Fundamental Group Fundamentals.
May 6, 2010
Prereqs: You might want some point-set topology, and a basic introduction to algebraic topology. This post is not good to learn from (for that sort of thing, I recommend the wonderful Hatcher text, which essentially starts where point-set leaves off, or the lectures by john baez which are wonderful but contain a little bit more category theory than is perhaps necessary to a beginner.) but is just a review for people who have seen algebraic topology but perhaps forget a little of what went on.
After the jump.
In Algebraic Topology, the first nontrivial thing that we investigate is the fundamental group, affectionately denoted . Give a topological space X, we can ask a lot about what properties X has. For example, does it look like a torus (a donut)? Does it look like a sphere? Can we squish it around and make something nice?
In topology, we call a set an open set if it has helped us define the topology on X; but what does this really mean? We use open sets to distinguish what points are “close” to one-another. If two points are in an open set together, we can say that these points are “close”, compared to points that may not be in this open set. This obviously makes more sense for proper open sets, but the entire space being open means that all of the points are “close” to each other in the way that they are all in the space X, which is nice to know. Let’s consider a very specific example: the real line, .
The open sets for the standard topology on are open intervals, arbitrary unions of open intervals, and finite intersections of open intervals — but, for now, let’s consider, specifically, the open sets which look like the interval . In this case, the points between the points a and b are considered “close” to each other. In fact, if we were to contract this interval to half its size, or expand it to triple its size, then we wouldn’t lose anything: these points are so “close together” that we can essentially substitute some points in the open set for other points in the open set; we can even “squish points together” to make our interval smaller. This is obvious when we consider the fact that the open interval is able to be continuously deforms into a point (!) or continuously extended to become the entire real line (!!).
Now, let’s go back to considering our space X. Given some relatively small open set of X, we can usually make it a little bit bigger or a little bit smaller in the same way. If the open set looks like a little circle, we can make it into a much bigger circle, and we can often even shrink it to a point.
But what happens is we take X to be a torus (that donut shape!) and we consider the open set to be a big rubber band shape that is fitting snugly on the torus and going through the hole in the center. Can we shrink this rubber band down to a point on the torus continuously? This is like asking the following: can we pull the rubber band off of the torus without cutting it or breaking the torus? No, we can’t. Try it if you don’t believe me.
Another, very similar, analogy is the following: pretend we’re on the xy-plane. Let’s say I put a rubber band down and hammer a nail in the hole in the middle of the rubber band. Can I get the rubber band away from the nail without picking it up? No. We can stretch it really far, but the nail is always holding it there.
This idea in topology makes it difficult to “contract” open sets down to the size of a point. Points in the open set are still “close”, but now they’re somehow not “close enough” to contract down to a single point continuously — the hole (or nail), which is not even part of the set!, is stopping us from doing this.
So what types of spaces have this kind of restriction? We can measure the number of restrictions (that is, the number of holes) by doing some clever calculations. The idea is to just take a lot of rubber bands (some which are very big!) and throw a lot of them on the surface. If there are any holes, we won’t be able to pull the rubber bands through them continuously. So, yes, in theory, we’d have to do this an infinite number of times to see if we’ve “gotten all the holes”, but we can do some clever calculations using some higher math and make it easier on ourselves. That symbol that I mentioned at the beginning of the post, tells us what kinds of ways we can manipulate our rubber bands.
The most basic calculations are:
where is the sphere, and is the circle. This is sort of obvious when you understand what it means: for any rubber band on the sphere, since there are no holes, we can pull the rubber band here, there, and everywhere. So any rubber band shape can be reformed into any other one on the sphere. If you have a rubber band on the sphere and you want it to be somewhere else, you can move it there — there are absolutely no restrictions! This is why we say that the fundamental group is equal to 0: it means that we can do just about anything to this rubber band because there are no holes.
With the circle (and note that this is not a filled in circle, it’s just the outside boundary), we can wrap the rubber band around no times, one time clockwise, two times clockwise, fifty times clockwise, one time counterclockwise, two times counterclockwise, and so forth. No times corresponds to the number 0, clockwise moves correspond to the numbers 1, 2, 3, 4, … in the obvious way, and similarly, counterclockwise moves correspond to the numbers –1, –2, –3, and so on. It’s nontrivial to prove, but it’s relatively obvious that we can’t take a rubber band that’s going around the circle twice clockwise and make it into one which is going around the circle once counterclockwise without ripping the rubber band or the circle — so all of these numbers correspond to distinct ways that we can wrap the rubber band. Note that when I say “wrap”, what I mean is that we sort of have a strip of rubber that we wrap around the space and after we’re done we glue the stray edges together to make a complete rubber band.
Let’s do one more before we complete this post. The torus, which is a donut: we can wrap the rubber band around through the center or around the top (check out here for the pictures of the two different ways to wrap the rubber band). So we can wrap each type of this around counterclockwise or clockwise any number of times. We can even combine these two types: maybe our rubber band goes through the center, then around, then through the center, then around, then around again, then through the center counterclockwise, and so on. Each kind of rubber band configuration corresponds to an ordered tuple where we have wrapped around a times and through the center b times — note that this is not an interval of the real line, it is an ordered set of points, like the ones we would graph on the xy-plane. Therefore, for example, (1, –3) means we wrap the band around 1 time clockwise and through the center counterclockwise 3 times. It’s nontrivial to prove that, say, (1, –5) isn’t the same as (2, –4) or any other ordered pair, but it’s true. So we say that the fundamental group of the torus is a tuple of integers: in other words, it’s the group of all things that look like where a and b are integers. Cool. So, we write that like this:
which means exactly what I said right before writing that.
Even with such a weak notion as “how many holes does something have”, we can prove a wide variety of theorems that are otherwise tedious or difficult to prove using other methods. We will prove the fundamental theorem of algebra using some of these methods at some point in the future, and the fixed-point theorem is a nice, easy way to apply this kind of thinking.
To sum up: the fundamental group of a space X, denoted , counts “how many holes” are in the space X, and reports this to us in a somewhat cryptic way: specifically, it tells us all possible ways we can put rubber bands, or loops, down on the surface to create different things which are not continuously deformable to things we already have. We also now have stated a few fundamental groups of spaces, but we haven’t proved their correctness: the sphere’s trivial fundamental group is straightforward, but the circle’s fundamental group takes a lot of work to find and prove correct rigorously; the torus’ fundamental group comes from a sweet and powerful theorem I will prove here soon!
- What is the fundamental group of the xy-plane minus two distinct points?
- A circle that is “filled in” is called a disk. What is the fundamental group of a disk?
- What about a sphere that’s “filled in” (that is, one that is not hollow)?
- What about a torus that’s “filled in” (that is, a non-hollow donut)?
- What do you notice about the filled-in torus and the circle? Why do you think this is?
- What is the fundamental group of the circle minus a single point?
- What is the fundamental group of the sphere minus a point? What about a sphere minus two points? What about a sphere minus three points? Which one of these is similar to the space in question 1? Why do you think that is?