# cohomology of surfaces with local coefficients

Let $$X$$ be an orientable surface (real manifold of dim 2). Let $$F$$ be a local system (a locally free sheaf of abelian groups) on $$X$$. Is is true that $$H^n(X,F)=0$$ for $$n>2$$? And that $$H^n(X,F)=0$$ for $$n \geq 2$$ if $$X$$ is non-compact?

(Here the cohomology used is either Grothendieck's sheaf cohomology or the standard singular cohomology with local systems -- they amount to the same thing).

The analagous statement is true for any $$n$$-dimensional CW complex $$X$$ equipped with a local system $$F$$. I am sure there is a sheafy proof which works in much more generality but I am less comfortable language so I won't try to find one.
Represent $$F$$ by a homomorphism $$\rho_F: \pi_1 X \to \text{Aut}(A)$$, where $$A$$ is some abelian group. Then the cohomology groups you are interested are given by taking your favorite model for $$C_*(\widetilde X;\Bbb Z)$$ as a free $$\Bbb Z[\pi_1(X)]$$-module, and taking the homology groups of $$\text{Hom}_{\pi_1 X}(C_*(\widetilde X;\Bbb Z), A).$$ In particular, if you take the cellular chain complex of $$X$$, one may use the corresponding cellular chain complex for the cell decomposition of the universal cover. In particular, in this model $$C^k(X;F) = \text{Hom}_{\pi_1 X}(C_k(\widetilde X;\Bbb Z), A) = 0 \;\;\;\;\;\; \text{for } k>n.$$
Therefore the cohomology groups $$H^k(X;F)$$ vanish for $$k > n$$.
• I think a Cech-to-sheaf cohomology spectral sequence for a sufficiently well-chosen open cover proves the desired result as well, so long as the cohomology of $\mathcal F$ vanishes in positive degrees on sufficiently small contractible neighborhoods of any point. This is not true for arbitrary sheaves, and is for locally constant sheaves. Maybe constructible is enough? – Mike Miller Jan 12 at 23:22