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实解析映射Milnor纤维化与Milnor-Lê纤维化的等价性

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

José Luis Cisneros-Molina, Aurélio Menegon

备注：
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
发表信息：
International Journal of MathematicsVol. 30, No. 14, 1950078 (2019)
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摘要：

在[22]米尔诺尔证明，一个真正的解析地图$f\colon (R^n,0) \to (R^p,0)$，其中$n \geq p$，与一个孤立的临界点在原点有纤维化管$f|\colon B_ε^n \cap f^{-1}(S_δ^{p-1}) \to S_δ^{p-1}$。构造一个向量场，使得（1）它横截球，（2）横截管，他将管“膨胀”到球，得到纤维化$φ\colon S_ε^{n-1} \setminus f^{-1}(0) \to S^{p-1}$，但投影不一定像复数情况那样由$f/ \|f\|$给出。在$f$具有孤立临界值的情况下，在[9]中证明，如果小管内的纤维横向于球体$S_ε$，则它在管上具有纤维化。同样在[9]中，定义了$d$ -正则性的概念，证明了$f$是$d$ -正则的当且仅当映射$f/|f\|\colon S_ε^{n-1} \setminus f^{-1}(0) \to S^{p-1}$是一个等价于管子上的纤维丛。在这篇文章中，我们证明了相应的事实，在一个更一般的设置：如果一个局部满射映射$f$有一个线性判别$Δ$和纤维化管$f|\colon B_ε^n \cap f^{-1}(S_δ^{p-1} \setminus Δ) \to S_δ^{p-1} \setminus Δ$，那么$f$是$d$ -正规的当且仅当该地图$f/ \|f\|\colon S_ε^{n-1} \setminus f^{-1}(Δ) \to S^{p-1} \setminus \mathcal{A}$（与$\mathcal{A}$ $Δ$上$S^{p-1}$的径向投影）是一个纤维丛等价于一个管。我们通过构造一个向量场$\tilde{w}$来实现这一点，该向量场以受控的方式将管膨胀到球，它满足类似于Milnor在复数环境下构造的向量场的性质：除了满足上面的（1）和（2）之外，它还满足$f/ \|f\|$在$\tilde{w}$的积分曲线上是常数。

英文摘要：

In [22] Milnor proved that a real analytic map $f\colon (R^n,0) \to (R^p,0)$, where $n \geq p$, with an isolated critical point at the origin has a fibration on the tube $f|\colon B_ε^n \cap f^{-1}(S_δ^{p-1}) \to S_δ^{p-1}$. 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 $φ\colon S_ε^{n-1} \setminus f^{-1}(0) \to S^{p-1}$, 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 $S_ε$, 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 $f/|f\|\colon S_ε^{n-1} \setminus f^{-1}(0) \to S^{p-1}$ 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 $Δ$ and a fibration on the tube $f|\colon B_ε^n \cap f^{-1}(S_δ^{p-1} \setminus Δ) \to S_δ^{p-1} \setminus Δ$, then $f$ is $d$-regular if and only if the map $f/ \|f\|\colon S_ε^{n-1} \setminus f^{-1}(Δ) \to S^{p-1} \setminus \mathcal{A}$ (with $\mathcal{A}$ the radial projection of $Δ$ 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 $\tilde{w}$.

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