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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?

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up vote 3 down vote accepted

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 $\{0,1,2\}$.

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@Arapura: Thanks a lot for the answer. If we replace homology by cohomology, can we get an affirmitive answer (under added conditions if necessary)? –  Jana Oct 15 '13 at 22:59
No, the ranks of cohomology and homology are the same by universal coefficients. –  Donu Arapura Oct 16 '13 at 14:30

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