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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.

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    $\begingroup$ The face relations act invertibly on one of your vector bundles, so this action factors through a localized category in which all these arrows have inverses. This new category is the groupoidificaiton of the face poset. If $X$ is connected, then this groupoid is connected, and hence equivalent to some group $G$. A vector bundle in your sense is then the same as a complex representation of $G$. $\endgroup$ Dec 8, 2019 at 17:53
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    $\begingroup$ @JohnWiltshire-Gordon with $G = \pi_1(X)$, right? $\endgroup$ Dec 8, 2019 at 19:00
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    $\begingroup$ Yes, I think so, since the face poset geometrically realizes to a subdivision of $X$. $\endgroup$ Dec 8, 2019 at 19:51
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    $\begingroup$ The "problem" with the definition is easily explained: You consider $GL_n(\mathbb{C})$ as a discrete group. It is also true in topological spaces that every map $X \to BGL_n(\mathbb{C})$ (where $GL_n(\mathbb{C})$ viewed with the discrete topology) factors homotopically over the one-truncation, which agrees with $B\pi_1(X)$ if $X$ is path-connected. This way, you only get vector bundles with flat connection. If you want all, you have to map your face-poset/simplicial set into the singular complex of the topological $BGL_n(\mathbb{C})$ or any Kan complex equivalent to that. $\endgroup$ Dec 9, 2019 at 8:17
  • $\begingroup$ @LennartMeier Sure, eventually I'd like to use a more complicated target category than the 1-truncation, but my impression is that the flat connection/local system case might be easier as a starting point. $\endgroup$ Dec 9, 2019 at 12:30

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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.

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  • $\begingroup$ Thanks, Mark! I've seen Segal's paper before but didn't think it constituted a roadmap towards actual machine computation. Time to give it another look... $\endgroup$ Dec 9, 2019 at 21:23

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