Classifying Space for What is the Infinite Unitary Group?

Question

There is the well known result that $$ \left[X\to Gr_n\left(\mathbb{C}^{\infty}\right)\right] = Vect_n(X)$$ That is, homotopy classes of maps from a topological space $X$ into the $n$-Grassmannian are bijective with the isomorphism classes of complex-vector-bundles of rank $n$ over $X$.

In fancy language that says that the Grassmannians are the classifying spaces for the principal bundle with the unitary group.

My question is, in analogy to the above equation, what is the right hand side of the following:

$$ \left[X\to U\left(\mathbb{C}^{\infty}\right)\right] = ???(X)$$

where $U\left(\mathbb{C}^{\infty}\right)$ is the infinite-unitary group, as in Bott-periodicity and the three question marks denote the object I am inquiring about.

Answer 1

This paper might help. Atiyah and Hirzebruch say that the kernel of then natural augmentation $\varepsilon\colon K^0(X)\to \Bbb Z$ — which sends an (isomorphism class of a) vector bundle to its rank — can be identified with $[X\to B_U]$, where $B_U$ is the classifying space of the infinite unitary group. I can't say much more than this because I am just learning it myself, but this paper is also well-sourced. So the answer to your question is that ??? is the group of (what the authors call) stable vector bundles $\ker\varepsilon$.