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The study of differentiable manifolds and differentiable maps. One fundamental problem is that of classifying manifolds up to diffeomorphism. Differential topology is what Poincaré understood as topology or “analysis situs”.
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A question related to fiber bundle
Note that $f$ is smooth as a map of varieties. If $f$ comes from a direct product $\mathbb{C}^3\approx S\times \mathbb{C}$ then $S\approx \mathbb{C}^2$. See Section 5.1 of $\mathbb{A}^1$-homotopy theo …