Suppose I have a smooth manifold $M$, and want to consider the $K$-theory $K^0(M)$. I could define this in the usual way (by taking the Grothendieck group of the monoid of equivalence classes of vector bundles) or in a ‘smooth’ way (considering only smooth vector bundles, and taking the Grothendieck group as usual).
I haven’t seen this discussed anywhere. Is there any difference between the two approaches? And are there any references in which this question is discussed?
(Even better: if $M$ is a $G$-space for a compact Lie group $G$, is the equivariant $K$-theory affected by taking only smooth bundles?)
The $K^0(M)$ based on continuous vector bundles is the same as that based on smooth vector bundles because of the fundamental (and perhaps not sufficiently advertised) result:
Theorem Every continuous vector bundle on a $C^\infty$ manifold has a compatible $C^\infty$ vector bundle structure. Such a structure is unique up to $C^\infty$ isomorphism.
You can find a proof in Hirsch’s Differential Topology , Chapter 4, Theorem 3.5, page 101.