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Suppose $X$ is a pointed and connected topological space and $L$ is a local system on $X$. By this I mean that $F$ is a locally constant sheaf on $X$, or equivalently, a representation of the fundamental group $\pi_{1}(X,x)$ of $X$.

A classical example of a local system is the sheaf

$$ U\mapsto L(U) = \{f:U\to \mathbb{C}\mid \mathcal{D}(f) = 0\} $$

of complex solutions to a system of differential operators $\mathcal{D}$ on $X$. This generalises to the sheaf of horizontal sections of a vector bundle with flat connection $(V,\nabla)$ on $X$. In fact, all local systems on $X$ arise in this way.

Given a local system $L$ on $X$, one can define cohomology groups $H^{n}(X,L)$ of $X$ with coefficients in $L$. Sadly, I only understand this theory in a formal way. The aim of my question is to understand cohomology with coefficients in a local system in elementary differential geometric terms. To this end I have two questions:

Question 1: What is a geometric interpretation of cohomology classes $\alpha\in H^{n}(X,L)$ when $L$ is interpreted as a vector bundle with flat connection?

Question 2: In terms on solutions to differential equations, what does it mean for cohomology to vanish in degree $n$? What does it mean for a map of local systems to induce the zero map on cohomology?

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  • $\begingroup$ A for $Q1$, geometers tend to prefer working with homology rather than cohomology, and attempting to represent homology classes by geometric subobjects. Its possible that working with homology with local coefficients might be more approachable for this question. $\endgroup$ – Tyrone Sep 11 '19 at 10:16
  • $\begingroup$ It depends on what you mean by "geometric". Do you find de Rham cohomology geometric? If so, then at least for manifolds and local systems given by (local) sections of flat bundles, you can define a "twisted" analogue of the de Rham complex. I suggest you clarify your question. $\endgroup$ – Moishe Kohan Sep 12 '19 at 3:59
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I don't have an answer for twisted complex systems, but if $\Bbb Z_w$ denotes the twisted system with twisting $w: \pi_1 X \to \{\pm 1\}$, then $X$ comes equipped with a canonical real line bundle $\Bbb R_w$, and geometric representatives for $H_*(X;\Bbb Z_w)$ are given by manifolds $M$ equipped with a map $f: M \to X$ and an isomorphism $\det(TM) \cong f^*\Bbb R_w$. This is called a twisted orientation.

This is useful when trying to geometrically understand Poincare duality on a non orientable manifold.

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