Introduction to Homotopy Theory
This post aims to explain how topologists tell spaces apart to a general audience. The only prerequisite is knowing that topology is geometry without an idea of size or angles: shapes are like clay, we can bend and stretch them as much as we want, but not cut or tear them. A natural question is how do we tell spaces apart in topology if things are so flexible? Let's look at a few different ways and see if we can form a general theory. Some of these ideas I'll pick up in later posts. So if you play around with shapes without cutting, tearing, or gluing them, you'll pretty quickly notice that we can't fill in holes/gaps. e.g. a circle is different to a disk, a doughnut (torus) is different to a sphere, two shapes can't be pushed together into one.
A coffee mug is equivalent to a torus.
A sphere is not equivalent to a torus.
So we might hope to classify spaces by how many holes they have. Now we need to consider:
-What do we mean by a hole?
-How do we do find holes?
-What topological information does this ignore? Are spaces fully explained by their holes?
Okay first: we're clearly talking about different things by holes depending on context. One way we could decide a hole's type is by looking at what we need to fill it in: a 0-dimensional hole is filled in by a line, a 1-dim hole is filled by a disk, etc. The pictures will explain more than words possibly could:
A zero dimensional hole is a gap which we can fill by adding a line. Another way to say this is a zero dimensional hole means our space is disconnected.
A one dimensional hole is one which we can fill by adding a disk. i.e. a one dimensional hole is filled by adding in a 2-dim shape to the gap.
A three dimensional hole is one which we can fill by adding a volume. i.e. a two dimensional hole is filled by adding in a 3-dim shape to the gap.
Wait so 1-dimensional spheres (circles) have one 1-dimensional hole, and 2-dim spheres have one 2-dim hole. What if an n-dim sphere captured the idea of an n-dim hole? It definitely works for 0-dimensions: x² = 1 has two points as solution, two points have a 0-dim hole between them. Why not try to use spheres to characterize holes? A common idea in math is to study objects by looking at maps into or out of them instead. So let's look at maps into spaces, in particular maps from n-dimensional spheres to topological spaces. What does this look like? We can think of a (continuous) map from space X to Y as placing a copy of X in Y, but squishing it around a bit. e.g. we can map a circle to a plane by drawing it as is, or we could twist it around a bit. Drawing the graph of a function is a special case of this, we take a copy of the real line and wrap it up in the plane. Here's some pictures of the idea:
Mapping a circle to a plane corresponds to drawing a loop on the plane. Note that we can unravel the loop in the second picture so that it is just a normal circle, and we can shrink this circle down to a point. A good way to think of this is having a piece of string on the surface.
Now, what do you notice? There's lots of different ways to map spheres into shapes, but most of them are topologically equivalent: if you draw a circle all twisted up on a page you can deform it into a point while staying on the page. What if there was a hole in the page?
There are two different sets of maps into the torus which aren't contractible to a point: if we wrap our imaginary piece of string around either ring of the torus we can't pull the string tight into a circle, it encloses the hole! This is a good sign that we've hit the right idea. One note is that we can wrap the string around the torus as many times as we want and it still isn't contractible.
Mapping the sphere into the torus is hard to draw, but hopefully you can see the idea that we can wrap the sphere around the "skin" of the torus.
So the claim is that (up to topological equivalence) these maps characterize the holes of spaces. e.g. for the torus any circle we draw on it can be deformed to a point, unless the circle encloses one of the two holes. Also note we have one of these sets in each dimension. So say we can work out the non-contractible maps of each dimension n for a space X, what do we have? First I really should say that each of these sets of equivalence classes of maps forms a group π_n(X) - we can add maps together by doing one then the other. For example in mapping the circle to itself we can wrap it around itself as many times as we want. We can also wrap it clockwise or counterclockwise: we say that π_1(S^1) is the integers. Can you see what happens for the sphere and torus? The easiest example to think of is the fundamental group π_1, this is often defined as the group of loops in a space.
For a circle we can wrap a loop around the circle as many times as we want, in either the clockwise or counterclockwise directions. For the complex analysis inclined it's worth thinking of the map z → z^n on the unit circle.
Just as for loops, we can wrap the sphere around itself as many times as we want. In general we have π_n(S^n) is the integers. This is exactly the claim that brought us here: an n-dimensional sphere has an n-dimensional hole.
We call these groups the homotopy groups. Now does knowing these groups classify topological spaces? (Un)fortunately not. As an example the cylinder and Möbius band have the same homotopy groups but are clearly different, the band has a twist. That sucks, but we do know if any homotopy groups of two spaces differ they aren't topologically equivalent. Another question: can we compute these groups easily? Nope. It's actually really hard when n>1, let's look at one reason why.
Generally for them to be easily computable we would like a set of building blocks which we know the homotopy groups of from which we can build up more general objects. In algebraic topology the usual building blocks are spheres. We started by trying to look at n-spheres as having one n-dim hole and nothing else, but n-spheres can actually have non-trivial homotopy groups for groups other than the n'th one! There are non-trivial maps from higher dimensional spheres to lower ones dimensional ones. In this way our idea of the homotopy groups only describing holes in the space falls through, they actually detect more. The first such example is the Hopf fibration from S^3 to S^2:
One way to view the Hopf fibration is to look at the image of circles under the map. What we see is a kind of spiral pattern.
This is a common way to view it, circles get twisted up around themselves.
This table shows that "higher" homotopy groups are actually quite common! This is a very good counterexample to the beauty and natural order of mathematics.
Well if everything is so complicated why bother?Three reasons:
1)Trying to work with these things has led to a whole lot of amazing math, and we've learned a lot about shapes through it.
2)π_1 is actually not *too* hard to compute and tells us a lot
3)Thinking this way is nice
I'll wrap this thread up with some comments and questions, but will continue the story later.
-See if you can figure out what π_1 should be for some spaces, also we call π_1 the fundamental group.
-Usually we look at maps with a basepoint i.e. we pick a point on the sphere and a point in our space and make sure the point on the sphere maps there. When do you think this matters? hint: think about 0-dimensional holes.
Also I never said why we should care that we have groups, even though it hurt me. Groups are beautiful and easy to work with. We can classify and play with groups much much easier than spaces, so seeing groups is exciting. Groups have a lot of things we can do with them: products, quotients, etc. Spaces also have this: we can take disjoint unions, smash products, quotients, etc. One thing that would be amazing is if we could show these operations transfer over using homotopy groups. e.g. if say, the homotopy groups of the (wedge) sum of two spaces was the (direct) sum of the homotopy groups. In category terms we are trying to relate limits, colimits, products, etc. between different categories with a collection of functors between them. Turns out there are lots of results like this, the first people usually learn about is Van Kampen's theorem, but that's for a different post.
Finally there is actually a version of this which is a lot easier to compute: homology groups. The core idea goes back to the start: an n-dim hole is a boundary of some n+1 dim space which isn't filled in, so the set of holes is the full set of boundaries minus ones which can be filled in in the space. More about these groups in the future!