How do we compute homology from the nerve $\text{Nrv}(\Sigma)$ for $\Sigma=\{S^n,\dots,S^n\}$ covering the CW complex $M$?

Question

Suppose a CW complex $M$ is given by the union of $n$-spheres, namely $M=\bigcup_{\alpha\in A}S^n$, without knowledge of intersections. The only requirement is that the union covers $M$. Let $\Sigma=\{S^n,\dots,S^n\}$ be a finite collection of sets, with cardinality $|A|$. The nerve consists of all subcollections whose sets have a non-empty common intersection, $\text{Nrv}(\Sigma)=\left\{X\subseteq\Sigma\big|\bigcap X\ne\emptyset\right\}$, which is an abstract simplicial complex. The nerve should look something like this (e.g. a Čech complex):

That is, we are allowed to arrange the spheres in a configuration of our choosing, so long as the configuration still covers $M$. (Indeed, we can "pull" the spheres apart as much as possible so that they still cover $M$-an optimal configuration-with the least amount of spheres used).

How do we compute homology groups $H_k(M;\mathbb{Z})$ from the nerve $\text{Nrv}(\Sigma)$ for $\Sigma=\{S^n,\dots,S^n\}$ covering the CW complex $M$?

Idea: Since we need some information on intersections, suppose we construct the following optimal configuration. Begin with two $n$-spheres attached at a base point, namely $S^n\vee S^n$. Then construct two other $n$-spheres that pass through the intersection point of $S^n\vee S^n$. Finally, we continue the process by constructing other spheres $S^n$ that pass through the intersection "points" of other spheres. We write "points" realizing that the intersection of two $n$-spheres is actually an $(n-1)$-sphere. The points we refer to are those corresponding to the two intersections in the $S^2$ orthographic projection onto a plane. We can of course vary the radius of the sphere under this construction.

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As mentioned by Mike Miller, the condition we want is that the $k$-fold intersections, for $k$ sufficiently large, are all empty or contractible. Then the Čech complex of this cover (with constant sheaf $\mathbb{Z}$) recovers the homology of the manifold. The general case where $k$-fold intersections are not contractible instead takes the form of a spectral sequence involving cohomology of the various intersections. I am not sure, however, how to make this mathematically precise.

Any help would be much appreciated. Thanks in advance!

Answer 1

Very interesting question, unfortunately this is not enough to know the nerve. For an explicit counterexample consider $M_1$ and $M_2$ given as the union of two circles, where the circles in $M_1$ intersect twice and the circle in $M_2$ intersect four times. The nerves are isomorphic but the homology groups are not isomorphic.

When the covering is given by open balls, in order to be able to compute homology using the nerve you need to assume strong conditions on the intersections (they should be all contractible or empty) on your cover. You need probably strong conditions here too but I can't think of a good condition right now.