Open book decompositions of $\mathbb{S}^{5}$ and real singularities

TL;DR

This paper investigates the topology of certain real analytic germs in three complex variables, showing their Milnor fibrations on the 5-sphere are often not realizable as complex singularity fibrations, despite their links being homeomorphic.

Contribution

It characterizes the links of these real singularities as Seifert manifolds and demonstrates that their open-book decompositions generally do not originate from complex singularities.

Findings

Links are homeomorphic to complex singularity links.

Most open-book decompositions do not come from complex singularities.

Links are described as Seifert manifolds.

Abstract

In this article, we study the topology of the family of real analytic germs $F \colon (\mathbb{C}^3,0) \to (\mathbb{C},0)$ given by $F(x,y,z)=\bar{xy}(x^p+y^q)+z^r$ with $p,q,r \in \mathbb{N}$, $p,q,r \geq 2$ and $(p,q)=1$. Such a germ has isolated singularity at 0 and gives rise to a Milnor fibration $\frac{F}{|F|} \colon \mathbb{S}^{5} \setminus L_F \to \mathbb{S}^{1}$. We describe the link $L_F$ as a Seifert manifold and we show that it is always homeomorphic to the link of a complex singularity. However, we prove that in almost all the cases 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 $\mathbb{C}^3$.