I am giving a presentation soon on the Classification of Complex Line Bundles and I would like to have some very "basic" visualizations to use as examples.

## Background and Context

I am considering the Cech-cohomology of a principal $ \mathbb{C}^{*} $ bundle, where my sheaf $\underline{\mathbb{C}}_M^{*}$ is the sheaf of smooth $\mathbb{C}^{*}$ valued functions on the manifold $M$. Using the exponential sequence of sheaves $$ 0 \to \mathbb{Z}(1) \to \underline{\mathbb{C}}_M \to \underline{\mathbb{C}}_M^{*} \to 0$$ we get an isomorphism (via properties of cohomology and the connecting homomorphism) $$H^1(M, \underline{\mathbb{C}}_M^{*}) \cong H^2(M, \mathbb{Z}(1)) $$

It turns out that $H^1(M, \underline{\mathbb{C}}_M^{*}) $ is also isomorphic to the group of isomorphism classes of principal-$\mathbb{C}^{*}$ bundles over $M$. Since the principal- $\mathbb{C}^{*}$ bundles are in one-to-one correspondence with the complex line bundles, it should be evident how this all relates to my title.

## My Questions

(1) Given the above information, and some knowledge of cohomology, there should be only a trivial principal- $\mathbb{C}^{*}$ bundle on the circle $S^1$. How can we see this visually?

*See my example/analogue below.

(2) Similarly, how can we visualize a non-trivial principal- $\mathbb{C}^{*}$ bundle on the standard 2-dimensional torus?

*Example/Analogue:

So consider a circle bundle on $S^1$, then we can consider a section of the bundle like so:

alt text http://www.cheynemiller.com/Math/Figures_files/SectionOnU.png

Now, given two sections on adjacent trivializations,

alt text http://www.cheynemiller.com/Math/Figures_files/TransitionUaUb.png

We can imagine deforming one section into another, to get our transition functions. Now, I can also believe that any such family of sections can be deformed into a global section, so again I *want to know why this necessarily doesn't work on the Hopf bundle* **via pictures**.

`$\mathbb{C}^{*}$`

-bundle! In particular, I want to see why I CAN define transition functions for a family of local sections, but why I could never manipulate these sections to form a global one! $\endgroup$