Let $f\colon E\to B$ be a fiber bundle with a connected fiber $F$, $f$ is proper. Let $\underline{\mathbb{C}}_E$ be the constant sheaf on $E$. Let $f_*(\underline{\mathbb{C}}_E)$ denote its direct image in the derived category $D(Sh_B)$ of sheaves of $\mathbb{C}$-vector spaces. Since $F$ is connected, it is clear that $\tau_{\leq 0}(f_*(\underline{\mathbb{C}}_E))=\underline{\mathbb{C}}_B$. Hence there exists an exact triangle in $D(Sh_B)$ $$\underline{\mathbb{C}}_B\to f_*(\underline{\mathbb{C}}_E)\to \mathcal{F},$$ where $\mathcal{F}\in D^{\geq 1}(Sh_B)$.

**Question.** Is it true that the above exact triangle splits? If not, under what conditions this is true?

**Remark.** I am aware of a situation when the above triangle splits, but it is too restrictive for my purposes; in fact much more it true in that case. Let $f\colon E\to B$ be a smooth morphism of smooth projective complex algebraic manifolds. Then it is a special case of the decomposition theorem due to Beilinson-Bernstein-Deligne-Gabber that not only the above exact triangle does split, but moreover $f_*(\underline{\mathbb{C}}_E)$ is isomorphic in $D(Sh_B)$ to the direct sum of its cohomology sheaves (which are necessarily shifted local systems).