Relationship between topological and Quillen's K-theory

Question

Up until now, I've taken it for granted that the topological k-theory of a space $X$ is equal to the K-theory of vector bundles on $X$. $K_0$'s of both coincide (Serre-Swan) however, is it the case that $K_i(\text{Vect}(X)) \cong K_{top}^i(X)$? ($\text{Vect}(X)$ being the category of (real/complex) vector bundles over $X$.)

I'm doubting this now, however, and I can't seem to find anything written up explicitly on the relationship between the two.

In what way, if at all, is topological K-theory a special case of Quillen's K-theory? What is the relationship?

Answer 1

I've taken it for granted that the topological K-theory of a space X is equal to the K-theory of vector bundles on X

This is not true even for a point: $K_1^{alg}(\mathrm{Vect}(pt))=K_1^{alg}(\mathbb C)=\mathbb C ^\times$.

(Of course, there is a map from $GL(\mathbb C)$ with discrete topology to ‘ordinary’ $GL(\mathbb C)$ inducing a map $K^{\text{alg}}(\mathbb C)\to K^{\text{top}}(pt)$; but this map is far from being isomorphism. That’s how I think about the difference in general case: in algebraic K-theory we forget about non-trivial topology [on our group]…)

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Looks like some results in the direction you want are discussed in Michael Paluch. Algebraic K-theory and topological spaces K-theory:

In this note we discuss the algebraic and topological K-theories of an admissible space X and demonstrate how one may recover the connective topological K-theory of X from the algebraic K-theory of a simplicial ring which encodes the topological structure of the Fréchet algebra of continuous functions on X (...)