Characteristic Homology Classes

[Jamma]

Roughly speaking, in cohomology theory, characteristic classes are elements of the cohomology of the base space of a fibre bundle which can tell you something about the nature of the fibre bundle.

In "Characteristic Classes" by Milnor, he mentions that characteristic homology classes for the tangent bundle of a smooth manifold have been defined. I've tried looking these up without much luck.

My question is: is there a simple, analogous notion of characteristic homology class to the regular one in cohomology? Would I be right in saying that these elements would live in the homology of the total space of a fibre bundle? Could they, for example, give obstruction to certain topological spaces being the base of a fibre bundle with the total space in question?

 

[lavinia]

Roughly speaking, in cohomology theory, characteristic classes are elements of the cohomology of the base space of a fibre bundle which can tell you something about the nature of the fibre bundle.

In "Characteristic Classes" by Milnor, he mentions that characteristic homology classes for the tangent bundle of a smooth manifold have been defined. I've tried looking these up without much luck.

My question is: is there a simple, analogous notion of characteristic homology class to the regular one in cohomology? Would I be right in saying that these elements would live in the homology of the total space of a fibre bundle? Could they, for example, give obstruction to certain topological spaces being the base of a fibre bundle with the total space in question?

Characteristic homology classes are the Poincare duals of characteristic cohomology classes - I think.

 

[Jamma]

Ok, thanks for the reply.

So it doesn't seem that there is much new there then. Thinking about it, there doesn't seem to be any obvious geometrical meaning to a characteristic class living in the total space of a fibre bundle- I suppose the characteristic class says how the fibres wrap around the base space but there would be no obvious way of explaining this from the perspective of the total space.

I still wonder, however, if given a fibre and a total space it would be possible to determine all of the different kind of base spaces you could have (or the properties such a space could have).

 

[lavinia]

Ok, thanks for the reply.

So it doesn't seem that there is much new there then. Thinking about it, there doesn't seem to be any obvious geometrical meaning to a characteristic class living in the total space of a fibre bundle- I suppose the characteristic class says how the fibres wrap around the base space but there would be no obvious way of explaining this from the perspective of the total space.

I still wonder, however, if given a fibre and a total space it would be possible to determine all of the different kind of base spaces you could have (or the properties such a space could have).

Characteristic classes tell you a lot about the topology and geometry of the base space. For instance the Euler class of the tangent bundle integrates to the Euler characteristic of the manifold. Although it is an integer cohomology class if can be expressed as an n-form in the curvature tensor of any Levi-cevita connection. This is the Gauss-Bonnet theorem in full generality.

Chern Classes can also be expressed in terms of curvature of a connection.

Stiefel-Whitney numers tell you whether a compact smooth manifold without boundary can be the boundary of a 1 higher dimensional manifold. The theorem is that a manifold bounds if an only if all of its Stiefel-Whiney numbers are zero. So for instance, the real projective plane is a 2 dimensional surface that is not a boundary of a three manifold. On the other hand the klein bottle is a boundary.

There are many more applications of Characteristic classes to the topology and geometry of manifolds.

One reason that Characteristic homology classes are interesting is that sometimes they they can be expressed combinatorially interms of a triangulation of the manifold. this is not only true of the Euler characteristic but is true of all of the Stiefel-Whitney homology classes and is also true of the Pontryagin classes of the tangent bundle.

The tangent bundle and its relatives seems to be the only bundle that has geometric information about every manifold. But for special manifolds other bundles also contain important information.

 

[Jamma]

Characteristic classes tell you a lot about the topology and geometry of the base space.

Yes, and this isn't that surprising. Their construction is based on the fibre bundles of that space, as K-theory is, which can express an awfully large amount of detail about the space.

Stiefel-Whitney numers tell you whether a compact smooth manifold without boundary can be the boundary of a 1 higher dimensional manifold. The theorem is that a manifold bounds if an only if all of its Stiefel-Whiney numbers are zero. So for instance, the real projective plane is a 2 dimensional surface that is not a boundary of a three manifold. On the other hand the klein bottle is a boundary.

I love this theorem! I've been reading characteristic classes by Milnor, and when this came up it was totally unexpected, yet delightful! I couldn't believe that a bunch of a few finite invariants which only take 2 values totally classify the unoriented cobordism classes.

One reason that Characteristic homology classes are interesting is that sometimes they they can be expressed combinatorially interms of a triangulation of the manifold. this is not only true of the Euler characteristic but is true of all of the Stiefel-Whitney homology classes and is also true of the Pontryagin classes of the tangent bundle.

I didn't know this, but will check that out, thanks.

On an (almost totally) different note, if you have a (branched) manifold, if its tangent bundle is trivial, does this tell you that much about the manifold in question? I have a bunch of manifolds I'm interested in and their tangent bundles are trivial, I was wondering if there is anything more I can say about them because of this.

 

[lavinia]

There is a paper by Sullivan that gives a combinatorial formula for the signature of a 4K manifold. I will look for the reference.

 

[lavinia]

On an (almost totally) different note, if you have a (branched) manifold, if its tangent bundle is trivial, does this tell you that much about the manifold in question? I have a bunch of manifolds I'm interested in and their tangent bundles are trivial, I was wondering if there is anything more I can say about them because of this.

Can you show me some the example you are looking at?

 

[lavinia]

BTW: If you are reading Milnor's characteristic classes you will see the combinatorial Pontryagin classes defined. These classes are associated to a triangulation of the manifold and are rational rather than integer classes. But in case the triangulation is smooth, they are the Pontryagin classes of the tangent bundle. An example is given of a triangulation of an 8 manifold that is not smoothable.

Here is a reference on Stiefel Whitney homology classes

http://www.jstor.org/pss/1970823

 

[Jamma]

Thanks for the info.

Can you show me some the example you are looking at?

I am doing some stuff on aperiodic tiling. It may be a bit involved to go into totally if you aren't familiar (although I'd recommend it, it's really interesting stuff) but basically I am looking at the approximants for the tiling space of an aperiodic tiling.

Basically, they tell you about all the ways you can position a patch of a tiling at the origin (a patch is a subset of tiles of the tiling). There is the central tile, and you get one copy for each kind of patch (each point tells you where you put it on the origin). You identify points where you can have either option in the same tiling (i.e. along the boundary). We can simply only use the tiles themselves as the patches, or the tile with all tiles touching it also, or the ones touching them too, or we could say all patches of radius r etc. etc. but the important thing is that if we go to larger and larger approximants, we can map each approximant into the "smaller" one by "forgetting" information about the outside of the patch and when we take the inverse limit we get the tiling space (a sort of moduli space of all the tilings we have).

Because of the nature of these approximants, I believe that they have trivial tangent bundle (which I'm sure can be defined for branched manifolds). In fact, this is easy to see, there are obvious lin ind vector fields induced from the tiles originally living in R^d.

[I'll rewrite what I said later because I'm in a hurry and won't be at a computer for a while, but would have forgotten to write it if I hadn't done it now! I appreciate that the above probably isn't very comprehensible at the moment]