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* . The idea of a fiber bundle is to generalize the notion of a product space . A fiber bundle is, loosely, something that locally looks like the projection from a product . 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 be a fiber bundle over a compact space . There is then, by definition, an open covering of such that the restricted fiber bundles is trivial, isomorphic to for 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**