# Computing K-theory for cellular vector bundles

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.

• 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 $G$. Dec 8, 2019 at 17:53
• @JohnWiltshire-Gordon with $G = \pi_1(X)$, right? Dec 8, 2019 at 19:00
• Yes, I think so, since the face poset geometrically realizes to a subdivision of $X$. Dec 8, 2019 at 19:51
• 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. Dec 9, 2019 at 8:17
• @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. Dec 9, 2019 at 12:30

## 1 Answer

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 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.

• 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... Dec 9, 2019 at 21:23