I've avoided talking about the difference between homotopy and homology because they're both quite technical, especially homology. I'll give a very brief description of each now. If you don’t get this, just skip to the end where I’ll answer your questions.
What I've understood is that the n-hole-order is the number of ways you can put an n-net-sphere around the object so that it forms loops to itself.
This is roughly the definition of homotopy. Rather than the number of ways (which is usually infinite), it's a group of ways that are unique up to deformation. You can add two loops together if they have the same starting point, by going around one loop and then the other, which gives you a new loop. This addition makes it a group. It's a simple enough definition, but unfortunately it comes with annoying surprises. For example it turns out that you can wrap a 3-sphere around a 2-sphere, any integer number of ways. Even worse, you can wrap a 4-sphere around a 2-sphere exactly 2 ways, and you can wrap a 14-sphere around a 4-sphere exactly 2880 ways. The group for 7-spheres around 4-spheres is the integers cross the 12th cyclic group. For large spheres it gets very complicated and difficult to calculate.
http://en.wikipedia.org/wiki/Homotopy_g ... ral_theoryTo get rid of this problem, mathematicians defined simplicial homology. There's also another kind of homology I didn't learn that involves cubes. But here's what you need to know about simplicial homology
1. It's much more complicated to define, involving free abelian groups, chain complexes and quotient groups.
2. It uses little simplices (nD analogues of triangles) instead of spheres.
3. It's easier to calculate and gives nicer results than homotopy groups, but in many simple cases it gives you the same results.
4. H
0 X = Z iff X is path connected.
5. I think that H
n X = Z if X is an nD surface and X is path connected.
Ok now I'll answer your questions.
Because there's one thing that I don't get, which is why the 2-hole-order of a torus is 1. You can't wrap a sphere around a torus.
This is because homology uses simplices instead of spheres. You can’t wrap a sphere around a torus, but you can cover it with triangles.
So the 0-net-sphere (point) can be placed anywhere in any toratope equivalently so the 0-hole-order is always 1.
This is basically right. Homology uses a set of integer multiples of maps, so when you cut out all the redundancy you just get the integers back. Generally H
0 gives you one Z for each path component. So if your shape was in two separate pieces, then H
0 X = Z + Z.
With the sphere, you can't put a circle (a 1-net-sphere) around it in such a way that it wraps, so the 1-hole-order is 0
It’s more that any closed loop on the sphere can be deformed into a point. So the group is the trivial group, which we call 0 even though it actually contains one element.
Also, as far as I can see, the duocylinder's 1-hole-order is 2 because there are two unique circles in it. But where did the 2-hole-order being 1 come from? (Probably the same place as the 2-hole-order of the torus).
That’s right. You can cover a duocylinder with little triangles. Also the duocylinder is homeomorphic to a torus, so we expect them to have the same homology groups.
and with the torus, you can put the circle in two unique places such that it wraps, so the 1-hole-order is 2
Rather, there are infinitely many ways to wrap the circle. You can wrap around the small circle 12 times and then around the big circle 27 times. This is a unique wrapping, different from wrapping around big 16 times and then small 32 times. However it doesn’t matter what order you wrap in, you can deform some wrappings to others. So to describe a wrapping you need two integers, and that’s where the 2 comes from.
Does all that help?