Let $\pi:\mathcal{X} \to T$ be a family of smooth projective complex varieties. Assume $T$ is quasi-projective, reduced, irreducible but not smooth and of positive dimension. Let $\mathcal{Z}$ be a codimension $p$ subscheme of $\mathcal{X}$ flat over $T$. Denote by $\pi': \mathcal{Z} \to T$. Under what conditions on $p$ or $\mathcal{Z}$, we can find an element $\xi \in H^{2p}(\mathcal{X},\mathbb{Z})$ such that $\xi_t \in H^{2p}(\mathcal{X}_t,\mathbb{Z})$ (obtained by the pull-back morphism from $H^{2p}(\mathcal{X},\mathbb{Z})$ to $H^{2p}(\mathcal{X}_t,\mathbb{Z})$ induced by $\mathcal{X}_t \hookrightarrow \mathcal{X})$ is the class of $[\mathcal{Z}_t]$ (under the cycle class map) where $\mathcal{Z}_t:=\pi'^{-1}(t)$ for all $t \in T$?

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goodtextbook in algebraic topology would do, but personally I learned it long ago and in Russian, so I cannot recommend any English text. (In fact, I just don't know any satisfactory English textbook :) $\endgroup$ – Alex Degtyarev Mar 1 '14 at 12:01