TITLE: A nontrivial principal bundle which satisfies Leray-Hirsch theorem
QUESTION [12 upvotes]: What is an example of a nontrivial principal bundle whose fibre space $G$, total space $P$ and base space $M$ are compact connected manifolds (the fiber $G$ is a compact Lie group) such that $$H^*(P,\mathbb{Q})=H^*(G,\mathbb{Q})\otimes H^*(M,\mathbb{Q})$$

REPLY [2 votes]: There is also a low-dimensional example. Consider the canonical map $\mathbb RP^3 \to \mathbb RP^{\infty} \to \mathbb CP^{\infty}$, classifying the unique non-trivial principal $S^1$-bundle with base $\mathbb RP^3$. Its rational Serre spectral sequences collapses. Of course, its integral Serre spectral sequence does not collapse. The total space of this bundle is the 4-manifold $E = S^1 \times_{\mathbb Z/2} S^3$ whose fundamental group is $\mathbb Z$, not $\mathbb Z \times \mathbb Z/2\mathbb Z$.