Let $\pi:\mathcal{X} \to U$ be a family of hypersurfaces (not necessarily smooth) in $\mathbb{P}^n$ for some $n \ge 3$. Assume that $U$ is simply connected (under analytic topology). For any pair $u,v \in U$ does there exist an isomorphism between their homology groups i.e., is $H\_k(X\_u,\mathbb{Z}) \cong H\_k(X\_v,\mathbb{Z})$ where $X\_u$ and $X\_v$ are fibers of $\pi$ at $u, v$, respectively?



Isomorphism of homology groups under deformation
If $\pi$ is smooth, then the answer is yes, because by a theorem of Ehresmann, the fibres are diffeomorphic. In general, however, the answer is no. To see this, take the family of all cubics in $\mathbb{P}^2$, the ranks of $H\_1(X\_u)$ will take all possible values in the set $\{0,1,2\}$.


