Higher frame toratopes

Discussion of shapes with curves and holes in various dimensions.

Higher frame toratopes

Postby PWrong » Thu Dec 10, 2009 4:55 am

Let's start with the maximum frame and use the frame operator as defined in the other thread. Suppose we can prove that, based on this definition, the frames of the product of balls of any dimension are given by a Pascal's triangle kind of pattern. e.g.

BxB
SxB U BxS
SxS

BxBxB
SxBxB U BxSxB U BxBxS
SxSxB U SxBxS U BxSxS
SxSxS

etc.

Then the easiest way to work out the homology groups of all frames would be to come up with some analogue of Mayer-Vietoris that uses unions of three or more shapes. I'll ask my lecturer if this is possible when I see him. In the meantime I can work out products of two balls.

The homology groups of X = (a+1)x(b+1) are
H X = h0
H fX = h0+ha+b+1 EDIT: fixed this equation
H f2X = h0+ha+hb+ha+b

This works for any a and b, even if a=0 or a=b.
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Re: Higher frame toratopes

Postby PWrong » Sun Dec 13, 2009 3:28 am

I've done it for 3 balls.

X = (a+1)x(b+1)x(c+1)

HX = [1]
HfX = h0 + ha+b+c+2
Hf2X = h0 + ha+b+1 + ha+c+1 + hb+c+1 + 2 ha+b+c+1
Hf3X = h0 + ha + hb + hc + ha+b + ha+c + hb+c + ha+b+c
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Re: Higher frame toratopes

Postby PWrong » Sun Dec 13, 2009 10:03 am

I think I've got something more general. First of all, I have a conjecture for cartesian products.

Define hi ⊗ hj = hi+j.
Write HA = a0 h0 + ... an hn and HB = a0 h0 + ... an hn
Then
H(AxB) = HA ⊗ HB = (a0 h0 + ... an hn) ⊗ (a0 h0 + ... an hn)
where we expand out the brackets, a bit like multiplying polynomials. This conjecture implies the other conjecture about min-frame rotopes, but it's stronger.

A consequence of this is that H(AxSk) = HA ⊗ (h0 + hk) = HA + HA⊗hk.
This implies Hq(AxSk) = HqA + Hq-kA
Negative homology groups are always the zero group, so if k>q then Hq(AxSk) = HqA.

Anyway using all this I can get homology groups of the frame of a cartesian product.
Let X = f(Y x Bk+1).
Then X = fY x Bk+1 U Y x Sk
Let A = fY x Bk+1 ~ fY and B = Y x Sk.
Then AnB = fY x Sk.

HqAnB = HqfY +Hq-kfY
HqA = HqfY
HqB = HqY +Hq-kY

I also worked out the maps from HqAnB to HqA+HqB. They use the kronecker delta function.

HqAnB --> HqA+HqB

HqfY +Hq-kfY ---> HqfY + HqY +Hq-kY
( x , y ) ---> ( x+δk,0 y , -δq,0 Σx , -δq,k Σy )

Σ simply means add up all the elements of x. I'm not 100% sure on this map, and it probably only works if Y is the sort of shape we expect it to be (so no Klein bottles or arbitrary wedge sums). There's more to come, I'll finish this tomorrow.
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Re: Higher frame toratopes

Postby PWrong » Sun Dec 13, 2009 11:38 pm

For k=0, we have

q=0:
H0fY +H0fY ---> H0fY + H0Y +H0Y
( x , y ) ---> ( x+y , -Σx , -Σy )
-->> 0 --> Z >--> H0 X

q=1:
H1fY +H1fY ---> H1fY + H1Y +H1Y
( x , y ) ---> ( x+y , 0 , 0 )
-->> H1fY --> H1Y +H1Y >--> H1X -->> 0

q>1:
HqfY +HqfY ---> HqfY + HqY +HqY
( x , y ) ---> ( x+y , 0 , 0 )
-->> HqfY --> HqY +HqY >--> H1X -->> Hq-1fY

So we have:
H0X = Z
H1X = 2H1Y
HqX = 2HqY +Hq-1fY for q>1

Note that all this only works when Y is not a min-frame. Otherwise there's problems with H0X.

EDIT: after checking some examples it looks like this is all wrong. :(
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Re: Higher frame toratopes

Postby PWrong » Mon Dec 14, 2009 5:08 am

Here's k>0.

q=0:
H0fY +H0fY ---> H0fY + H0Y +H0Y
( x , y ) ---> ( x , -Σx , 0 )
H1 X-->> 0 --> Z >--> H0 X

0 < q < k
HqfY + 0---> HqfY + HqY + 0
( x , y ) ---> ( x , 0 , 0 )
Hq+1X -->> 0 --> HqY >--> HqX -->> 0

q=k:
HkfY +H0fY ---> HkfY + HkY +H0Y
( x , y ) ---> ( x , 0 , Σy )
Hk+1X -->> H0fY - Z --> HkY >--> HkX -->> 0

q=k+1:
Hk+1fY +H1fY ---> Hk+1fY + Hk+1Y +H1Y
( x , y ) ---> ( x , 0 , 0 )
Hk+2X -->> 0 --> Hk+1Y +H1Y >--> Hk+1X -->> H0fY - Z

q>k+1:
HqfY +Hq-kfY ---> HqfY + HqY +Hq-kY
( x , y ) ---> ( x , 0 , 0 )
Hq+1fY-->> 0 --> HqY +Hq-kY >--> HqX -->> 0


Conclusion: For X = f(YxBk+1) where Y is not a min-frame rotatope:
H0X = Z
HqX = HqY for 0 < q < k
HkX = HkY
Hk+1X = Hk+1Y + H1Y + H0fY - Z
HqX = HqY + Hq-kY for k < q

Now we just have to check all these with some simple examples.
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