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Real singularities and open-book decompositions of the 3-sphere. (English) Zbl 1049.32031
If \(f:(\mathbb{C}^n,0)\to (\mathbb{C},0),\) is an analytic germ with an isolated critical point at \(0\in \mathbb{C}^n\) and \(L=f^{-1}(0)\cap S^{2n-1}_{\varepsilon}\) be the link of the singularity, where \(S^{2n-1}_{\varepsilon}\) denotes a sufficiently small \((2n-1)\)-sphere with radius \(\varepsilon \) centered at \(0\) in \(C^n.\) Then the map \(f/| f| :S^{2n-1}_{\varepsilon}-L\to S^1\) is a \(C^{\infty }\) locally trivial fibration which defines an open book decomposition of \(S_{\varepsilon}^{2n-1}\) with binding \(L.\) The authors prove that the open-book fibration provided by \(f(z_1,z_2)=\lambda _1z_1^p\bar z_2+\lambda _2z_2^q\bar z_1\) induces the “negative” orientation around two components of the link, i.e. the map \(f/| f| \) has degree \(-1\) restricted to small meridians of two components of \(L\) and degree \(+1\) for the other components. This fibration is described topologically, the genus of the fiber and the monodromy are computed. It is shown that \(f\) is not topologically equivalent to a holomorphic germ, whereas its link \(L\) is isotopic to the link of the complex singularity \(z_1z_2(z_1^{p+1}+z_2^{q+1})\).

MSC:
32S20 Global theory of complex singularities; cohomological properties
57R45 Singularities of differentiable mappings in differential topology
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