I have been planning to say a few words about local cohomology, and I stand by that still–but I got sidestracked.

I would now like to explain an interesting phenomenon that we talked about today in topology class. In algebraic topology, one often considers fiber bundles ${p: E \rightarrow B}$. The idea of a fiber bundle is to generalize the notion of a product space ${F \times B}$. A fiber bundle is, loosely, something that locally looks like the projection from a product ${F \times B \rightarrow B}$. The reason that we only require it to look locally like a product is, for instance, that we want the tangent bundle to be considered a fiber bundle. However, as is well known, the tangent bundle to a manifold may admit no nonzero global sections—which means that it cannot be trivial. Nonetheless, given a fiber bundle, one may ask in some sense how strongly it fails to be nontrivial.

A detailed study of such questions belongs to the theory of characteristic classes, which I am not currently ready to talk about. Nonetheless, let us start with a very simple question. Let ${p: E \rightarrow B}$ be a fiber bundle over a compact space ${B}$. There is then, by definition, an open covering ${\{U_i\}}$ of ${B}$ such that the restricted fiber bundles ${p^{-1}(U_i) \rightarrow U_i}$ is trivial, isomorphic to ${F \times U_i \rightarrow U_i}$ for ${F}$ the fiber. Clearly we can take this cover finite. One might naturally ask how small we can take this cover.

We shall describe the answer in a specific important case. One of the standard examples of a fiber bundle is the Hopf bundle $\displaystyle S^1 \rightarrow S^{2n-1} \stackrel{p}{\rightarrow}\mathbb{CP}^{n-1}.$ (more…)