Introduction to Manifolds and Surfaces

There tends to be a lot of confusion about what exactly a manifold is on a first, second, or even third encounter with them. This post was originally written as a Twitter thread attempting to give an intuitive understanding what we we mean when we say manifold by viewing them as a generalization of surfaces. At the end we take a quick look at the classification of surfaces and some analogous questions for manifolds.This is intended to be for a completely general audience, but be ready to stop at think about some concepts, and don't be discouraged if some ideas are challenging the first time.

As you'd expect mathematicians care about shapes a whole lot. Manifolds are an incredibly important type of shape, but we need to motivate them a little to see this. Just like we don't really care about the difference between congruent triangles, in different contexts we ignore details about what we're studying to look at general properties. The most common thing we do is look at objects "up to homeomorphism" - this means that we effectively pretend everything is made of clay. This is the realm of topology. We can bend and stretch things as much as we want, but not cut, tear, or glue. So a sphere is not homeomorphic to a doughnut as we would need to poke a hole in the sphere. However we consider a coffee cup equivalent to a doughnut. We denote homeomorphism by ≅ .

Usually we want a little more structure: we only want to look at things that "locally" look like normal space: e.g. we can make a map of any part of the shape, but not necessarily a "good" map of the whole thing. This is like how world maps distort shapes. If we try map the whole world things get messed up but if we pick a small area, say a city, we can map the area out without distorting it. The roundedness and weirdness of the entire globe doesn't really matter when you're zoomed in! This is a nice condition to have in general. Shapes like this are called topological manifolds.

A nicer thing to work with sometimes is smooth manifolds- these are just like topological manifolds except we add a structure which lets us take derivatives of functions. Equivalence of smooth manifolds are called diffeomorphisms. A key difference is that they care about corners. We can form a homeomorphism between the triangle and circle by squishing the edges so the shape becomes round, so they are topologically the same. However they are not diffeomorphic. Think about a function around the corners - there's going to be a jump - so the derivative isn't defined. For most of this post I'm not going to worry about the difference between topological and smooth manifolds, but it's worth knowing that there are different ideas of manifold we can work with.

Okay so we can classify all topological and smooth manifolds right? Well.. no. The fact we can't leads to most of modern geometry/topology. Let's try anyway though! Some rules: we'll only look at connected manifolds, there's no reason not to. By connected I mean that we're looking at one shape, not two separate shapes. Plus one more technical thing: we'll only look at "closed" manifolds. These are manifolds which have no boundary and are compact: you can think of this as making sure the shape is finite, but it's not quite right. There are a few other details I'm ignoring but none of them would come up in any intuitive example in your head, so let's ignore for now.

Now in maths we like to start with the simple cases, so we'll start looking at one dimensional smooth manifolds then move up. In one dimension things are actually very easy! We can have a circle or an interval (line segment). All we have to do is straighten out the kinks in the curve. Try to convince yourself there's no other options!

In two dimensions we get a little more interesting. We call two dimensional manifolds surfaces, just like in everyday use. There's 3 "building blocks" for surfaces. Every other surface can be pieced together by gluing copies of these together. Before I give the list try think of what you think they might be, remember in topology we can stretch and bend as much as we'd like. Ready? Here they are:

-A sphere.

-A hollow doughnut (torus).

-The real projective plane.

Sticking these together gives us every possible closed, connected surface. Sticking two doughnuts together gives a doughnut with two holes for example. Okay so by sticking together I mean "cut a disk out of both and stick them together along the boundary" this is the connected sum. I'm skipping out on details but I'd really recommend you draw some pictures and play around with it.

Now what's the real projective plane? It's the space of lines through the origin in R^3... not the easiest to visualize. It looks something like a disk with its boundary glued to itself. The picture at the top of this blog post is an attempt to visualize it, I also have a blog post here motivating projective space in general.

Okay, three dimensions? I admittedly know less about this case than others but three is where things get rough. There's no tidy set of building blocks here. The Poincaré conjecture (2003) proved "Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere". This case is still a hugely active area of research, 3 dimensions is really interesting, so different tools need their own posts.

Now you'll notice I've stopped talking about topological vs smooth for all this, but why did I do that? Well it turns out in 1, 2, and 3 dimensions topological manifolds are equivalent to smooth ones! If a manifold is topological it has a unique smooth structure. In higher dimensions this does not hold... ever. In 4 dimensional manifolds are wild, we know very very little. As an example there are infinitely many "exotic R^4's". This is a manifold homeomorpic to four dimensional space, but not diffeomorphic. In every other dimension there is only one differential structure on the n-dimensional reals. However in dimensions greater than 4 something really weird happens: things actually become a whole lot more understandable! The extra wiggle room of those extra dimensions lets us do something called surgery to stitch manifolds together and classify them. A post for another time maybe.

So that's roughly where things stand now. We mostly understand things in 1,2,3?,n>4 dimensions, but 4-dimensions is unknown. So what do we do? What I'm working on is exotic manifolds - manifolds with weird smooth structure. This thread motivates a whole lot of directions though. For example algebraic topology uses algebra to understand shapes, Riemannian geometry adds the idea of length and angle to look at curvature, Morse theory adds analysis to the mix (and leads to the classification in n>4), knot theory plays with knots! usually in 3-dimensions. You can really go lots of directions after this, and I hope you do!

Some further reading:

-various wikipedia pages, starting on the surfaces page and moving forward from there.

-Milnor's Topology from the Differential Viewpoint.

-Weeks' The Shape of Space.