Computing K-theory for cellular vector bundles

Question

One of the most computationally convenient properties of singular cohomology $X \mapsto H^\bullet(X;\mathbb{Z})$ is the fact that one can extract it from a good cover $\{U_i\}$ of $X$ via Cech cohomology, thus bypassing any scary spectral sequences. Is there a similar algorithmic shortcut available for other cohomology theories? I'm mostly asking about the possibility of computing the complex $K$-theory of a finite simplicial complex $X$ in the presence of a good cover (without using the Atiyah-Hirzebruch SS).

There has been at least one old question about this with no answers in sight, which doesn't sound too promising. Perhaps it will help if I propose a candidate definition for vector bundles on $X$:

Def: A vector bundle $V$ (of rank $n$) over a simplicial complex $X$ is a functor from the poset of simplices in $X$ to the group $\text{GL}_n(\mathbb{C})$. In other words, each face relation $\sigma < \tau$ is assigned an invertible complex $n \times n$ matrix $V_{\sigma<\tau}$ so that (1) the matrix $V_{\sigma < \sigma}$ is the identity for every simplex $\sigma$, and (2) the relation $V_{\tau < \gamma} V_{\sigma < \tau} = V_{\sigma < \gamma}$ holds across any triple $\sigma < \tau < \gamma$ of simplices.

Let's say I don't care about honest vector bundles on the geometric realization of $X$, and focus exclusively on these discrete guys which are constructible with respect to the fixed simplicial decomposition. Is there a non-spectral-sequencey way to compute the obvious notion of $K_\bullet(X)$ for the collection of all such vector bundles? Maybe we can at least compute Chern classes? I'm happy to assume that $X$ is a triangulated smooth manifold if that helps in any way.

Answer 1

The fact that singular cohomology agrees with Cech cohomology of a good cover $\mathcal{U}$ is a sort of generalised Mayer-Vietoris principle, as explained in Section II.8 and III.15 of Bott and Tu's "Differential Forms in Algebraic Topology". You get a double complex $C^*(\mathcal{U},S^*)$ of Cech-singular cochains, and the resulting Mayer-Vietoris sequence degenerates in two ways to give the result.

To generalise this to other cohomology theories, you'd need to use the spectral sequence of the pre-simplicial space $$ X \leftarrow \bigsqcup_{i_0} U_{i_0} \stackrel{\leftarrow}{\leftarrow}\bigsqcup_{i_0<i_1} U_{i_0}\cap U_{i_1}\stackrel{\leftarrow}{\stackrel{\leftarrow}{\leftarrow}}\ldots $$ as explained for example in this answer. A friendly reference is

Segal, Graeme, Classifying spaces and spectral sequences, Publ. Math., Inst. Hautes Étud. Sci. 34, 105-112 (1968). ZBL0199.26404.

This doesn't completely avoid spectral sequences of course, but I believe if you follow it through it should give a way to compute K-theory just from the combinatorics of the good cover.