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- W2980454081 abstract "In [22] Milnor proved that a real analytic map $fcolon (R^n,0) to (R^p,0)$, where $n geq p$, with an isolated critical point at the origin has a fibration on the tube $f|colon B_epsilon^n cap f^{-1}(S_delta^{p-1}) to S_delta^{p-1}$. Constructing a vector field such that, (1) it is transverse to the spheres, and (2) it is transverse to the tubes, he inflates the tube to the sphere, to get a fibration $varphicolon S_epsilon^{n-1} setminus f^{-1}(0) to S^{p-1}$, but the projection is not necessarily given by $f/ |f|$ as in the complex case. In the case $f$ has isolated critical value, in [9] it was proved that if the fibres inside a small tube are transverse to the sphere $S_epsilon$, then it has a fibration on the tube. Also in [9], the concept of $d$-regularity was defined, it turns out that $f$ is $d$-regular if and only if the map $f/|f|colon S_epsilon^{n-1} setminus f^{-1}(0) to S^{p-1}$ is a fibre bundle equivalent to the one on the tube. In this article, we prove the corresponding facts in a more general setting: if a locally surjective map $f$ has a linear discriminant $Delta$ and a fibration on the tube $f|colon B_epsilon^n cap f^{-1}(S_delta^{p-1} setminus Delta) to S_delta^{p-1} setminus Delta$, then $f$ is $d$-regular if and only if the map $f/ |f|colon S_epsilon^{n-1} setminus f^{-1}(Delta) to S^{p-1} setminus mathcal{A}$ (with $mathcal{A}$ the radial projection of $Delta$ on $S^{p-1}$) is a fibre bundle equivalent to the one on the tube. We do this by constructing a vector field $tilde{w}$ which inflates the tube to the sphere in a controlled way, it satisfies properties analogous to the vector field constructed by Milnor in the complex setting: besides satisfying (1) and (2) above, it also satisfies that $f/ |f|$ is constant on the integral curves of $tilde{w}$." @default.
- W2980454081 created "2019-10-25" @default.
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- W2980454081 date "2019-11-20" @default.
- W2980454081 modified "2023-10-16" @default.
- W2980454081 title "Equivalence of Milnor and Milnor-Lê fibrations for real analytic maps" @default.
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- W2980454081 doi "https://doi.org/10.1142/s0129167x19500782" @default.
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