Introduction to Manifolds and Surfaces

It's the start of a new term so time for some quick posts!

Here's something that always annoyed me as an undergrad. We talk a lot about the difference between the category of smooth manifolds and the category of topological manifolds but the exact meaning gets brushed over. You should probably know about topological manifolds and some calculus going into this. Here's my post on topological manifolds.

Okay, so quick recap on topological manifolds: topological manifolds are "Locally Euclidean, Hausdorff, 2nd countable topological spaces" a mouthful, but the point is that at every point we can zoom in so that it looks like R^n. Now let's be a little more precise about what we mean by zooming in- around any point we can continuously map a patch into a region of R^n. Think of this as literally making a map, you can stretch and twist it but you can't skip a chunk or suddenly jump all over the place.

An important property is that we can go either way with these maps, they are continuous maps which have continuous inverse. We’re going to call these maps into R^n coordinate charts, or charts, or sometimes patches. Lots of terminology for the same thing.

That's us for topological manifolds. Now we want to do calculus on them. Think about how this could work given we know how to do calculus on R^n. You'll come across an issue - if something is smooth in one chart how do we know it's smooth in another? The trick is to add a "smooth compatibility condition" between the charts. The picture here helps a lot. We have two maps from patches of our manifold to regions of R^n - but if we invert one of these maps and compose them we have a normal function from R^n to itself!

Follow the diagram to check you know what's happening. Now, the composition between charts (transition map) is a real function. We say the charts are smoothly compatible if the transition function is smooth with a smooth inverse (a diffeomorphism). If we cover the manifold in smoothly compatible charts we get an atlas for the manifold. That is, we have a chart over each point which has a smooth transition map with every other chart it overlaps with in the collection. Let's look at a couple atlases over the real numbers:

We'll call A the atlas for R given by the single chart (R,id) - we just map every point to itself. Check for yourself that this is a real atlas. Now consider B where we take as charts an interval around each point for all points of R. This is an atlas. What about A U B?

From those examples it seems like there's going to be a whole lot of atlases which are very similar, so let's restrict a little bit. A smooth structure is a maximal smooth atlas. What we do is lump in every possible compatible chart we can! So A and B give the same smooth structure.

Now, finally - a smooth manifold is a topological manifold equipped with a smooth structure. All that means is that we take a topological manifold and tell you how to patch together the charts to make functions on it smooth. But how many smooth structures does a manifold have?

One last example from R. Let C be the smooth structure given by the single chart R and the coordinate map given by x -> x³. This structure isn't compatible with A and B above - the transition map is the cube root function, which isn't smooth at 0! It actually gets much worse.

Not ideal if we want to classify smooth manifolds! We need a looser definition of when two smooth manifolds are equivalent. Most courses cover the next bit but don't go back to explain the connection. First I'll give a quick motivation for what we want to do then give some definitions. So we had two charts, φ:U→R, θ:V→R, and said we needed φoθ^{-1} to be smooth. That is we map from R^n into M with the inverse of θ, then go back to R^n with φ.What if instead we could squish around our copy of M in the middle of the composition? We use the inverse of θ to go from R^n to M, take F:M→M to mix up M, then use φ to get back to R^n. You can view the original smooth compatibility condition as the case with F the identity. We can't just use any map, we need an idea of smooth maps, then we'll have a good idea of equivalence between smooth manifolds.

A continuous map F:M→N between two smooth manifolds is smooth if we can take any point p and produce charts (U,φ) around p and (V,θ) around F(p) such that the composition θoFoφ^{-1} is a smooth real function. Really we just lift the definition for smooth real functions up to manifolds.

Right, so you can probably guess the definition now - a diffeomorphism is a smooth map with smooth inverse, and any two manifolds are considered to be equivalent if there is a diffeomorphism between them. Now we have the intuitive idea of sticking F in the middle of the transition function from before formalised. We can see that the two distinct smooth structures for R from earlier give diffeomorphic manifolds- just use the diffeomorphism F(x) = x³ check the details to see exactly how it works!

Suddenly our many different smooth structures on each topological manifold collapse into only a few smooth manifolds. Now we can actually ask the question - are there topological manifolds with multiple smooth structures which give non-diffeomorphic smooth manifolds? We call these exotic manifolds. In dimensions 1-3 there are no exotic structures, however in 4 dimensions we have many exotic structures. There are uncountably many exotic R^4s. I work with exotic spheres, the first one that was discovered was in 7 dimensions, but since many more have been discovered. One cool example is E_8. This is a 4 dimensional manifold with no smooth structure. There are lots more examples with some really interesting tools used to study them. I'll probably write about them another time.