An Example (part 2)
We follow yesterday’s example of an interesting differential form with a (much simpler) example of some -chains. Specifically, we’ll talk about circles!
More specifically, we consider the circle of radius around the origin in the “punctured” plane. I used this term yesterday, but I should define it now: a “punctured” space is a topological space with a point removed. There are also “twice-punctured” or “-times punctured” spaces, and as long as the space is a nice connected manifold it doesn’t really matter much which point is removed. But since we’re talking about the plane it comes with an identified point — the origin — and so it makes sense to “puncture” the plane there.
Now the circle of radius will be a singular -cube. That is, it’s a curve in the plane that never touches the origin. Specifically, we’ll parameterize it by:
so as ranges from to we traverse the whole circle. There are two -dimensional “faces”, which we get by setting and :
When we calculate the boundary of , these get different signs:
We must be very careful here; these are not vectors and the addition is not vector addition. These are merely points in the plane — -cubes — and the addition is purely formal. Still, the same point shows up once with a positive and once with a negative sign, so it cancels out to give zero. Thus the boundary of is empty.
On the other hand, we will see that this circle cannot be the boundary of any -chain. The obvious thing it might be the boundary of is the disk of radius , but this cannot work because there is a hole at the origin, and the disk cannot cross that hole. However this does not constitute a proof; maybe there is some weird chain that manages to have the circle as its boundary without crossing the origin. But the proof will have to wait.