arxiv tag for k-theory
Relation between higher algebraic K-groups and topological K-groups
Let X be a compact Hausdorff space. Swan's theorem provides an equivalence between the category of (say real) vector bundles on X and the category of finitely generated projective modules over the ring C(X,R) of continuous functions from X to the real numbers. This relates the topological K0 to the algebraic K0 of a ring, i.e. the group completion of the semiring with elements finitely generated projective modules.
Higher topological K-Groups are defined as K0 of suspensions, i.e. Kn(X)=K0(Sn(X)) and in particular Kn(X)=K0(C(Sn(X),R)). Higher algebraic K-groups are defined as certain homotopy groups of a space, given by topological constructions like the +-construction, Q-construction or Waldhausen's S-construction which remind one of the group completion.
Is there a kind of analogous statement to Swan's theorem for higher K-groups? To be more precise, is it possible to construct a functor Rn(-) from well behaved topological spaces to the category of rings such that Kn(X)=Kn(Rn(X))?