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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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‘Sur les groupes de tresses’ by Egbert Brieskorn
I have a very specific question. How does one check the following map
${\mathbb C}^n-\bigcup_{i\neq j} \{z \mid z_i= \pm z_j\}\to {({\mathbb C}^*)}^{n-1}-\bigcup_{i\neq j}\{w \mid w_i= w_j\}$ defined …
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Implicit function theorem and its consequence
Let $f:{\Bbb C}^{n+1}\to {\Bbb C}^n$ be a map defined by homogeneous polynomials. There is a point $p\in f^{-1}(0)$ and a neighbourhood $U$ of $p$ in $f^{-1}(0)$, such that $d(f)$ has rank $n$ at ever …
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Thom's first isotopy lemma
Thom's first isotopy lemma says that given $f:M\to P$ a smooth map between smooth manifolds and a closed Whitney stratified subset $S$ of $M$, such that
$f|_S:S\to P$ is proper and $f|_X:X\to P$ is a …