Homology groups of toratopes

Discussion of shapes with curves and holes in various dimensions.

Homology groups of toratopes

Postby PWrong » Tue Nov 10, 2009 6:22 am

Hi everyone, sorry I've been gone so long! I've just finished my honours in pure maths, and decided to come back here to talk about some ideas I've had.

So I learned about homotopy groups and homology groups in Algebraic topology this year, and the whole time I was thinking about applying the ideas to toratopes. They're basically ways of talking about the number of holes in an object in a precise way. In the course we didn't learn very much about how to actually calculate them, but we spent a lot of time defining them. However I think I've developed an intuition about how they work, and a conjecture about the homology groups of every toratope.

Homotopy groups are easy to define but they give you nontrivial, nonintuitive answers. Homology groups have an extremely complicated definition, but they have simpler answers. I think the best way to explain them is with lots of examples, and I'll do that in my next post. Does anyone know what homotopy and homology are?
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Re: Homology groups of toratopes

Postby Keiji » Tue Nov 10, 2009 2:45 pm

Welcome back PWrong, the forums were a bit dead without you. ;)

All I know on the subject is that two shapes are homologous if their genii are the same, but that seems too obvious to be useful.

As for counting the number of holes, I believe it's rather impossible in dimensions greater than 3 (and irrelevant in dimensions less than 3), since in 4D you get at least two different types of holes, and I can't begin to imagine what 5D has to offer.

In any case, some more research on the toratopes would definitely be appreciated! :)
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Re: Homology groups of toratopes

Postby PWrong » Wed Nov 11, 2009 12:17 am

Hi Keiji.

As for counting the number of holes, I believe it's rather impossible in dimensions greater than 3 (and irrelevant in dimensions less than 3), since in 4D you get at least two different types of holes, and I can't begin to imagine what 5D has to offer.

I think this is probably right, which is why we replace the idea of holes with the idea of homology groups. Actually even shapes like the klein bottle have an odd sort of hole which is different to the torus.

The idea of homotopy groups is a bit like "how many ways can you wrap an n-sphere around this object?" Technically, pin(X) is the group of maps from an n-sphere to the space X, where two maps are equivalent if one can be deformed into the other. There's a way to add these maps, basically by doing one and then the other, so this thing is a group.

So for example, pi1 S^1 is the group of ways you can wrap a circle around itself. You can wrap around clockwise once, or wrap around halfway, go back a bit, then go the rest of the way. These wrappings are equivalent, or homotopic. However if you wrap around twice and get back to your starting point, and this wrapping can't be deformed into the one that just goes around once. Also, you can wrap around backwards, or backwards 3 times, or forwards 217 times. So for any integer n you can wrap that many times. So pi1 S^1 is the set of integers, Z.

Some more examples:
pi1 of the torus is Z + Z, the set of pairs of integers, because we can wrap around the big circle or the small circle.
pi1 S^2 = 0, since any closed curve on the sphere can be deformed to a point. Here zero is the trivial group with one element, 0.

Unfortunately this definition doesn't always lead to what you'd expect. For some reason you can wrap a 3-sphere around a 2-sphere in a nontrivial way.
http://en.wikipedia.org/wiki/Homotopy_g ... opy_groups

Homology groups of spheres have a much nicer table, but it's really hard to define. It involves free groups and chain complexes and other stuff. So I'll just give some examples.

If X is path connected, H0X is always Z. If X is in two pieces, then H0X is Z + Z.
For q > 0, the spheres are quite simple.

H_q S^n = Z if q = n
H_q S^n = 0 otherwise

So this basically means that an n-sphere has one n-hole, and no other holes.

The torus has these homology groups

H_0 T = Z
H_1 T = Z + Z
H_2 T = Z
H_3 T = 0
H_4 T = 0
etc.

I've only just noticed that H_2 = Z. I had a conjecture about the homology groups of toratopes, but H_2 = Z is a counterexample :(. So that's a shame, but maybe we can improve the conjecture.

