If one starts off with the notion that a hole is a missing patch (in the fabric of space)
We begin with the notion that an object without holes is piecewise construction of spherous patches. Spherous means 'topologically convex', and peicewise means that you add peices of consecutive dimensions, eg vertex+edge, edge+hedron, hedron+choron. The root form is the nulloid+vertex.
The notion of "topologically convex" allows for things like dints, inward curves etc, but not holes. Basically, it means that any set of lines drawn between two points, can be continuously merged into one, without crossing the surface. The term in the PG is 'spherous'.
The patching of the torus would be to add a circle to stretch across the hole, a second one to stretch across the tube. The intersection of these new circles creates two edges and a point, the whole is a figure for which no space of any dimension admits non-vanishing loops of any kind.
Add a sphere in the "tube"
Hyperballs (point, disc, ball, solid glome, etc) all have [1] as their hole-sequence. When you add a hyperball to a shape you subtract the hole-sequence of the corresponding hypersphere from the hole-sequence of the shape. But couldn't you just deform it so you only needed to add a hyperball of lesser dimension?
In any case, I can try G2: adding two discs makes it into a torus, so it's [1, 4, 1].
PWrong wrote:I don't think so. Take the torus, let the larger radius be 1, and call the smaller radius r. The idea is to make r so close to 1 that instead of adding a disk you can just add a point. If r = 1, it's not really a torus anymore, because it's already connected in the middle. If r < 1, you still need a disk to patch it. There's nowhere in between where you can just add a point. You could try pinching it so that you only had to add a line, but that would have the same problem.
What's G2?
In any case, I can try G2: adding two discs makes it into a torus, so it's [1, 4, 1].
PWrong wrote:Filling in the hole and the tube on the same side would give you a torus.
If you fill in both the holes, you get something equivalent to a sphere with two poles.
Filling in the part of the tube in the middle separates the shape so you get a wedge sum of two torii. This has [1,4,2], so the original shape must have been [1,5,2].
As for the missing patch theory, I think I can recover it. Our problem was with things like adding a disk to the inside of a sphere. The difference between that and adding a disk to the inside of a torus is the following. You can take the disk and deform it, without moving the boundary, until it lies entirely on the sphere. You can't do that with the patches we were talking about on the torus. Thus, I claim it doesn't count. With this restriction, I think the patch method will work as a way to count homology groups, at least for toratopes.
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