Equivalence of Milnor and Milnor-Lê fibrations for real analytic maps

Abstract

In [22] Milnor proved that a real analytic map $f:(R^n,0)\to (R^p,0)$, where $n\ge p$, with an isolated critical point at the origin has a fibration on the tube $f|:B_{\epsilon }^n\cap f^{-1}(S_{\delta }^{p-1})\to S_{\delta }^{p-1}$. 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 $\phi :S_{\epsilon }^{n-1}\setminus f^{-1}(0)\to S^{p-1}$, but the projection is not necessarily given by $f/\Vert f\Vert$ 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 $f/|f\Vert :S_{\epsilon }^{n-1}\setminus f^{-1}(0)\to S^{p-1}$ 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 $f|:B_{\epsilon }^n\cap f^{-1}(S_{\delta }^{p-1}\setminus \Delta )\to S_{\delta }^{p-1}\setminus \Delta$, then $f$ is $d$-regular if and only if the map $f/\Vert f\Vert :S_{\epsilon }^{n-1}\setminus f^{-1}(\Delta )\to S^{p-1}\setminus \mathcal{A}$ (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 $f/\Vert f\Vert$ is constant on the integral curves of $\tilde{w}$.

23 pages. This is a corrected version of the article published in Internat. J. Math., 30(14):1950078, 1-25, 2019. https://www.worldscientific.com/doi/abs/10.1142/S0129167X19500782

References in corpus (1)

• Fibration theorems à la Milnor for differentiable maps with non-isolated singularities

Cited by in corpus (2)

• Fibration theorems à la Milnor for differentiable maps with non-isolated singularities
• Results on Milnor Fibrations for mixed polynomials with non-isolated singularities