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

TL;DR

This paper investigates the topology of real analytic germs with isolated critical points, describing their links as graph manifolds, analyzing their Milnor fibrations, and exploring cases where their open book decompositions differ from complex singularities.

Contribution

It provides a detailed topological description of real suspension singularities of the form $f ar{g}+z^n$, including their links, monodromy, and Milnor fibers, extending understanding beyond complex singularities.

Findings

Links are described as graph manifolds via open book decompositions.

Milnor fibrations are characterized and related to monodromy invariants.

Some open book decompositions do not originate from complex singularities.

Abstract

In this article we study the topology of a family of real analytic germs $F \colon (\mathbb{C}^3,0) \to (\mathbb{C},0)$ with isolated critical point at 0, given by $F(x,y,z)=f(x,y)\bar{g(x,y)}+z^r$, where $f$ and $g$ are holomorphic, $r \in \mathbb{Z}^+$ and $r \geq 2$. 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 $\frac{F}{|F|} \colon \mathbb{S}^5 \setminus L_F \to \mathbb{S}^1$. 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 $\mathbb{C}^3$.