Introducing Topology

Back when GM/BM first moved to ScienceBlogs, we were in the middle of a poll about the next goodmath topic for me to write about. At the time, the vote was narrowly in favor of topology, with graph theory as a very close second.

We're pretty much done with category theory, so it's topology time!

So what's topology about? In some sense, it's about the fundamental abstraction of *continuity*: if I have a bunch of points that form a continuous line or surface, what does that really mean? In particular, what does it mean *from within* the continuous surface?

Another way of looking at is as the study of what kinds of *structures* are formed from continuous sets of points. This viewpoint makes much of topology look a lot like category theory: a study of mathematical structures, what they mean, and how we can build them and create mappings between them.

Let's take a quick look at an example. There's a famous joke about topologists; you can always recognize a topology at breakfast, because they're the people who can't tell the difference between their coffee mug and their donut.

It's not just a joke; there's a real example hidden in there. From the viewpoint of topology, the coffee mug and the donut *are the same shape*. They're both toruses. In topology, the exact shape doesn't matter: what matters is the basic continuities of the surface: what is *connected* to what, and *how* they are connected. In the following diagram, all three shapes are *topologically* identical:

i-23e69bd7401c0691ff4136bc133b2776-toruses.jpg

If you turn the coffee mug into clay, you can remold it from mug-shape to donut-shape *without tearing or breaking*. Just squishing and stretching. So in topology, they *are* the same shape. On the other hand, a sphere is different: you can't turn a donut into a sphere without tearing a whole in it. If you've got a sphere and you want to turn it into a torus, you can either flatten it and punch a hole in the middle; or you can roll it into a cylinder, punch holes in the ends to create a tube, and then close the tube into a circle. And you can't turn a torus into a sphere without tearing it: you need to break the circle of the torus and then close the ends to create a sphere. In either case, you're tearing at least one whole in what was formerly a continuous surface.

Topology was one of the hottest mathematical topics of the 20th century, and as a result, it naturally has a lot of subfields. A few examples include:

1. **Metric topology**: the study of *distance* in different spaces. The measure of distance and related concepts like angles in different topologies.
2. **Algebraic topology**: the study of topologies using the tools of abstract algebra. In particular, studies of things like how to construct a complex space from simpler ones. Category theory is largely based on concepts that originated in algebraic topology.
3. **Geometric topology**: the study of manifolds and their embeddings. In general, geometric topology looks at lower-dimensional structures, most either two or three dimensional. (A manifold is an abstract space where every point is in a region that appears to be euclidean if you only look at the local neighborhood. But on a larger scale, the euclidean properties may disappear.)
4, **Network topology**: topology in the realm of discrete math. Network topologies are graphs (in the graph theory sense) consisting of nodes and edges.
5. **Differential Topology**: the study of differential equations in topological spaces that have the properties necessary to make calculus work.

Personally, I find metric topology rather dull, and differential topology incomprehensible. Network topology more properly belongs in a discussion of graph theory, which is something I want to write about sometime. So I'll give you a passing glance at metric topology to see what it's all about, and algebraic topology is where I'll spend most of my time.

One of the GM/BM readers, Ofer Ron (aka ParanoidMarvin) is starting a new blog, called [Antopology][antopology] where he'll be discussing topology, and we're going to be tag-teaming our way through the introductions. Ofer specializes in geometric topology (knot theory in particular, if I'm not mistaken), so you can get your dose of geometric topology from him.

[antopology]: http://antopology.blogspot.com/

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Yesterday, I introduced the idea of a *metric space*, and then used it to define *open* and *closed* sets in the space. (And of course, being a bozo, I managed to include a typo that made the definition of open sets equivalent to the definition of closed sets. It's been corrected, but if you're not…
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The Poincarė conjecture has been in the news lately, with an article in the Science Times today. So I've been getting lots of mail from people asking me to explain what the Poincarė conjecture is, and why it's a big deal lately? I'm definitely not the best person to ask; the reason for the recent…

If you want, I can write the intro to point-set topology - basically, how the real numbers are generalized to metric spaces, and how the concept of open spaces is then generalized.

