In practice, we wish to know if the topology of the base manifold and that of general fibres can somewhat "control" the topology of the total space.
Precisely, let $\pi: X\to B$ be a flat morphism between projective complex manifolds where the base space $B$ and fibres are supposed to be connected. By Sard's theorem, one knows that general fibres of $\pi$ are smooth, and diffeomorphic to each other by Ehressmann's theorem.
On knowing the Betti numbers of the base space $B$ and those of a general fibre $F$, does one have something controlling the Betti numbers of the total space $X$ ?
P.S. In the case where $\pi: X\to B$ is furthermore supposed to be smooth (submersive), by Ehressmann's theorem, $\pi$ is in fact a topological fibration. So one may utilise things like Leray-Serre's spectral sequence to control the Betti numbers of $X$. For example, in the case when $B$ is simply-connected, one has $$b_k(X)\leq \sum_{p+q=k}b_p(F)b_q(B),$$ where $b_i$ signifies the Betti numbers.