I found this example in the book Differential Forms in Algebraic Topology of Bott and Tu :
Exercise 10.7 (Cohomology with twisted coefficients). Let $\mathscr{F}$ be the presheaf on $S^1$ which associates to every open set the group $\mathbb{Z}$. We define the restriction homomorphism on the good cover $\mathfrak{U} = \{U_0, U_1, U_2\}$ (Figure 10.1) by: $$\begin{align} \rho^0_{01} & = \rho^1_{01} = 1,\\ \rho^1_{12} & = \rho^2_{12} = 1,\\ \rho^2_{02} & = -1,\; \rho^0_{02} = 1, \end{align}$$ where $\rho^i_{ij}$ is the restriction from $U_i$ to $U_i \cap U_j$. Compute $H^*(\mathfrak{U}, \mathscr{F})$. (Cf. presheaf on an open cover, p. 142.)
I found that $H^1(\mathfrak U, \mathcal F) = 0$. Is there any geometric interpretation of this result (and motivation for defining such restriction map)?