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