Anyway I'll talk more about this later.
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Re: Homology groups of toratopes

Postby PWrong » Sat Nov 14, 2009 11:21 am

So I haven't learned enough about homology groups to actually calculate them for complicated shapes like toratopes. It involves a thing called the Mayer-Vietoris sequence, but there's a step in the process that I haven't yet figured out. But here's a problem that hopefully we'll all understand, and I think it will be relevant later.

Say we punch a hole into a torus, so there's one point missing. Then we can smoothly deform the punctured torus into a pair of circles that are connected at a point. Try visualizing this.

This is called the wedge sum of the two circles. So we can write:

T \ {.} ~ S1 v S1

Or in our notation:
(21) \ {.} ~ 2 v 2

Here \{.} means we're taking out a point, ~ means the spaces are homotopic or we can deform one into the other, and v means the wedge sum.

Today I was doing this to all the closed (i.e. with brackets around them) toratopes up to dimension 6.

A punctured sphere of any dimension except 0 is a ball in one dimension lower, which can be contracted to a point. So S^n \ {.} ~ {.}
I don't think this fits the pattern I found with the other toratopes, so I left spheres out of the list.

Torus:
(21) \ {.} ~ 2 v 2

Toraspherinder:
(31) \ {.} ~ 3 v 2

Toracubinder:
(211) \ {.} ~ 2 v 3

Tiger:
(22) \ {.} ~ 2 v 2 v 2

3-torus:
((21)1) \ {.} ~ 2 v 2 v 2

I used the pattern to complete the list for 5D and 6D, and I noticed something interesting about the number of unique shapes resulting. What does everyone think of all this?
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Re: Homology groups of toratopes

Postby PWrong » Sun Nov 15, 2009 10:05 am

Ok forget about most of that last post. I've worked through the Mayer-Vietoris sequences for a heaps of shapes, and I've found a new conjecture, just as simple as my old one.

First of all, my conjecture will imply that (31) and (211) and 32 are homeomorphic, and so are (22) and ((21)1) and 222. That is, there's a smooth map between them with a smooth inverse, so in some sense they're the same shape but twisted around. The relationship between them is a bit like the one between (21) and 22.

Now I guess you can think of the homology group H_q X as the number of q-holes the shape X has, times by Z. Kind of. A lot of mathematicians probably wouldn't like me saying that.
We have H0 X = Z whenever Z is path connected. So in some sense every path connected shape has one 0-hole. Also it seems that every n-dimensional surface has one n-hole, or at least that holds for closed toratopes. A torus has two 1-holes: the main obvious one, and one going around the inside.

I'm going to write 1 for Z, 2 for Z + Z, 3 for Z+Z+Z etc. and separate different homology groups by commas. So since the torus has:

H0 (21) = Z
H1 (21) = Z + Z
H2 (21) = Z
Hn (21) = 0 for n > 2

I will write H(21) = 1,2,1,0,0, ...

In the following list I'll just stop at the last nonzero number. This is what I've calculated with Mayer Vietoris sequences so far. Each of the 4D shapes took about half an hour, probably an hour for 5D shapes. I haven't done most of the rotatopes and open toratopes. They should be trivial, but you have decide if, for example, 21 means a pair of circles, a tube, a pair of disks or a pair of disks and a tube. For all shapes here I've assumed we're talking about the surface with the smallest possible dimension.

2D:
H2 = H S1 = 1,1

3D:
H3 = H S2 = 1,0,1
H(21) = H S1 x S1 = 1,2,1

4D:
H22 = 1,2,1,0
H4 = 1,0,0,1
H(31) = 1,1,1,1
H(22) = 1,3,0,1
H(211) = 1,1,1,1
H((21)1) = 1,3,0,1

5D:
H5 = 1,0,0,0,1
H(2111) = 1,1,0,1,1

I'm currently in the middle of calculating H(221).

