Cohomology algebra of a fibration whose spectral sequence degenerates in the second term

Question

Let $f\colon E\to B$ locally trivial bundle of 'nice' topological spaces (say finite CW-complexes) with fiber $F$. Assume also that the base $B$ is simply connected.

Assume that either the cohomology spectral sequence (with coefficients in a field) degenerates in the second term $H^*(B)\otimes H^*(F)$ or that the push-forward in the derived category of the constant sheaf is isomorphic (in derived category) to the direct sum of its cohomology sheaves (I think that the second condition implies the first one, but I am not sure about the converse).

Is it true that there exists an isomoprhism of graded algebras $$H^*(E)\simeq H^*(B)\otimes H^*(F)$$ which is compatible with the pull-back $f^*\colon H^*(B)\to H^*(E)$ and with restriction to fiber $H^*(E)\to H^*(F)$?

Answer 1

The statement is false, here is a counterexample. First note that for a Lie Group $G$ and its closed subgroup $H$, we have a fibration $G/H\to BH\to BG$. $BG$ and $BH$ are not finite, but they are almost just as good, if you really want finite CW complex, you can restrict the fibration over a finite skelton of $BH$. Now, let's take $G=S^3$, $H=S^1$. Then $G/H$ is homeomorphic to $S^2$. If we restrict this fibration to the $2$-skelton of $BH$, one gets the example mentioned in the comment by @Gustavo Granja, but let's look at the whole thing. We have $BG=CP^{\infty }$, $BH=HP^{\infty }$ so, the $E^2$ term of the spectral sequence is $$P[z]\otimes \Lambda (y)$$ with $y$ in degree 2, $z$ in degree 4, whereas the cohomology of the total space is $P[x]$ with $x$ in degre 2.