Homeomorphisms of toratopes

[PWrong]

I've been trying to find explicit homeomorphisms between shapes like (31) and (211) or (22) and ((21)1). I think I've figured it out. A homeomorphism is a continuous bijection with a continuous inverse. There doesn't have to be a deformation. First we want a map from the toratope to its expanded rotatope. For example,

(21) -> 22
(31) -> 32
(211) -> 32
(22) -> 222
((21)1) -> 222

Now we have a method for finding parametric equations for any toratope, I think there's a thread somewhere about that. A set of parametric equations is really a homeomorphism between the shape and a cube of parameters with some equivalence relation. Like the equations for a sphere map from a square [0,2pi]x[0,pi] onto the sphere, except that some values of theta and phi are equivalent. When you look at the equations for tiger and ditorus, the three angles have the same rules about which points are equivalent. So I think this can be the basis for a proof that all toratopes are homeomorphic to their expanded rotatopes.

[PWrong]

Anyway I'm pretty convinced that these homeomorphisms exist. In which case we can forget about homology groups for toratopes. The homology of a toratope is always the same as its expanded rotatope.

Perhaps on the list of toratopes on the wiki, rather than having a hole sequence we should just have the expanded rotatope. Then on the rotatope pages we can have all the homology groups for all frames.

[Keiji]

Well, I've changed the hole-sequence column in the list of toratopes to an expanded rotatope column. I've also constructed this table listing the min-frame hole-sequence for the min-frame of each rotatope, and rotatopes that are homologous to the other frames. a^b is the wedge sum of b a-spheres. To find the hole-sequence for the other frames, look up the min-frame hole-sequence of the corresponding rotatope.

So far there are three known entries on each row: the min-frame which is itself, the max-frame which is a point (0), and the max-but-one-frame which is a hypersphere of the same dimension.

I've also written in 2^5 for the 1-frame cube and I'm guessing 2^(2n-1) for the 1-frame n-cube.

Questions:
1. Is that 1-frame hypercube guess correct?
2. How can we find the other missing values?
3. Since it's apparently not correct to just count the numbers in the expanded rotatope, how is the hole-sequence found? I see a pattern that appending 1 to the expanding rotatope multiplies the elements in the hole-sequence by two, and I've seen a few other patterns, but I can't see anything completely general.
4. Can we generate an expanded rotatope from a hole-sequence? Does every hole-sequence have an associated shape, even if it isn't a rotatope?

[PWrong]

The table looks great, I don't know if I can complete it though. Deforming the shape into wedge sums is easy for low dimensional shapes, but it gets really hard in higher dimensions, especially for frames in the middle. I'll edit it a bit after I answer your questions. It's worth keeping, but a table of homology groups might be more useful and easier to calculate.

1. Is that 1-frame hypercube guess correct?

Not quite, but the 5,7,9 sequence is in there. The 3-frame tesseract is 3^7. You can get this by drawing the net for the tesseract and counting all the cubes but one. Similarly the 4-frame penteract is 4^7.

2. How can we find the other missing values?

The others are hard to calculate but I found some of them. I can tell you that the 1-frame tesseract has 17 lines, and a few other things about hypercubes that I'll put into the table. All of my work on this is currently in a stack of unordered loose pages .

3. Since it's apparently not correct to just count the numbers in the expanded rotatope, how is the hole-sequence found? I see a pattern that appending 1 to the expanding rotatope multiplies the elements in the hole-sequence by two, and I've seen a few other patterns, but I can't see anything completely general.

It's very difficult . I do have recursive formulas for the following:

1. Cartesian product of two hyperspheres (it's basically what you'd expect)
2. The product of a hypersphere with many lines (in any frame)
3. The n-torus T^n.
4. 3T^n, 33T^n and 4T^n.

I think when I get a few more formulas I'll start seeing the pattern. I'll post all of these formulas in a separate thread later so we have them all in one place.

4. Can we generate an expanded rotatope from a hole-sequence? Does every hole-sequence have an associated shape, even if it isn't a rotatope?

Good question. I think we'll have to wait till we have a general formula to find out. Certainly not every hole-sequence has a shape. For instance you can't have H₀X = 0, unless X is the empty set. I don't know if there are less trivial counterexamples though.

[Keiji]

1. Is that 1-frame hypercube guess correct?

Not quite, but the 5,7,9 sequence is in there. The 3-frame tesseract is 3^7. You can get this by drawing the net for the tesseract and counting all the cubes but one. Similarly the 4-frame penteract is 4^7.

Did you mean the 2-frame tesseract is 3^7 and the 3-frame penteract is 4^9? The (n-1)-frame of an n-cube is always homologous with n.

I can tell you that the 1-frame tesseract has 17 lines

As in the wedge sum of 17 circles?

Certainly not every hole-sequence has a shape. For instance you can't have H₀X = 0, unless X is the empty set. I don't know if there are less trivial counterexamples though.

