Relation between higher algebraic K-groups and topological K-groups

Question

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))?

Answer 1

I am quite sure that such a construction exists (for complex topological K-theory): Start with a space X, from this produce the C^*-algebra of continuous complex-valued functions A:=Cont(X,C) - there you already have a ring encoding your space and I am sure this is step one.

The C^ *-algebra-K-theory of A is the topological K-theory of X. But the C^ *-K-theory is algebraic K-Theory made homotopy invariant (see Example 2.1.3 and what follows in the Notes of Cortinas you find here). Now I think one could enforce homotopy invariance somehow on a ring level, but can not recall how.

If you take the topological tensor product of A with K:=kernel of the map from the Toeplitz algebra to Cont(S^1,C) (see start of section 2.3 in Cortinas' notes), then algebraic and topological K-theory coincide on the result (Thm 3.4.1.3 in Cortinas' notes) - if this leaves topological K-theory unchanged for C^ *-algebras coming from spaces then that is what you do. If not then it is something else, which you might find in those notes. Actually I thought it sounded less complicated than what I wrote, something like, pass to infinite matrices over A...

There also is an article comparing algebraic and topological K-theory, from the handbook of K-theory, which might contain what you look for: here

Answer 2

Let $F =\mathbb{R}$ or $F=\mathbb{C}$. There is a close connection between the algebraic K-groups $K_i(F)$ and the topological K-groups $K^{-i}_F(P)$, where $P$ denotes the one-point space. I'm trying to learn this stuff at the moment so I hope someone can fill in the details here (post is community wiki), but I believe the statement is that if you take 2-completions the groups will be isomorphic.