The first Chern class can be obtained from the exponential sequence:
$$0 \rightarrow \mathbb{Z} \rightarrow \mathbb{C} \stackrel{\text{exp}}{\rightarrow }\mathbb{C}^\times \rightarrow 0$$
Note that $\text{exp}$ is a local homomorphism and a universal cover. Its fibers are isomorphic to $\mathbb{Z}$, and hence the Puppe sequence shows that $\mathbb{C}^\times$ is a $K(\mathbb{Z}, 1)$
Now, $E = \mathbb{C}^\infty - \{ 0 \}$ is contractible - a fact interesting in its own right since it seems like it's got an "infinite dimensional hole", but an argument in the style of Hilbert's Hotel allows us to show that it is contractible. It's also got an action of $\mathbb{C}^\times$ on it, and there is then an exact sequence:
$$0 \rightarrow \mathbb{C}^\times \rightarrow E \rightarrow \mathbb{C}P^{\infty} \rightarrow 0$$
This time the Puppe sequence shows that $\mathbb{C}P^{\infty}$ is a $K(\mathbb{Z}, 2)$.
Somehow, $\mathbb{C}P^\infty$ also classifies $\mathbb{C}^\times$ bundles, though the proof is still not completely clear to me. One approach involves the connecting map of the exponential sequence.
Let $X$ be a Riemann surface.
- Is it true that $\mathbb{C}P^\infty$ also classifies $\mathbb{C}P^1$ bundles? That is, are maps of Riemann surfaces $X \rightarrow \mathbb{C}P^1$ in correspondence with $\mathbb{C}P^1$ bundles on $X$? It would seem that we can fill in two missing points in the fiber of each $\mathbb{C}^\times$ bundle.
I am interested in this because it seems to be related to meromorphic functions on $X$.
Maybe folk here could give a reference to a gentle source explaining (in terms of Riemann surfaces, number rings, or varieties) the significance of projective bundles. In algebraic geometry, it seems we like to take the fraction field of a variety. From this we can read off the original points using valuations, given that the variety is normal. And in early courses, this is defined using primes and localization. But something interesting about this view is that we seemed to arrive at the fraction field e.g. of a Dedekind domain $D$ over $\mathbb{F}_p[x]$ using only considerations involving projective space.
Another important fact is about how Cartier divisors are the quotient $\mathcal{M}^\times / \mathcal{O}^\times$ where $\mathcal{M}$ is the fraction field of the global sections $\mathcal{O}$ of an irreducible variety. And here there is a projective bundle (in which points $0$ and $\infty$ were added in) which has $\mathcal{M}$ as its sections. It seems, then, that there is an exact sequence of quasi coherent sheaves over $X$, and that divisors are to do with homming with $k^\times$ in the right slot. But is there some source where this is phrased entirely in terms of projective bundles? And perhaps Weil divisors also have an interpretation in terms of projective bundles?
So it would be nice to see a source which lays out the story of $k \mathbb{P}^1$, the one dimensional multiplicative group, line bundles, $K(\mathbb{Z}, 1)$, $K(\mathbb{Z}, 2)$, and $E k^\times \rightarrow B k^\times$ rather nicely.