Geometric results following from sections of a bundle being a module Suppose $\pi:E\to B$ is a smooth bundle. Its set of sections $\Gamma(E)$ is a $C^\infty(B)$ module. Are there interesting geometrical facts that derive mainly from the algebraic structure of a $C^\infty(B)$ module? In other words are there algebraic theorems about modules that give geometrical insight about the bundle?
 A: You actually can get some powerful stuff from this module structure! Assume that $M$ is compact and choose a Riemmanian metric on $M$, $g$. Taking a finite covering of $M$ such that the bundle is trivial, we can extend the sections given this covering by a partion of unity. This gives in injection $f:\mathbb{R}^n\times M\to E$. Now we have an exact sequeunce of vectorbundles $0\to ker(f)\to \mathbb{R}^n\times M\to E\to 0$, and we can take the projection of $\mathbb{R}^n\times M$ onto $ker(f)$ given by $g$ to split this sequence. Then we have that $$\Gamma(E)\oplus \Gamma(F)=\Gamma(E\oplus F)=\Gamma(\mathbb{R}^n\times M)=C^{\infty}(M)^n$$ Thus $\Gamma(E)$ is a finitely generated projective $C^{\infty}(M)$-module! This means that our bundle infact defines an element in $K_0(C^{\infty}(M))$! This infact was the inspiration for Atiyah to introduce topological $K$-theory. Especially since we can define a map $K_0(C^{\infty}(M))\to \{\mathrm{Vector bundles on }M\}/\{\mathrm{Stable isomorphisms}\}$, take is infact an ismorphism, which means that isomorphisms of the bundle are almost entirely determined by the algebraic structure of this module. This fact is known as Swans theorem.
