two dimensions. A sphere needs three. A “real projective plane” needs four.

So how do we know it exists? Well, let me describe it to you.

Imagine you have a disk, which is a filled-in circle. A disk is made out of sheet-material, but it’s not a manifold, because of all the points around the edge. But if you have two disks, you can carefully stitch them together along their edges until they become one shape without any edges at all. They become a manifold.

In this case, that manifold is a sphere, which isn’t very helpful since we already know about the sphere. But this basic idea is very useful: You can take two almost-manifolds with the same boundary, and stitch them together to get an actual manifold.

So now imagine you have a thin band of sheet-material with a single twist in it. This shape might look like it has two boundaries, but it only has one, because of the twist. Follow the edge with your finger and you see it loops all the way around the top and bottom and back to where it started.

Here’s the plan. The boundary of a disk is shaped like S-one (a circle). The boundary of this twisted strip is shaped like S-one. Let’s stitch them together to build a new manifold.

If you try to imagine this in your head or simulate it with your hands, you run into problems pretty quickly. The disk has to twist around and pass through itself, which isn’t allowed (no special points). But if you had four dimensions to work in, you’d have no problem.

How’s that? Think about a figure-eight. It intersects itself if you draw it on a flat piece of paper, but if you could lift up one of the crossing lines into the third dimension, off the page, it wouldn’t intersect itself. Think that, but one dimension up. The weird, twisty manifold we just made intersects itself when we’re stuck in three dimensions, but if you could “lift it up” through the fourth dimension, you’d get a perfectly nice, smooth, nonintersecting sheet-manifold.

It’s bizarre. This is the real projective plane, or RP-two for short, and it’s unique and confusing in a couple ways. A sphere and a torus have an inside and an outside, but a real projective plane just has one side that twists to the inside and out. If you write the letter R on a sphere or torus, and slide it around through the space, it’ll always come back looking like an R. But if you slide an R around on a real projective plane, it could come back looking like an .

It’s a manifold, though, and it fits all our rules, so we have to add it to the list. There’s the sphere, plane, all the tori, and the real projective plane. Is that it?

Still no. The real projective plane comes with its own infinite family of twisty, unimaginable spaces. Just like you can smush two tori together to get a double torus, you can smush two real projective planes together to get a new manifold called a Klein bottle, which also needs four dimensions to exist without intersecting itself. Or you can smush three of them together, or four, and so you get a whole infinite family of these odd, twisted spaces.

And that, finally, is the complete list of all possible sheet-manifolds.¹

Okay, ready to move up another dimension? No, me neither. The next dimension is manifolds made of dough-like material, and even the simplest ones are impossible to imagine. Like the hypersphere, S-three, whose cross-sections are spheres. So let’s not.

You can see how classifying all manifolds could end up being one of the hardest unsolved math problems of all time. The surprising thing is just how little we know. It’s not like we made it to dimension ten and got stuck—not even close. Beyond the two dimensions we just looked at, there are question marks all over the place.

The third dimension, dough-type manifolds, is pretty well understood at this point, though it took a hundred years and a million-dollar prize to get there, and we still don’t have a totally neat and clean classification like the lower dimensions. In dimensions five and up, topologists use a set of techniques called “surgery theory” to operate on manifolds and construct new ones.

That just leaves dimension four.

I wish I could tell you what’s going on in dimension four. I’m not sure there’s anyone who really knows. It’s a weird boundary case: too many dimensions to do visually, but not enough to use sophisticated surgery tools. There are entire textbooks dedicated to what little we know about four-manifolds, and I couldn’t make sense of anything past the opening pages. A professional topologist once told me she’d wanted to work on four-manifolds as an undergraduate but was advised to steer clear.

This is particularly eerie, because many physicists think our universe is best modeled as a four-dimensional manifold, including time as a fourth dimension. If they turn out to be right, that puts some pressure on topologists to get their act together with dimension four. It’s not just that we don’t know the shape of the universe—until we finish classifying the four-manifolds, the universe might be a shape we haven’t thought of yet.