Holes of Toratopes

Discussion of shapes with curves and holes in various dimensions.

Holes of Toratopes

Postby PWrong » Thu Nov 19, 2009 8:12 am

Ok since homology groups and homotopy groups are complicated, I'll talk about them in the other thread, and in this one I'll define an intuitive kind of hole and talk about my conjectures. We won't be able to really prove my conjectures without real algebraic topology, but I'll try doing that with the help of lecturers at uni.

Take a toratope A. Replace the numbers with groups of 1s in brackets, e.g. 4 = (1111).
Now for each integer k > 1, count the number of pairs of brackets in A with k objects in them. Call this number A_k. We could call this the number of k-holes in A.

Conjecture 1:
If A_k = B_k for all k > 1, then A is homeomorphic to B. That is, there is a continuous bijective map from A to B with a continuous inverse.

Example: A = (211), B = (31)
Rewrite as A = ((11)11), B = ((111)1)

A_2 = 1, since only the inside bracket has two objects in it.
B_2 = 1 because only the outside bracket has two objects.
A_3 = 1 because only the outside bracket has three objects in it
B_3 = 1 because only the inside bracket has three objects in it
A_4 = B_4 = 0

Here's a list of which toratopes are homeomorphic if my conjecture is true.

(21) ~ 22 (this we already know)
(211) ~ (31) ~ 32 (I've tried to find an actual map between these, so far I haven't been completely successful. However I've shown they have the same homology groups, so they must be homeomorphic)
((21)1) ~ (22) ~ 222 (I had an argument with wendy about this years ago. Turns out she was partially right, the shapes are homeomorphic but they're not exactly the same object.)

(2111) ~ (41) ~ 42 (possibly others)
(221) ~ (32) ~ ((21)11) ~ ((211)1) ~ ((31)1) ~ 322 (quite likely others)


Conjecture 2:
Let A be an n-dimensional toratope (not a toratope embedded in n dimensions). The homology groups of A are as follows

H0A = Z
H1A = A2 Z (that is, A2 copies of Z summed together)
HkA = Ak+1 Z for 1 < k < n
HnA = Z
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Re: Holes of Toratopes

Postby Keiji » Thu Nov 19, 2009 11:17 am

Aha, now I understand :)

I'll calculate the holes for all the closed toratopes up to 5D. Open ones aren't worth bothering with as they are the same as their base. Each sequence starts counting from 2, and obviously ends with an infinite number of zeroes.
(II) ~ [1] (circle)

(III) ~ [0, 1] (sphere)
((II)I) ~ [2] (torus)

(IIII) ~ [0, 0, 1] (glome)
((II)II) ~ [1, 1] (toracubinder)
((III)I) ~ [1, 1] (toraspherinder)
((II)(II)) ~ [3] (tiger)
(((II)I)I) ~ [3] (ditorus)

(IIIII) ~ [0, 0, 0, 1] (pentasphere)
((III)II) ~ [0, 2] (toracubspherinder)
((II)III) ~ [1, 0, 1] (toratesserinder)
((IIII)I) ~ [1, 0, 1] (toraglominder)
((III)(II)) ~ [2, 1] (cylspherintigroid)
(((II)I)II) ~ [2, 1] (toracubtorinder)
(((II)II)I) ~ [2, 1] (toracubindric torus)
(((III)I)I) ~ [2, 1] (toraspherindric torus)
((II)(II)I) ~ [2, 1] (toraduocylinderinder)
(((II)I)(II)) ~ [4] (cyltorintigroid)
(((II)(II))I) ~ [4] (tigric torus)
((((II)I)I)I) ~ [4] (tritorus)

As you can see I've ordered them by their sequences. Questions:
1. Is it possible to find a sensible secondary order for closed toratopes with the same hole-sequence?
2. Is it possible to count the number of n-dimensional closed toratopes with a given hole-sequence for each dimension (and therefore check whether any exist)?
3. Is it possible to list all the hole-sequences for each dimension which apply to at least one closed toratope?
4. Is it possible to derive the group notation from the hole-sequence and an additional meaningful property? If so, what property can be used for this purpose?

Some simple observations about construction:
If you construct a n-oprism R from closed toratopic bases P1...Pn, HS(R) = Σ1≤i≤nHS(Pi) where HS(A) is the hole-sequence of closed toratope A and the sum of sequences is the sequence of sums.
If you then construct a tigroid T from R, HS(T) = HS(R) + K where K is a sequence with a 1 in its nth entry (counting from 2, of course) and zeroes everywhere else.
If you construct a torus T from a base P, HS(T) = HS(P) + [1].
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Re: Holes of Toratopes

Postby PWrong » Fri Nov 20, 2009 11:58 am

1. Is it possible to find a sensible secondary order for closed toratopes with the same hole-sequence?
2. Is it possible to count the number of n-dimensional closed toratopes with a given hole-sequence for each dimension (and therefore check whether any exist)?
3. Is it possible to list all the hole-sequences for each dimension which apply to at least one closed toratope?
4. Is it possible to derive the group notation from the hole-sequence and an additional meaningful property? If so, what property can be used for this purpose?


These are great questions, but I don't know what the first one means. I'd also add another question.

5. How many unique toratopes are there in nD up to homeomorphism? I've worked it out up to 7D on paper, and so far it's the partition function P(n-1). Which is interesting but I'm not sure why that should be the case.
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Re: Holes of Toratopes

Postby Keiji » Fri Nov 20, 2009 12:42 pm

PWrong wrote:
1. Is it possible to find a sensible secondary order for closed toratopes with the same hole-sequence?


These are great questions, but I don't know what the first one means.


Basically I'm asking whether there's any pattern in toratopes with the same hole-sequence, and if there is, what kind of properties can be used to determine an ordering for those toratopes once they are already ordered by hole-sequence.
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