The Boundary of the Milnor Fibre and a Linking Invariant of Finitely Determined Germs

ABSTRACT The image of a finitely determined holomorphic germ $\Phi$ from $\mathbb {C}^2$ to $\mathbb {C}^3$ defines a hypersurface singularity $(X,0)$, which is in general non-isolated. We show that the diffeomorphism type of the boundary of the Milnor fibre $\partial F$ of X is a topological invariant of the germ $\Phi$. We establish a correspondence between the gluing coefficients (so-called vertical indices) used in the construction of $\partial F$ and a linking invariant L of the associated sphere immersion introduced by T. Ekholm and A. Szűcs. For this we provide a direct proof of the equivalence of the different definitions of L. Since L can be expressed in terms of the cross cap number $C(\Phi )$ and the triple point number $T(\Phi )$ of a stable deformation of $\Phi$, we obtain a relation between these invariants and the vertical indices. This is illustrated on several examples.