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. Sz\H{u}cs. 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.