Yeah, obviously if H₀ = 0 then H_q = 0 for all q. But with wedge sums and "clones" (many disjoint instances of a certain shape) it's possible to get more than just what we have listed in the table so far. There are infinitely many of those per dimension.

[PWrong]

Sorry, I don't know what I was thinking.

Did you mean the 2-frame tesseract is 3^7 and the 3-frame penteract is 4^9? The (n-1)-frame of an n-cube is always homologous with n.

Yes, you're right.

As in the wedge sum of 17 circles?

Yes, that's what I meant to say. It actually has 32 lines and it can be deformed into 17 circles.

Yeah, obviously if H0 = 0 then Hq = 0 for all q. But with wedge sums and "clones" (many disjoint instances of a certain shape) it's possible to get more than just what we have listed in the table so far. There are infinitely many of those per dimension.

You can get [1,1,1] by patching a torus with a disk in the middle, like in the other thread, but without a disk in the tube. This is also like a sphere with a pole down the middle.

If a is sufficiently large, you can get [a,b,c] from a union of points, circles and spheres. (a-b-c) points U b circles U c spheres. Similar argument for [a,b,c,d] etc.

So you probably can get every possible hole-sequence.

[PWrong]

There are objects whose homology groups aren't just copies of Z by the way.

The projective plane has
Z, Z₂, Z, 0, 0, ...

and the klein bottle has
Z, Z⊕Z₂, 0, ...

[PWrong]

You can get [1,1,1] by patching a torus with a disk in the middle, like in the other thread, but without a disk in the tube. This is also like a sphere with a pole down the middle.

If a is sufficiently large, you can get [a,b,c] from a union of points, circles and spheres. (a-b-c) points U b circles U c spheres. Similar argument for [a,b,c,d] etc.

I forgot about wedge sums. You can get all this stuff from wedge sums easily.

[Keiji]

You can get [1,1,1] by patching a torus with a disk in the middle, like in the other thread, but without a disk in the tube. This is also like a sphere with a pole down the middle.

I like this

A ball minus a smaller ball is a sphere [1, 0, 1] and a ball minus a torus is the shape you just described [1, 1, 1].

What about a ball minus two smaller balls?

If a is sufficiently large, you can get [a,b,c] from a union of points, circles and spheres. (a-b-c) points U b circles U c spheres. Similar argument for [a,b,c,d] etc.

So you probably can get every possible hole-sequence.

What about when a is not sufficiently large? Sure, you've just shown that you can get all n-dimensional sequences [a, b₁, b₂, ..., b_n-1] where a ≥ Σ_∀bb, but there are infinitely many sequences where this is not the case - and out of those, we know infinitely many are possible and there are infinitely many we don't know are possible.

I also added the 6D extended rotatopes to the table. Are my hole-sequences for 43 and 52 correct?

[PWrong]

What about a ball minus two smaller balls?

[1, 0, 2]
You could easily deform this to a wedge sum of two spheres.

What about when a is not sufficiently large? Sure, you've just shown that you can get all n-dimensional sequences [a, b1, b2, ..., bn-1] where a ≥ Σ∀bb, but there are infinitely many sequences where this is not the case - and out of those, we know infinitely many are possible and there are infinitely many we don't know are possible.

You can get any sequence with wedge sums.

e.g. [4,2,3] = S¹ v S¹ v S² v S² v S² plus 3 separate points.

[Keiji]

I assume the wedge sum adds the elements of the hole-sequences and then subtracts 1 from the first element?

[PWrong]

Add the elements, but H₀ is always just 1, because a wedge sum is connected.
So 2v2v3v4 has [1,2,1,1]


43 and 52 are correct. I'll figure out the rest soon. I get a lot of this stuff done on weekends.

[Keiji]

I've just updated the List of toratopes and Table of expanded rotatopes pages, using data generated from a program I've been writing over the past few days. Seems I couldn't count properly for quite a few of the 6D extended rotatopes, huh.

If I extend this to 9D (the maximum possible before we run into problems with base 10 ), will it help find patterns?

[PWrong]

It might not be a good idea to go much further, since I've apparently made a mistake. I'm no longer sure if H222 is 1,3,0,1 or 1,3,3,1.

EDIT: turns out it's 1,3,3,1, as I've shown in another thread.

[Keiji]

I just added the 7D expanded toratopes and it turns out that 333 is the only one so far that we need to work out.

Edit: Or not since you just worked that out. How about the Cartesian product of something with a sphere?

[PWrong]

I tried that, but I'm not sure how to do it in general. For A3 we need to find this map:

H_q-1 A ⊕ H_q A --> H_q ⊕ H_q A

I don't know if we can find this in general since H_q-1 A and H_q A are different.

I can probably work out any combination of 3's and 2's easy enough though. The next shapes after that are 433, 4333, 443, stuff like that.