The boundary of the Milnor fibre of complex and real analytic non-isolated singularities

Abstract

Let $$f$$ and $$g$$ be holomorphic function-germs vanishing at the origin of complex analytic germs of dimension three. Suppose that they have no common irreducible component and that the real analytic map-germ $$f\bar{g}$$ has an isolated critical value at 0. We give necessary and sufficient conditions for the real analytic map-germ $$f\bar{g}$$ to have a Milnor fibration and we prove that in this case the boundary of its Milnor fibre is a Waldhausen manifold. As an intermediate milestone we describe geometrically the Milnor fibre of map-germs of the form $$f\bar{g}$$ defined in a complex surface germ, and we prove an A’Campo-type formula for the zeta function of its monodromy.