The boundary of the Milnor fiber for some non-isolated singularities of complex surfaces

Abstract

We study the boundary $L_{t}$ of the Milnor fiber for the non-isolated singularities in $\mathbf{C}^{3}$ with equation $z^{m} - g(x,y) = 0$ where $m \geq 2$ and $g(x,y)=0$ is a non-reduced plane curve germ. We give a complete proof that $L_{t}$ is a Waldhausen graph manifold and we provide the tools to construct its plumbing graph. As an example, we give the plumbing graph associated to the germs $z^{2} - (x^{2} - y^{3})y^{l} = 0$ with $l$ odd and $l \geq 3$. We prove that the boundary of the Milnor fiber is a Waldhausen manifold new in complex geometry, as it cannot be the boundary of a normal surface singularity.