Carrousel in family and non-isolated hypersurface singularities in $C^3$

TL;DR

This paper investigates the topology of the boundary of the Milnor fiber for certain complex hypersurface singularities in three variables, introducing a new carrousel technique to analyze their structure as graph manifolds.

Contribution

It introduces a novel carrousel method depending on a parameter to prove that the boundary is a graph manifold, facilitating comparison with the normalization link.

Findings

The boundary $L_t$ is a graph manifold.

The new technique allows comparison of $L_t$ with the link of the normalization.

Provides insights into the topology of non-isolated hypersurface singularities.

Abstract

We study the boundary $L_t$ of the Milnor fiber for the reduced holomorphic germs $f:(\Bbb C^3,0) \rightarrow (\Bbb C,0)$ having a non-isolated singularity at $0$. We prove that $L_t$ is a graph manifold by using a new technique of carrousel depending on one parameter varying on a circle. Our results enable one to compare the topology of $L_t$ and of the link of the normalization of $f^{-1}(0)$.