Homology of m-frame n-cubes

[PWrong]

So I thought that homology groups of simple shapes like cubes and cylinders would be easy, and they are most of the time. But if you think about it, something like a wire-frame cube has a lot of holes in it. Today I was looking at homology groups of things like a wireframe cube, and there's a lot of interesting complexity. So I've done all the homology groups for m-frame cubes up to 4D, and also worked out all the frames of the 5D cube except for 3-frames.

Recall that the 0-frame of an nD cube is just a union of 2^n points, the 1-frame or wire-frame is a union of lines, the 2-frame is a union of squares etc. There's two ways you can calculate these things. You can use the Mayer-Vietoris sequence, which I can't really explain. The other way is to deform the cube into something easy. For a wire-frame, it's possible to deform the shape into a whole bunch of circles, like "OOOOOOO", except each one is attached to the next at a point. This is called a wedge sum of circles, S^1 V S^1 V ... V S^1. It's easy to work out the homology of these shapes.
I'll show how to do this with the wireframe 3D cube, then I'll give a table of what I've got so far.

Think of the cube as being made of actual bits of wires, and you can drag the endpoints of each wire along the other wires but you can't detach them. Draw the cube as if you're looking at it face down, with a big square on the outside. Now take the wire at the top, and drag one endpoint along the other wires to the other endpoint, so you get a little loop at one corner. Now do the same to the bottom wire. You should now have three squares attached to each other, and the square on the left has two circles attached. Now the three squares can be deformed into three circles attached at points, so you get a shape that looks like
OOOOO.
This is the wedge sum of five circles. This has the same homology groups as the original shape because we used continuous deformations. So the homology groups are as follows:
H₀ X = Z
H₁ X = Z+Z+Z+Z+Z = 5 Z
H₂ X = 0

By the same method, you can show that H₁ of the wireframe tesseract is 17 Z.
By looking carefully at the net for the tesseract, you can deform it into a wedge sum of spheres, and you might be able to convince yourself that H₂ of the sheet-frame tesseract is 7 Z.

I've worked these out both by deforming into circles, and using the Mayer-Vietoris sequence. Most of the following table is either trivial or I only used Mayer-Vietoris.

3D cube:
0-frame: 8,0,0,...
1-frame: 1,5,0.
2-frame: 1,1.
3-frame: 1.

4D cube:
0-frame: 16,0,0,0
1-frame: 1,17,0,0
2-frame: 1,0 ,7,0
3-frame: 1,0 ,0,1
4-frame: 1.

EDIT: fixed H₁ of the 2-frame

5D cube: I may have made a mistake with this one, and I haven't done the 3-frame yet. I'll finish this tomorrow.
0-frame: 32,0 ,0 ,0 ,0
1-frame: 1 ,49,0 ,0 ,0
2-frame: 1 ,0 ,31,0 ,0
3-frame: 1 ,? ,? ,? ,0
4-frame: 1 ,0 ,0 ,0 ,1

[Keiji]

Are you sure it's valid to move a node to the other side of another node?

With the wireframe cube, I ended up with three connected circles in a row, and two extra lines connecting the first and last circles. To get OOOOO, I'd have to move one end point of each of these lines past two other nodes.

[PWrong]

Are you sure it's valid to move a node to the other side of another node?

That's fine, it's still a continuous deformation.

If you look at the cube face down and squash it flat, you can also just count the obvious holes.