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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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Whitney stratifications of hypersurfaces

Suppose that $X$ is a Whitney stratified algebraic variety with strata $\{S_i\}.$ Suppose that $Z$ is a hypersurface of $X$ which transversely intersects all strata of $X$, i.e. $S_i \cap Z$ is a smoo …
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