# What is the pullback morphism on sheaf cohomology of local systems in terms of representations of the fundamental group?

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).

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.
• To make this compatible with that resolution, you can map by multiplication by $1 + T + T^2 + \dots + T^{d-1}$ on the earlier copy of $R$. I think this shows that the multiplication-by-$1 + T + T^2 + \dots + T^{d-1}$ map on coinvariants is the map on $H^1$. Commented Oct 13, 2021 at 18:31
• @EduardodeLorenzo One reference which I like is Brown Cohomology of groups. Everything goes through with $\mathbb{Q}$-coefficients. Commented Oct 14, 2021 at 12:40