I was not intending on writing a post about spheres anytime soon. I was planning on writing a bit about cobordisms, in particular 2-cobordisms. As I was working on some notes a question occurred to me, are knots objects in 2Cob? Topologically they’re homeomorphic to . It’s just a different embedding in . On the other hand what such a cobordism look like? It’s fairly easy to grasp the idea of a oriented compact 2-manifold with boundary linking together oriented compact 1-manifolds (essentially surfaces from disjoint unions of circles to other disjoint unions of circles). But knots are all tangled….how could disjoint unions of knots be boundaries of surfaces?
Mind you I’m aware of Siefert surfaces, but a given a knot, K, there exists a surface S such that which is different than starting with a disjoint union of knots. Turns out there is a notion of cobordism between knots and in fact this establishes equivalence classes of knots which form an abelian group. It looks pretty interesting and I’ll probably put up a post about it at some point though I won’t go into it very deeply. In the process of looking at the material there are some references to which I was having trouble picturing. I thought I’d take this opportunity to discuss spheres and their product with the interval, essentially: cylinders.
I tend to use analogies from Flatland when I think about higher dimensional objects. If we start with the circle and it’s cylinder we can drop them through the plane and think about how a flatlander would see them. Let’s start with the circle:
The cylinder is a bit trickier and brings up an important point when talking about manifolds. Manifolds don’t exist inside another space. They are entities in their own right. We often embed manifolds in to visualize them better but this is necessarily artificial. What does the cylinder look like? It depends. We might imagine attaching a circle to each point of the unit interval or we might imagine attaching the unit interval to each point of the circle. The ways in which we draw this are rather arbitrary as far as topology.
In both cases we have a compact 2-manifold with boundary (equal to the disjoint union of circles). One of these ‘fits’ into and one of them needs . Of course the one on the left is what we usually think of when we talk about cylinders and the one on the right is an annulus (you’ll notice that I had to do a little stretching to make the annulus work). Let’s drop these through flatland.
On the left is what A. Square would see for a cylinder falling through his world. On the right is what he’d see when the annulus falls through it. This is dependent on the orientation relative to Flatland. Keep these in mind. We’re due to go up a dimension.
Dropping the 2-sphere through flatland,
This was how A. Square saw it in the classic Flatland! What does the cylinder made with the 2-sphere look like? In principle we should be able to view this, unnaturally, embedded in but there should also be a way that requires ‘extra’ room like the cylinder formed from the circle which was embedded in even though there’s an embedding into the plane. Just as before we could take the unit interval and attach a sphere to each point along it or take the 2-sphere and attach a unit interval at each point. Just as before, the way these are depicted or drawn is very arbitrary, topology says nothing about sizes or angles.
As you can see on the left I’m a little stuck on the version of that needs an extra dimension (so would be embedded in ) and so the ‘rest’ of the spheres are in a fourth dimension. There is an embedding into which is essentially an annulus in 3D. We can still get a feeling for the 4D version by dropping it through but what we’ll see is just a sequence of 2-spheres just like A. Square saw a sequence of circles as we dropped the cylinder made with the circle through his plane. Let’s drop the 3D annulus through the plane and see what that looks like.
So A. Square sees cross sections of the 3D annulus which go from a point, to a disk, to a series of annuli and back again. Likewise we can drop the other cylinder through which will look like a constant sphere (for the period of time it takes the cylinder to finish passing through our space). With these analogies in hand let’s tackle . This is harder to draw since there’s no embedding in 3-space but it’s not hard to think about. Take a look around you, that’s basically as long as it’s large compared to you. But if you tend to take long walks (or intergalactic voyages) you might find yourself returning to your point of departure unintentionally. If we drop through 3-space we’ll see a sequence of spheres:
Once again we can make cylinders. One of them will be the interval with a copy of attached at each point. This 4-manifold will sit in so I’ve got nothing. We can drop it through 4-space and it’ll be a constant series of identical copies of but that’s not terribly easy to see. So we’ll consider the four dimensional annulus where we attach a copy of the interval to each point of . Now we drop that through 3-space:
Essentially the ‘bottom’ of is a series of balls and then we’d start seeing the 3D annuli come through. This is important intuition for understand knot cobordisms. In that context we’ll view a knot as an embedding of in (Not sure why other than is sort of like a compact version of ). Then defining the cobordism involves submanifolds of that are homeomorphic to . This is the un-generalized version from The Encyclopedia of Mathematics entry on the topic.
This is hardly the last word on spheres which is why I labeled the post “Spheres-1.” Spheres of very many dimensions turn up in statistical mechanics, in the phase space of gases for example. By very many dimensions I mean stuff on the order of and the geometry of these spheres is very interesting. In fact there is a lot of nonintuitive geometric curisousities in vector spaces of many (but finite) dimensions. If the spheres are infinite in their dimension we have other interesting results such as the fact that many (all?) Banach spaces are homeomorphic to their unit ball!
Spheres are also interesting algebraically, though I know almost nothing here. can be thought of as . This has lots of consequences that I’ve yet to follow up on yet. I know that you can also think of as the unit quarternions for example and that forms a double cover of the group of rotational symmetries of . So much to learn!
EDIT (1/27/2014): Knots are not in 2Cob. I’ve only just appreciated that a knot is a pair: (space embedded, embedding). For example, what if you embedded the circle in ? There’d be no knot. We can only knot circles in 3-manifolds (and I can say little outside of the 3-manifolds and . So, to talk about compact 1-manifolds are to talk about disjoint unions of circles. To distinguish between circles and knots we need to talk about spaces and functions that are not relevant to 2Cob. I’m certainly not done with this train of thought, but it’s evolving.