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.

  • 4
    $\begingroup$ 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
  • 1
    $\begingroup$ @JohnWiltshire-Gordon with $G = \pi_1(X)$, right? $\endgroup$ Dec 8, 2019 at 19:00
  • 1
    $\begingroup$ Yes, I think so, since the face poset geometrically realizes to a subdivision of $X$. $\endgroup$ Dec 8, 2019 at 19:51
  • 1
    $\begingroup$ 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

1 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.

  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.