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).



On push-forward of the constant sheaf for fibrations
No, it is not true. For example, let $E=\mathbb C^2\backslash \{ 0\}$, $B=\mathbb C\mathbb P^1$ (with the obvious map $f$). Then the pushforward as a complex of sheaves on $\mathbb C\mathbb P^1$ has the following cohomology: constant sheaf in degree 0 and constant sheaf in degree +1. If the triangle you are asking about were split, then the cohomology of the pushforward would be $H^\*(\mathbb C\mathbb P^1)\oplus H^\*(\mathbb C\mathbb P^1)[-1]$. But it the cohomology of the pushforward is the same as $H^\*(\mathbb C^2\backslash \{ 0\})$, which has total dimension 2 and not 4.


