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The topology of real suspension singularities of type \(f\bar{g}+z^n\)

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Abstract

In this article we study the topology of a family of real analytic germs \(F :(\mathbb {C}^3,0) \rightarrow (\mathbb {C},0)\) with isolated critical point at the origin, given by \(F(x,y,z)=f(x,y)\overline{g(x,y)}+z^r\), where \(f\) and \(g\) are holomorphic, \(r \in \mathbb {Z}_{}^+\) and \(r \ge 2\). We describe the link \(L_F\) as a graph manifold using its natural open book decomposition, related to the Milnor fibration of the map-germ \(f\bar{g}\) and the description of its monodromy as a quasi-periodic diffeomorphism through its Nielsen invariants. Furthermore, such a germ \(F\) gives rise to a Milnor fibration \(\frac{F}{|F|} :\mathbb {S}^{5} {\setminus } L_F \rightarrow \mathbb {S}^{1}\). We present a join theorem, which allows us to describe the homotopy type of the Milnor fibre of \(F\) and we show some cases where the open book decomposition of \(\mathbb {S}^{5}\) given by the Milnor fibration of \(F\) cannot come from the Milnor fibration of a complex singularity in \(\mathbb {C}^3\).

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Acknowledgments

I am most grateful to Anne Pichon and José Seade for their supervision and comments on this work. I also want to thank Professor Walter Neumann and Patrick Popescu-Pampu for all their suggestions which improved greatly the present article, José Luis Cisneros-Molina for helpful conversations and remarks, and the referee for his comments and suggestions. Research partially supported by grants U55084 and J49048-F of the Consejo Nacional de Ciencia y Tecnología (CONACyT, Mexico), by grant M06-M02 of the Coopération Scientifique France- Amérique Latine (ECOS, France–Mexico), by the Laboratorio Internacional Solomon Lefschetz of the Centre National de la Recherche Scientifique and the Consejo Nacional de Ciencia y Tecnología (CNRS-CONACyT, France and Mexico) and by a postdoctoral fellowship of the Consejo Nacional de Ciencia y Tecnología (CONACyT, Mexico). The hospitality at Columbia University is greatly appreciated.

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  1. Department of Mathematics, Columbia University, New York City, NY, USA

    Haydée Aguilar-Cabrera

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  1. Haydée Aguilar-Cabrera
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Aguilar-Cabrera, H. The topology of real suspension singularities of type \(f\bar{g}+z^n\) . Math. Z. 277, 209–240 (2014). https://doi.org/10.1007/s00209-013-1251-9

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