Alon:

Thanks for the offer. I'd rather try to do it myself; feel free to correct me if I mess it up :-)

I'm most interested in structural stuff, and the basic introduction to point-set topology and metric topology can be presented that way: metric topology is built up from a generalization of the structure of the real numbers.

No problem... well, that's pretty much how I was going to present it: real numbers -> metric spaces -> open sets and continuity -> topological spaces.

Arthur Winfree started with a torus and developed the Phase-Response Curve (Look at a whole series of posts form last week's Wednesday and Thursday on my blog) from it and discovered singularity in it which earlier researchers missed, then tested it in Drosophila and stopped their clocks entirely!

I'll have to brush up on his stuff before I read your series, as I come to his stuff from outside (the biological rhythms), but I remember I managed to follow the math throughout his books ('Geometry of biological time' and "When time breaks down') so I should be able to follow your blogposts (hopefully) as well. Which subdiscipline was he in?

Does antopology have an rss feed?

Oh, Perelman is just weird, and I'm saying this as someone who uses Erdös as a yardstick. Apparently he not only refused the prize, but also disappeared at one point so that nobody knows where he is; I had a link a few days ago, but unfortunately can't find it anymore.

Perelman story was 'tracked down' by an English newspaper last weekend, Link, free sub required.

A maths genius who won fame last week for apparently spurning a million-dollar prize is living with his mother in a humble flat in St Petersburg, co-existing on her £30-a-month pension, because he has been unemployed since December.

The Sunday Telegraph tracked down the eccentric recluse who stunned the maths world when he solved a century-old puzzle known as the Poincaré Conjecture.

Grigory "Grisha" Perelman's predicament stems from an acrimonious split with a leading Russian mathematical institute, the Steklov, in 2003. When the Institute in St Petersburg failed to re-elect him as a member, Dr Perelman, 40, was left feeling an "absolutely ungifted and untalented person", said a friend. He had a crisis of confidence and cut himself off.

By Neutral Observer (not verified) on 22 Aug 2006 #permalink

Forget topology, graph theory kicks ass move onto that.

Mark: When you write "From the viewpoint of topology, the coffee mug and the donut are the same shape" you mean a coffee mug with a "handle" (a hole) and the donut, as you have pictured there in, right?

Do continue with topology rather than graph theory. Let it wait. I wanted to read topology for a long time. Now is my chance.

One small point-- Continuity (homeomorphism, actually) turns out, IMO, to be a rather weak equivalence. A lot of point-set topology is devoted to sifting through the many varieties of continuous-but-not-metric (or continuous-but-not-differentiable) classes of spaces. Once you have a metric, of course, a lot of these wierd spaces go away.

I don't understand how a coffee mug is topologically a torus. If the handle is glued on to the mug, we end up with a wall within the hollow ring of the torus. If the handle is not glued on (but is teased from the surface of the coffee mug itself), we end up with a hole in the surface of the torus. Is closing this hole legal?

Which of these two kinds of coffee mugs did you mean to illustrate?

Thanks in advance. This is all fascinating and I'm looking forward to the topology posts.

RS:

Don't think of the coffee mug as a solid body. Think of the two dimensional outer surface. The body of the cup is just a stretched out area with a depression in it. It's really a closed surface with one hole - the handle.

If you've got a sphere and you want to turn it into a torus, you can either flatten it and punch a hole in the middle; or you can roll it into a cylinder, punch holes in the ends to create a tube, and then close the tube into a circle.

Wait a sec. What does making the cylinder a tube have to do with the hole in the middle? You've created the hole by joining the ends, regardless of whether it's hollow or not. The join is just a tear in reverse.

Do topologists categorize hollow objects differently than solids? I would guess so, since a hollow 3D object has two discontinuous 2D surfaces

jackd:

The topology of the torus/coffee mug is a 2-d topology - that is, it describes *only the surface*.

