New Open-book Decompositions in Singularity Theory

TL;DR

This paper investigates the topology of specific real analytic germs in three variables, revealing new open-book decompositions that often do not originate from complex singularities, thus expanding understanding of singularity topology.

Contribution

It introduces new open-book decompositions in singularity theory and demonstrates that these often cannot be derived from complex singularities, highlighting novel topological phenomena.

Findings

Link $L_F$ described as a Seifert manifold

Many open-book decompositions do not come from complex singularities

Provides new insights into the topology of real analytic germs

Abstract

In this article, we study the topology of real analytic germs $F \colon (\C^3,0) \to (\C,0)$ given by $F(x,y,z)=\overline{xy}(x^p+y^q)+z^r$ with $p,q,r \in \N$, $p,q,r \geq 2$ and $(p,q)=1$. Such a germ gives rise to a Milnor fibration $\frac{F}{\mid F \mid} \colon \Sp^5\setminus L_F \to \Sp^1$. We describe the link $L_F$ as a Seifert manifold and we show that in many cases the open-book decomposition of $\Sp^5$ given by the Milnor fibration of $F$ cannot come from the Milnor fibration of a complex singularity in $\C^3$.