Kiyoshi Oka proved in 1938 that topological line bundles over closed, complex submanifolds of complex affine space admit unique holomorphic structures. Nearly twenty years later, Hans Grauert proved the same theorem for topological vector bundles of any rank. I will examine these results from the point of view of K-theory, and explain the proofs, which are strikingly similar to the proofs of some fundamental theorems in homology theory, for example the Jordan separation theorem. Oka’s theorem is in some sense “commutative,” since it concerns the abelian Lie group GL(1,C), whereas Grauert’s theorem concerns the non-abelian groups GL(n,C). But there are further extensions of both theorems into the realm of noncommutative geometry (in the sense of Alain Connes), and as I shall explain these extensions have interesting links to representation theory.