Torsion and homology: examples

[Physics Monkey]

Hi everyone,

What I'm looking for are good examples demonstrating torsion in homology. The basic example I know is RP^2, but I suspect there are many more good examples out there. I am interested in the topic both for fun and as part of my research. In addition, I will be teaching a class soon which will involve discussions of homology (in physics). I am primarily a physicist and will be speaking primarily to physicists, so the best examples for my purposes are relatively geometrical constructions, but I'll take anything you can give. Also, I would personally be interested in excellent (pedagogical or otherwise) discussions of the topic even if they are primarily mathematical.

Thanks very much!

 

[wofsy]

Hi everyone,

What I'm looking for are good examples demonstrating torsion in homology. The basic example I know is RP^2, but I suspect there are many more good examples out there. I am interested in the topic both for fun and as part of my research. In addition, I will be teaching a class soon which will involve discussions of homology (in physics). I am primarily a physicist and will be speaking primarily to physicists, so the best examples for my purposes are relatively geometrical constructions, but I'll take anything you can give. Also, I would personally be interested in excellent (pedagogical or otherwise) discussions of the topic even if they are primarily mathematical.

Thanks very much!

perhaps the most vivid examples are unorientable manifolds of which Rp^2 is an example. These manifolds all have Z/2 integral cohomology in the top dimension.

The Klein bottle for example is an easy one and can be demonstrated with a picture. You can triangulate it and show that the simplices can not be oriented to cancel all of the edges.

RP3 is another great example. It is orientable but it has Z/2 fundamental group and Z/2 second cohomology. One can show the torsion directly if one first thinks of it as SO(3) and then let it act transitively without fixed points on the tangent circle bundle of the 2 sphere to see that it is diffeomorphic to the tangent circle bundle of S^2. You can then demonstrate a 2 torsion loop in the tangent circle bundle with pictures. It is also a bit surprising since the 2 sphere itself is simply connected and orientable.

Many flat manifolds have all kinds of homology torsion and I could give you some examples if you like. Their homology is a little hard to compute but I will try if you want - say a couple of 3 manifolds. They can be pictured as cubes with faces identified and are easy to picture. If you have the patience you could illustrate how the torsion arises with pictures.

 

[Physics Monkey]

Hi wofsy,

Thanks for your reply, it was very helpful. I will try the klein bottle and RP^3 myself. Regarding the flat manifolds, what you have in mind are things like 3-tori but with twisted identifications?

 

[wofsy]

Hi wofsy,

Thanks for your reply, it was very helpful. I will try the klein bottle and RP^3 myself. Regarding the flat manifolds, what you have in mind are things like 3-tori but with twisted identifications?

They are quotients of flat 3 tori by a finite group of isometries just as the Kelin bottle is the quotient of the flat 2 torus by an action of Z/2.

 

[Physics Monkey]

Hi wofsy,

Thanks for the examples.

 

[wofsy]

Hi wofsy,

Thanks for the examples.

your welcome.

Here is a famous flat 3 manifold.

Start with the standard lattice in R^3. The quotient of R^3 by this lattice is a flat 3 torus.

Add to this lattice the following isometries. (x,y,z) - > (x + 1/2,-y,-z)
(x,y,z) -> (-x, y+1/2, -z+1/2) and the product, (x,y,z) -> (-x + 1/2, -y + 1/2, z+1/2).

The group of isometries that these generate covers the Hansche-Wendt manifold. It is the quotient of the 3 torus by an action of Z/2 + Z/2.


You can picture it as the standard 3 cube of edge length 1/2 with identifications. It is easy to picture.


This manifold is orientable because all of the covering transformations are orientation preserving.

It has first Betti number zero because each axis is reflected by one of the covering transformations. thus its first homology is entirely torsion ( 2 torsion). By Poincare duality its second cohomology is entirely torsion.

Also it is easy to check that the first integer cohomology is zero by using the isomorphism H^1(Hansche-Wendt manifold:Z) = Hom( group of covering transformations:Z) and checking that there are no homomorphisms.

The torsion arises from the reflections of the axes from the added elements of the group of covering isometries. If you draw the cube with identifications you can see Klein bottles out the wazoo inside of this manifold.