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

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We study the boundary Lₜ of the Milnor fiber for the non-isolated singularities in C³ with equation zᵐ - g(x,y) = 0 where g(x,y) is a non-reduced plane curve germ. We give a complete proof that Lₜ 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 z² - (x² - y³)yˡ = 0 with l an interger >1. 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.