# Sheaves on solenoids

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.

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.