Difficulty with the pullback bundle

[xaos]

i attempted the first 100 pages of Walter Poor's Differential Geometric Structures, taking me about two months, and the most difficulty i had was trying to understand the utility(?) of the pullback bundle. its possible that its just a very difficult subject and the only thing that's needed is some time to grasp the ideas, or at least the notation, but its still a very slow subject and i wonder what prerequisites are needed to make it go smoother. it takes a half page just to set up every new idea.

he uses the idea in several constructions and several commutation diagram isomorphisms, but i still don't see their full nature. the only intuition i have on the idea is to see it as a kind of 'pre image' of the fibre bundle structure. for example, the structure of a cube might be seen as a pre image of a family of tangent planes along a curve inside a manifold. but why not just do this (a mapping between preimage and image), instead of constructing a commutation diagram and then building catagorical isomorphisms between the diagrams? or am i asking the wrong question? it seems strange that wikipedia describes this stuff way too lightly, as if its such constructions are self evident. thankyou for any help.

 

[lavinia]

I am not sure if you just don't understand the construction of the pull back bundle or whether you are asking - why do I need to learn about it? What is is goof for?

 

[xaos]

maybe i simply don't have enough background experience with it to know what-when-where-why its designed purpose is. at this point, I'm able to read an entire book on this stuff, but when i close the book and someone asks me, "what have you learned?", i stratch my head and go blank. i think this is only partly because i don't understand it, but because the subject requires a global explanation that makes it impossible to separate it into any subelements that are easier to understand. I'm thinking maybe i should return to Spivak and try again in a few months.

 

[lavinia]

The induced bundle is a natural way to create a new bundle from an old one.

One could ask the question which vector bundles are induced and which are not. The remarkable answer is that they all are and they are all pullbacks of a single vector bundle over a single space. This universal space is called the Grassmann manifold of n-planes in R$^{∞}$ and the bundle is called the universal n-plane bundle. This space is similar to an infinite dimensional projective space.

So in some sense the study of bundles over a space is the study of maps into this Grassman manifold. Further if two such maps are homotopic then the induced bundles are isomorphic so one is really studying homotopy classes of maps into the Grassmann manifold.

Because of this, the cohomology of the Grassman manifold is extremely important since it's pullback under the map reflects the bundle itself. the study of this cohomology and it's pullbacks is one approach to the study of Characteristic classes. These classes contain information about the structure of the bundle e.g. whether the bundle has a non-zero section. For smooth manifolds there characteristic classes tell you things about the tangent bundle of the manifold.

Further the universal bundle has a universal connection and all connections on all smooth vector bundles are pullbacks of the universal connection. this means that all Rimannian geometries are pull backs of a single universal geometry on the universal bundle. A famous theorem related wedge products of the curvature 2 form matrix to the characterisitc classes of the bundle so there are universal geometric invariants of smooth manifolds (and also all smooth bundles over a smooth manifold.) One example for oriented bundles with a Riemannain metric is the real Euler class whose integral over the manifold is its Euler characteristic.

Here are a couple of examples for you to think about.

- Suppose you have a smooth surface - like a torus - embedded in 3 space. The surface is orientable (proof?) and so has a well defined unit normal. The unit normal to each point on the surface assigns a point on the sphere of radius 1 centered at the origin. So it defines a mapping of the surface into the unit sphere. It is called the Gauss map. (You should convince yourself that the mapping is smooth.) Is the tangent bundle the induced bundle under this map? What bundle map do you get if you parallel translate each tangent vector to the surface to the tangent plane to the sphere at the point determined by the unit normal? Note that this map differs in general from the differential of the Gauss map which may have singularities or may distort the tangent plane.

- Suppose a differentiable map is a local diffeomorphism. Is the induced bundle of the tangent bundle the tangent bundle?

- Suppose an oriented vector bundle has a non-zero section i.e. a non-zero vector field. Do all bundles induced from it also have a non-zero section?

- What is the difference between the Hopf fibration and the tangent circle bundle of the 2 sphere? Does the Euler class distinguish them?

 

[mathwonk]

what a great answer! a tiny additional remark: not only are all bundles pullbacks of the universal bundles, but it can also render a bundle simpler to pull it back further. E.g. one reason to study pullbacks is that (I believe) all bundles split into sums of line bundles after appropriate pullback, which makes studying their chern classes simpler, i.e. it reduces to understanding chern classes of line bundles.

another reason to study pull backs is that the existence of the pull back operation implies that bundles form a functor, like cohomology, and in fact can be turned into an "extraordinary" cohomology functor. Thus one can compare manifolds by comparing their bundles.

 

[xaos]

wow. I've been suspecting these deep connections. it seems like everything learned up to now is a prerequisite to be able to see into the next lesson.