Finding the right map between cohomology with local coefficients and Čech cohomology

Question

Let $X$ be a space which is paracompact, Hausdorff, and sufficiently nice that it has a universal covering space (and map) $p:\tilde{X}\to X$. Also, let $\pi:=\pi_1(X)$ and $A$ some $\mathbb{Z}[\pi]$-module, or if preferred, some abelian group on which $\pi$ acts. Under these conditions, there is a notion of cohomology with local coefficients which can be computed using cochain groups $C^n_\pi(X,A)=Hom_{\mathbb{Z}[\pi]}(C_n(\tilde X),A)$ where $C_n(\tilde{X})$ is the singular chain groups on $\tilde X$ and has an action of $\pi$ on it "inherited" from the action on $\tilde X$ itself by post-composition, that is, for $g\in \pi$, $\sigma \in C_n(\tilde X)$, $g\cdot \sigma$ is just $\sigma$ followed by the action of $g$. Let's call this cohomology $H^n_\pi(X,A)$.

On the other hand, there is another notion of cohomology with local coefficients whereby one defines a locally constant sheaf $\mathcal A$ on $X$ using the constant sheaf $A_\tilde{X}$ on $\tilde X$ and the action of $\pi$ on $A$, and takes the sheaf cohomology of that. It appears to be well known that (1) these two theories are essentially the same, and (2) for paracompact Hausdorff spaces, the Čech cohomology of a sheaf is the same as the sheaf cohomology of that same sheaf. That means that the Čech cohomology $\check{H}^n(X,\mathcal A)$ should be isomorphic to $H^n_\pi(X,A)$.

With all of this background out of the way, I can begin to describe my actual question. First of all, I cannot comprehend Sheaf cohomology, no matter how hard I try. Every reference I have found that describes it just seems impenetrable to me. On the other hand, I can more or less make sense of Čech cohomology, at least a lot better than "direct" sheaf cohomology. So: I would like to define some explicit map, $\check H^*(X ,\mathcal A)\to H^*_\pi (X, A)$ (or going the other way), which can exhibit this isomorphism more directly.

What appears natural for a map like this is to perhaps restrict to some convenient cofinal collection of covers of $X$ to get maps $C^*(\mathscr U,\mathcal A)\to C^*_\pi(X,A)$. A good candidate comes from the property of a universal cover, whereby every element $x\in X$ has a neighborhood $U_x$ so that $p^{-1}(U_x)$ composed of a disjoint union of open sets, each of which $p$ makes homeomorphic to $U_x$. It's not hard to show covers of the form $\mathscr U=\lbrace U_x : x\in X \rbrace$ with each $U_x$ a neighborhood like this is a cofinal collection, and then $p^{-1}(\mathscr U)$ is an open cover of $\tilde X$. There is a possibly useful property of Singular cohomology, too, which for any open cover $\mathscr V$ of $\tilde X$ makes the inclusion of the subcomplex $C_n(\mathscr V)=\langle \sigma: \text{im} (\sigma) \subset V\text{ for some }V \in \mathscr V \rangle$ a chain equivalence. That carries over to a cochain equivalence between $Hom_{\mathbb{Z}[\pi]}(C_n(\tilde X),A)$ and $Hom_{\mathbb{Z}[\pi]}(C_n(\mathscr V),A)$, and I'll call the latter group $C^n_\pi (\mathscr V , A)$.

My most promising candidate after several attempts, using these ideas, is to consider covers of $X$ as above, and for each one make a map $\psi_\mathscr{U}: \check{C}^n(\mathscr U , \mathcal A) \to C^n(p^{-1}(\mathscr U),A)$ by first applying a map $\iota_\mathscr{U}:\check{C}^n(\mathscr U , \mathcal A)\to \check{C}^n(p^{-1}(\mathscr U ), A_\tilde{X})$ defined by $(\iota_\mathscr{U} f)(p^{-1}(U_{x_0}),...,p^{-1}(U_{x_n}))=f(U_{x_0},...,U_{x_n})$, then picking out for each $\sigma \in C_n(p^{-1}(\mathscr U))$ the open set $U_\sigma:=U_{p(\sigma(v_0))}\cap ... \cap U_{p(\sigma(v_n))}$ and setting $(\psi_\mathscr{U} f)(\sigma):=[(\iota_\mathscr{U} f)(p^{-1}(U_\sigma))](\sigma(v_0))$. I say this is promising because evaluating at some point is needed in order to get the $\pi$-equivariance expected of an element of $C^n_\pi(p^{-1}(\mathscr U),A)$, and because it seems likely to give "injectivity" and "surjectivity" at least at the cohomology level after taking the direct limit defining Čech cohomology.

