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Zbl 0682.32006
Perron, B.
Les cycles évanescents sont dénoués. (Vanishing cycles are unknotted). (English)
[J] Ann. Sci. Éc. Norm. Supér. (4) 22, No. 2, 227-253 (1989). ISSN 0012-9593

Let $f: {\bbfC}\sp n\to {\bbfC}$ be a holomorphic function with an isolated singularity at the origin. Fix $\eta \ll \epsilon \ll 1$ and consider the Milnor construction $f: B(\epsilon)\cap (f\sp{- 1}(D\sb{\eta})\to D\sb{\eta}.$ \par Let $f\sb t$ be a morsification of f, that is to say a $C\sp{\infty}$ family of holomorphic functions (parametrized by an open set containing 0 in ${\bbfC})$ defined in a neighbourhood of $\overline{B(\epsilon)}$ and satisfying 1) $f\sb 0=f$ near $\overline{B(\epsilon)}$; 2) for $t\ne 0$ small enough $f\sb t$ has no critical point on $\partial B(\epsilon)$; 3) for $t\ne 0$ small enough each critical point of $f\sb t$ in B($\epsilon)$ is a Morse point with critical value in $D\sb{\eta}.$ \par Then joining a base point * fixed in $\partial D\sb{\eta}$ to the critical values of $f\sb t$ (for $t\ne 0$ small fixed) $c\sb 1...c\sb{\mu}$ by disjoint paths in $D\sb{\eta}$ allows to push the vanishing (n-1)-spheres associated to the corresponding Morse points of $f\sb t$ into the $(2n-1)$ sphere $\partial B(\epsilon).$ \par The result of this paper is that for $n=2$, any f and for a suitable choice of paths, the corresponding circles in $S\sp 3$ are unknotted (so they bound an embedded disc). \par For $n\ge 3$ $S\sp{n-1}$ cannot be knotted in $S\sp{2n-1}$ by a general result of Whitney.
[D.Barlet]

MSC 2000:: *32S05 Local singularities (analytic spaces)
32S30 Deformations of singularities (analytic spaces)
57M25 Knots and links in the 3-sphere

Keywords: vanishing cycles; morsification; unknotted

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