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Let $(X_n)$ be a tower of finite covering maps of compact smooth manifolds, with $f_{s,t} : X_t\to X_s$ the maps, and $\Lambda_n := f_{n,0}^{-1}\Lambda$, with $\Lambda$ the constant abelian sheaf on $X_0$ with value $\Lambda$, an abelian group.

Clearly $\Lambda_n$ is the constant sheaf on $X_n$ with value $\Lambda$ too.

There are, then, transition maps $f_{n+1,n}^{-1}\Lambda_n\to\Lambda_{n+1}$.

A base for the topology on $X := \varprojlim_{n\ge 0}X_n$ is given by opens of the form $U:= \varprojlim_{m\in I}U_m$, $I\subset\mathbf{N}$, $U_m\subset f^{-1}_{m,k}(U_k)$ open, $m,k\in I$, and the assignment:

$$U\mapsto \varinjlim_{m\in I}\Lambda_m(U_m)$$

uniquely determines a sheaf $\mathcal{F}_{\Lambda}$ on $X$.

One can give $X$ the structure of a solenoid in the sense of Smale (see here and the book by Moore and Sochet).

Say $\Lambda = \mathbf{C}$ the sheaf of locally constant smooth functions with value in the complex numbers. On $X$ one can talk about the sheaf $\mathcal{F}’$ of continuous complex valued functions that are smooth along the “leaves” and continuous along the transversals (these are copies of the Cantor set).

Are $\mathcal{F}_{\Lambda}$ and $\mathcal{F}’$ the same sheaf on $X$? Do they at least have the same cohomology?

In case this question has a negative answer, then:

Is $\mathcal{F}_{\Lambda}$ the constant abelian sheaf on $X$ with value $\Lambda$?

I expect the answer to this last question is “no”.

Related: this seems a related question, esp the answer by Ian Agol.

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I think that the sheaf $\mathcal{F}_\Lambda$ is the constant sheaf with stalk $\Lambda$.

A basis of open sets in a solenoidal space is given by the sets $f_k^{-1}(U_k)$ where $U_k \subseteq X_k$ is open and $f_k \colon X \to X_k$ is the canonical projection. Now if you take $U_k \subseteq X_k$ so small that it is regularly covered, that is $$ f_{k,m}^{-1}(U_k) \cong F_m \times U_k $$ for all $m > k$ and finite sets $F_m$, then $$f_k^{-1}(U_k) \cong F \times U_k$$ where $F = \varprojlim F_m$ is a profinite set. Then the group of locally constant $\Lambda$-valued functions on $f_k^{-1}(U_k)$ is the direct limit of the locally constant $\Lambda$-valued functions on $F_m \times U_k$ and this is, by your definition, the group of sections of the sheaf $\mathcal{F}_\Lambda$ over $f_k^{-1}(U_k)$.

The sheaf $\mathcal{F}'$ is much larger and moreover, it seems to me that it is a soft sheaf. This implies that $\mathcal{F}'$ is acyclic, i.e. trivial higher cohomology groups. If you want to compute the cohomology of $\mathcal{F}_\mathbb{C}$ then the canonical map $\mathcal{F}_\mathbb{C} \to \mathcal{F}'$ is a useful first step in constructing a resolution of $\mathcal{F}_\mathbb{C}$ by acyclic sheaves.

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