A fibre bundle $ \pi \colon E \to M$ is a surjective submersion between manifolds with the property that the domain $E$ has extra structure. Smooth maps that go in the opposite direction are—from the point of view of fibre bundles—uninteresting unless they respect this extra structure. Those that do are called sections.

In the case of a vector bundle, the space of sections is a module over the ring of smooth functions. The main result of today’s lecture is the so-called Hom-Gamma Theorem[1] that relates vector bundle homomorphisms and operators on the space of sections that are linear over the smooth functions:

$$ \Gamma \big( \mathrm{Hom}(E, F) \big) \cong \mathrm{Hom}\big( \Gamma(E) ,\Gamma(F)\big).$$

Along the way we introduce the notion of a point operator and a local operator between spaces of sections.

The lecture concludes with another algebraic interlude, which covers the rudiments of sheaf theory. This material is for interest only, and can safely be skipped — it will not be used in the remainder of the course.

Roughly speaking, a presheaf is a way to assign data locally to open subsets of a topological space in such a way that it is compatible with restrictions. A sheaf is a presheaf for which it is possible to go backwards and reassemble global data from local data.

After giving the relevant definitions, we give a “sheaf-theoretic” definition of a manifold and a vector bundle, and outline the proof that there is an equivalence of categories between the category of vector bundles over $M$ and the category of finite rank locally free $C^{ \infty}(M)$-modules.

  1. Where by “so-called” what I actually mean is “called by me”. I made the name up ¯\_(ツ)_/¯ ↩︎

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