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 $$H^\*(P,\mathbb{Q})=H^\*(G,\mathbb{Q})\otimes H^\*(M,\mathbb{Q})$$



A nontrivial principal bundle which satisfies Leray-Hirsch theorem
Let $P$ be any $SU(2)$-bundle on $X$ with vanishing second Chern class $c\_2(P)$. The hypotheses of the Leray-Hirsch theorem are satisfied if there is a class in $H^3(P)$ which restricts to the generator of $H^3(SU(2))$. This happens if and only if in the Leray spectral sequence, the map $d\_3: H^0(X,H^3(SU(2))) \to H^4(X,H^0(SU(2))$ vanishes (since $H^i(SU(2))$ is concentrated in degrees $0$ and $3$, so the only nontrivial differentials are on the third page). This map is exactly the top Chern class.


There are nontrivial rank 2 complex bundles with vanishing Chern classes. The vanishing of $c\_1$ implies that the $U(2)$ structure may be reduced to $SU(2)$, at which point the above argument shows that the hypotheses of Leray-Hirsch are satisfied.