A hollow 2-d object does *not* have two discontinuous surfaces. It has one continuous surface enclosing its three-dimensional volume.

Describing things in terms of clay was probably a bad idea, because it gives you the image of the three-dimensional body - the *contents* of the shape - as being the thing that we're talking about. But what we're looking at is only the surface.

So to create a torus from a sphere, you can punch a hole in the middle; or you can stretch it into a cylinder. If you do the cylinder, when you try to close the torus, you'll have two surfaces separating the two ends unless you punch holes. Those surfaces separating the ends means that you *don't* have a continuous body - it's closed at the ends.

Don't think of the coffee mug as a solid body. Think of the two dimensional outer surface. The body of the cup is just a stretched out area with a depression in it. It's really a closed surface with one hole - the handle.

Thanks, that was helpful. I now see that a coffee mug with a glued-on handle is a closed surface with a hole in it. What's inside the volume enclosed by the closed surface is not important. This type of mug is a torus.

And when the handle is not glued on (when the coffee can flow into the handle), we do not have a closed surface. Therefore this type of mug is not a torus. Please correct if wrong.

RS,

The your second example is a little more complicated but it still has at least one hole, you can still stick you finger through the handle and lift up the cup. However, if I'm visualizing this properly, you actually added a second hole through the handle. So its not a torus but some more complicated shape whose name I forget.

Drekab:

The your second example is a little more complicated but it still has at least one hole, you can still stick you finger through the handle and lift up the cup. However, if I'm visualizing this properly, you actually added a second hole through the handle. So its not a torus but some more complicated shape whose name I forget.

True that the second example has a hole in it, but it's not a closed surface with a hole in it. So as you said, it's not a torus.

I believe it can be reduced to two rings sitting side-by-side.

RS:

Yes, you're correct. If you have a hollow handle, so that the coffee inside the mug can flow into the handle, then you don't have a torus, you've got something else.

Hmm. In the hollow-handle coffee cup, I think it reduces to a sphere with two holes.

Let's morph it a bit. The body of the cup is unnecessary, so we can shrink that out. What we're left with is essentially an oval with a hollow tube stuck into it. If you straighten out the tube, it becomes a hollow stick with an arch, which can then be shrunk into a small loop on a hollow tube. Since we already know that a hollow tube is equivalent to a torus, we can morph that and end up with a torus with an additional loop stuck on to it, or a 3-sphere with two holes.

Xanthir:

Hmm. In the hollow-handle coffee cup, I think it reduces to a sphere with two holes.

There are no closed surfaces in a hollow-handle cup to begin with, so you cannot enclose volume in what you end up with.

Take a long straight hollow tube, grab each end with one hand, and bend the tube to make the ends touch and fuse partly. That's the handle, and that's what we're dealing with here.

I believe it reduces to two side-by-side rings.

There are no closed surfaces? Does a normal, non-hollow-handle coffee cup have a closed surface?

A hollow-handle coffee cup is just a normal coffee cup with a tunnel bored through the handle, making it hollow. Heck, with that definition the entire problem is trivial. A coffee cup is reducible to a torus, a hollow-handle coffee cup is a coffee cup with one additional hole, so it reduces to a torus with an additional hole, or a sphere with two handles (holes)

((Sorry I didn't put this in my last comment.))

In other words, if a hollow-handle coffee cup reduces to *two* shapes, then either it can't hold liquid, or part of it must be secured to the cup in some other manner. Since neither of these are true, the hollow-handle coffee cup is a single object. (And is reducible to a two-handled sphere.)

Xanthir:

There are no closed surfaces? Does a normal, non-hollow-handle coffee cup have a closed surface?

Yes.

A hollow-handle coffee cup is just a normal coffee cup with a tunnel bored through the handle, making it hollow.

By doing this you remove the closedness of the surface. Note that the cup is not considered to have a thickness. It is a two-dimensional surface in three-dimensional space.

I wrote:

Note that the cup is not considered to have a thickness. It is a two-dimensional surface in three-dimensional space.