The only problem is that I'm not entirely sure this actually gives a cochain map. Running through the calculations, I've found that:

$(\partial\psi_{\mathscr{U}}f)(\sigma)-(\psi_{\mathscr{U}}\partial f)(\sigma)=[(\iota_{\mathscr{U}}f)(p^{-1}(U_{p(\sigma(v_{1}))}),...,p^{-1}(U_{p(\sigma(v_{n+1}))}))](\sigma(v_{1}))-[(\iota_{\mathscr{U}}f)(p^{-1}(U_{p(\sigma(v_{1}))}),...,p^{-1}(U_{p(\sigma(v_{n+1}))}))](\sigma(v_{0}))$

This is more or less because the '0 boundary' of the simplex sends $v_0$ to the same place the original simplex sends $v_1$ to, while the rest of the boundaries send $v_0$ to $\sigma(v_0)$. Of course, this needs to be $0$ to get a cochain map, but I'm not sure these two terms are the same (or if they are, how to show it).

So: Is this the wrong map again? If it is, what's the right map? Or, is there some really simple way to see that those two terms are the same and the difference really is $0$? Are there any references that have attempted something similar?

Answer 1

Like I explained in the comments, there is no reason to expect the existence of a particularly explicit direct map between the Cech and singular complexes, since we naturally obtain a zig-zag $$ \check C^*(\{U_i\}_{i\in I},\mathcal A)\to \check C^*(\{U_i\}_{i\in I},\operatorname{Sing}^\bullet(\mathcal A))\leftarrow \operatorname{Sing}^\bullet(\mathcal A)(X) = \operatorname{Hom}_{\mathbb Z\pi}(\operatorname{Sing}^\bullet(\widetilde X),A) $$ where $\operatorname{Sing}^\bullet(\mathcal A)(-)$ is the presheaf which assigns to $U\in X$ the $\pi$-equivariant maps from $p^{-1}(X)$ to $A$ and $p:\widetilde X\to X$ is the universal cover. Note that the individual terms $\operatorname{Sing}^\bullet(\mathcal A)(-)$ are not sheaves: if you know the values on small simplices, i.e. those inside an open set of a cover, you can't recover the values on all simplices. However, the inclusion of small simplices is a quasiisomorphism since every simplicial chain is homologous to its barycentric subdivision, which will eventually be small. This shows that the left-pointing map is a quasiisomorphism; the right-pointing map is a quasiisomorphism if all intersections are empty or contractible, i.e. $\{U_i\}_{i\in I}$ is a good cover. For nice spaces (e.g. CW complexes) the Cech cohomology groups are computed by the Cech complex of a good cover. In that case, it is possible to get an explicit chain map from the singular cochain complex to the Cech complex as follows:

The good cover $\{U_i\}_{i\in I}$ determines an abstract simplicial complex $S_U$, i.e. a collection of finite subsets of $I$ (namely, those for which the intersection is nonempty) which is closed under taking subsets. We may inductively choose for $J\in S_U$ a simplex $\sigma_J:\Delta^{|J|-1}\to X$ whose image lands in $\bigcup_{j\in J} U_j$ and whose restriction to a face is given by the previous choice in the obvious sense (send the barycentre to a point in $\bigcap_{j\in J} U_j$ and iteratively "cone off" the maps on the lower-dimensional simplices, which is possible by contractibility). These singular simplices form a subcomplex of the singular chains, and it's straightforward to show that restricting singular cochains to this subcomplex gives a cochain map from these to the Cech complex.

In the other direction, one would have to pick a chain homotopy inverse to the inclusion of small simplices, which is not possible to do explicitly as far as I know.