Whenever we have a continuous map of topological spaces $f: X \to Y$ and a sheaf $\mathcal{F}$ on $Y$ (of abelian groups for example), we get an induced pullback map
$$ H^n(Y, \mathcal{F}) \to H^n(X, f^\* \mathcal{F}), $$
which is described for example in Iversen, *Cohomology of Sheaves*, II.5.1. I would like to have an explicit description of this map in one particular case.


Suppose $X = Y = S^1$ and further suppose that the sheaf is a local system $\mathcal{L}$. Then its pullback $f^\* \mathcal{L}$ is also a local system. We have a nice description of local systems in terms of representations of the fundamental group, and the pullback is easily describable in this language: if
$$ \pi\_1(Y) \to \mathrm{Aut}(\mathcal{L}\_t) $$
is the representation defining $\mathcal{L}$, then $f^\* \mathcal{L}$ is described by the composition
$$ \pi\_1(X) \xrightarrow{f\_\*} \pi\_1(Y) \to \mathrm{Aut}(\mathcal{L}\_t). $$


The cohomology groups of a local system on $S^1$ are also nicely described in terms of representations: if we denote by $T \in \mathrm{Aut}(\mathcal{L}\_t)$ the image of a generator of $\pi\_1 (S^1)$, then
$$ H^0(S^1, \mathcal{L}) \cong \mathrm{Ker}(T - id) $$
$$ H^1(S^1, \mathcal{L}) \cong \mathrm{Coker}(T - id) $$
$$ H^n(S^1, \mathcal{L}) = 0), \quad \text{for any } n>1 $$
This is proved using Cech cohomology.


Now the homotopy type of our map $f: S^1 \to S^1$ is determined by its degree $d$, and the automorphism of the stalk that defines $f^\* \mathcal{L}$ is just $T^d$. Since $\mathrm{Ker}(T - id) \subset \mathrm{Ker}(T^d - id)$ it seems tempting to say that the pullback map on $H^0$ is the inclusion. However I don't know how to prove this, because I haven't found any result about some possible functoriality of Cech cohomology with respect to the covering. Furthermore, I don't know what the map on $H^1$ could look like. The first candidate I can think of would be some projection between the cokernels, but $\mathrm{Im}(T - id) \subset \mathrm{Im}(T^d - id)$ seems unlikely because of the first isomorphism theorem (if the kernels get larger then the images should get smaller).



What is the pullback morphism on sheaf cohomology of local systems in terms of representations of the fundamental group?
The circle $S^1$ is a $K(\mathbb{Z},1)$ space, so the cohomology of a local system on it can be identified with group cohomology. Let $R=\mathbb{Z}[T,T^{-1}]$ be the group ring of $\mathbb{Z}$. Given a local system $\mathcal{L}$,
$$H^i(S^1,\mathcal{L})= Ext\_R^i(\mathbb{Z}, \mathcal{L})$$
This can be computed as using the formulas you wrote, because
$$0\to R\xrightarrow{T-1}R\to \mathbb{Z}\to 0$$
is free resolution.


The degree $d$ map on the circle corresponds to $f:R\to R$ by $T\mapsto T^d$. This determines a map on group cohomology. This does not seem compatible with the above resolution, bu you can compute it in other ways. Using derivations mod inner derivations for $H^1$ is probably better for what you want.