Xanthir, I now realize that it was wrong of me to think of the cup as having no thickness. Your transformations are correct if the cup has thickness, and that is the kind of cup considered here. That is the kind of cup considered here because a cup without thickness would not satisfy the original transformation to a torus which started the discussion.

Here's a summary:

Has thickness, handle not hollow: torus
Has thickness, handle hollow: two toruses stuck together
No thickness, handle hollow: two rings stuck together
No thickness in body of cup, handle not hollow: weird torus with a gash in its surface and walls within its ring which confused me in the first place because I made the no-thickness assumption.

Thanks.

Hey, it can still be a 2-surface, just so long as it curves around and fills out a shape that *would* have thickness if we started giving it volume. Just like a torus is simply a 2-surface, but we can interpret it as being 'filled' with a volume.

Were you interpreting the coffee cup as a half-open cylindrical shell with a tube stuck onto it? If so, then I understand your confusion.

On a partially unrelated note - wasn't it proven that all n-surfaces reduce to an n-sphere with some number of handles (holes) and at most one cap? A cap occurs when you get something like a mobius strip, where you sort of twist across the surface. Hard to visualize, because I don't think it can happen in our physical 3-space, but simple mathematically. I believe that two caps can always be reduced to a handle, meaning that you can't ever have more than one cap in a completely reduced surface.

Were you interpreting the coffee cup as a half-open cylindrical shell with a tube stuck onto it? If so, then I understand your confusion.

Yes I was. Thanks for pursuing it.

Mark, I appreciate your patience, but let me ask something similar to what I asked before. Maybe this will clear things up for me.

You said: So to create a torus from a sphere, you can punch a hole in the middle; or you can stretch it into a cylinder. If you do the cylinder, when you try to close the torus, you'll have two surfaces separating the two ends unless you punch holes. Those surfaces separating the ends means that you *don't* have a continuous body - it's closed at the ends.

So once you punch the hole, there's no joining needed, right? You can squeeze the cylinder lengthwise until it's shaped like a washer, then round it off to a torus.

I think I understand why *one* hole is necessary; what has me confused is that your second construction seems to have two.

Thanks again for your time.

JackD - If I understand your confusion properly, you're asking why, when you take the cylinder, you have to punch a hole in both ends, right?

The answer is that, if you only punch one hole, it's no longer a manifold. A 2-manifold, like a torus, always locally appears 2-dimensional. If you take a cylindrical 2-manifold, this holds. If you punch two holes (and join them with a tube, thus making a torus), this holds. If you only punch one hole, creating a 'cup' in effect, you no longer have a manifold, because on the edge around the hole you punched space now appears 1-dimensional, not 2-dimensional. If you travel from the 'outside' of the cup to the 'inside', along the edge of the hole you make an infinitely sharp turn, which means that at points on the edge of the hole space only extends in a single direction.

So, you have to punch a hole in the surface at two points and then join the holes with a tube. This reduces it back to a single hole, and it makes the space a smooth manifold again.

At least, I think that this is right. I've only studied topology casually, so I'm shooting from the hip here. If I'm incorrect, somebody please correct me. If I'm right, does this imply that sharp corners are impossible on a manifold? Must edges always be rounded to some degree?

Xanthir, I'm not going to claim to know anything about manifold theory, but maybe it's possible to define an open manifold - that is, remove the hole as well as its boundary. For instance, if you take a sphere and remove all points north of the equator inclusive, then you still get a manifold, because for each point on the southern hemisphere, you can find a suitably small neighborhood that is contained entirely in the hemisphere.

Hmm. That's possible. I'm not sure if it would appear locally 2-d, though. I'm certain there's a strict mathematical definition of that somewhere, but I don't know it and can't apply it to this situation.

Xanthir, thanks. You did clarify something I wasn't quite picking up on, which is the nature of the "puncture" of the 2D surface of the cylinder. It does take two of them and the "join" to create the tube. Now it seems to me that the tube is equivalent to a torus already. Instead of a circular cross-section, it has an oval with very long straight sides and tight curves at the ends.