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))?


2 Answers 2


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 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


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.