Now I'm pretty sure any two shapes with the same homology groups must be homeomorphic. So that would imply the homeomorphisms I mentioned at the start. So it looks like the different types of holes aren't as complicated as we thought. In nD there are n types, and two of them are trivial. So there is one interesting type of hole in 3D, and like Keiji said, there are two interesting types of holes in 4D.
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Re: Homology groups of toratopes

Postby Keiji » Mon Nov 16, 2009 11:53 am

After staring at it for hours, I think I'm beginning to understand your notation, though I'm not sure how you come up with it for the higher dimensional toratopes. 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. So the 0-net-sphere (point) can be placed anywhere in any toratope equivalently so the 0-hole-order is always 1. 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, and with the torus, you can put the circle in two unique places such that it wraps, so the 1-hole-order is 2.

Am I right, thus far? 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.

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).
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Re: Homology groups of toratopes

Postby PWrong » Tue Nov 17, 2009 9:51 am

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_theory

To 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. H0 X = Z iff X is path connected.
5. I think that Hn 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 H0 gives you one Z for each path component. So if your shape was in two separate pieces, then H0 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?
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Re: Homology groups of toratopes

Postby Keiji » Tue Nov 17, 2009 12:24 pm

Well, that helps, but now... how are you supposed to cover an n-dimensional shape with a p-simplex where p < n-1?
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Re: Homology groups of toratopes

Postby PWrong » Tue Nov 17, 2009 12:46 pm

Oh, sorry. You don't have to cover the shape, I'm not sure why I said that. Also I've remembered another, completely different way to look at it that I'll post when I work out the best way to say it.
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Re: Homology groups of toratopes

Postby PWrong » Wed Nov 18, 2009 6:17 am

Ok, another way to look at this is to talk about cycles and boundaries. I don't know if this will help, maybe it'll make it even worse.

Look at the images of simplices on X. The boundary of one of these is not exactly what you'd expect, but let's pretend it is. So the boundary of a path from A to B is simply the pair of points {A,B}. The boundary of the image of a triangle on X is the image of each side of the triangle. Note that the boundary of a boundary is always zero, or the empty set. Now we define a cycle as a thing that has no boundary, or rather something whose boundary is zero. For example a closed loop is a boundary. Now since the boundary of a boundary is zero, obviously when we take the boundary of something we always get a cycle. But that doesn't mean that every cycle is a boundary of something.

For example, take a circle drawn on a sphere. This is a cycle because its boundary is zero, and the circle itself is the boundary of a disk. In fact every 1-cycle on the sphere is a 1-boundary.
However, if we draw a loop around the small circle of a torus, this loop is a cycle but it's not the boundary of anything. You can't draw a disk whose boundary is that loop.

We can do technical stuff to make it possible to add up the images of simplices. This gives us a group of 1-cycles and a group of 1-boundaries. The 1st homology group is then the quotient group (which is hard to explain) of the cycles by the boundaries.
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Re: Homology groups of toratopes

Postby Keiji » Thu Nov 19, 2009 11:22 am

PWrong wrote:For example, take a circle drawn on a sphere. This is a cycle because its boundary is zero, and the circle itself is the boundary of a disk. In fact every 1-cycle on the sphere is a 1-boundary.
However, if we draw a loop around the small circle of a torus, this loop is a cycle but it's not the boundary of anything. You can't draw a disk whose boundary is that loop.


Why not?

Every circle bounds a disc...
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Re: Homology groups of toratopes

Postby PWrong » Fri Nov 20, 2009 11:28 am

I meant a disk on the manifold. Also when I say torus I always mean the surface of it. So you can draw a circle on a torus that isn't the boundary of any disk on the torus.
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Re: Homology groups of toratopes

Postby Keiji » Fri Nov 20, 2009 12:51 pm

Ah, now I understand. I'll think about this more when I get home.
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