Let $\pi: M^{n+k} \to N^n$ be a fibre bundle with fibre $F$ between compact smooth manifolds. What are “mild” sufficient conditions on the topology of $M$, $N$ and $F$ so that given a closed $p$-submanifold $\Sigma^p \subset N$ which is nontrivial in $H\_p(N; \mathbb{Z})$, the preimage $\pi^{-1}(\Sigma)$ is nontrivial in $H\_{p+k}(M;\mathbb{Z})$? I guess spectral sequences may help, but I am not familiar with them.



Conditions under which the preimage of a submanifold in nontrivial in homology
A sufficient condition with $\mathbf{Q}$ coefficients should be that the fiber $F$ has non-vanising Euler characteristic.


Consider the Gysin map (aka as fiber integraion) $$\pi^{!} = \int\_{\pi}\colon H^{\ast}(M;\mathbf{Q}) \to H^{\ast-k}(N;\mathbf{Q});$$ then $\langle \alpha , \pi^{-1}[S]\rangle = \langle \pi^{!} \alpha, [S]\rangle$ for $\alpha \in H^{p+k}(M)$.


So if we can ensure that the RHS is non-zero, it follows that $\pi^{-1}[S] \neq 0$.


But this is the case if $\chi(M) \neq 0$ as with $T\_{\pi}$ the vertical tangent bundle of $\pi$, the fact that $\pi^{!}$ is a $\text{H}^{\ast}(N)$-module homomorphism implies that $$\pi^!(e(T\_{\pi})\pi^{\ast}(\alpha)) = \pi^!(e(T\_{\pi}))\alpha = \chi(F)\alpha,$$
so $\pi^!$ is (rationally) surjective in this case.


