The boundary of the Milnor fibre and a linking invariant of finitely determined germs

TL;DR

This work shows that the boundary of the Milnor fibre ∂F, associated with the non-isolated hypersurface defined by a finitely determined germ Φ: C^2→C^3, is a topological invariant of Φ. It builds a bridge between local complex singularity theory and immersion topology by relating the boundary construction to the Ekholm–Szűcs invariant L, notably via the fundamental relation $L(Φ|_{\mathfrak{S}})=C(Φ)−3T(Φ)$. A key outcome is a topological reformulation of the vertical indices $\mathfrak{vi}_j$ in terms of nearby embedded 3-manifolds, which yields a direct algebraic-topological identity and proves the topological invariance of the boundary ∂F. The paper develops a componentwise generalization of L for stable immersions, provides a new direct proof that $L_1(f) = -L_2(f)$, and expresses L in terms of the cross-cap and triple-point counts (C and T) alongside double-point data, enabling computable relations and illustrative examples. Altogether, the results unify invariants from singularity theory and immersion topology, offering new avenues to compute Milnor-fibre boundaries from purely topological data and to investigate the invariance under topological left-right equivalence.

Abstract

The image of a finitely determined holomorphic germ $Φ$ 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 $Φ$. 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 $C(Φ)$ and the triple point number $T(Φ)$ of a stable deformation of $Φ$, we obtain a relation between these invariants and the vertical indices. This is illustrated on several examples.

Figures (6)

• Figure 1: The structure of some of the results related to this paper
• Figure 2: The boundary of the Milnor fiber on a transverse slice $\mathbb C^2 \subset S^5_{\epsilon}$ at a point $q \in \Upsilon$: the wedge of the two coordinate discs $\{xy=0\}$ is replaced by the cylinder $\{xy= \delta\}$
• Figure 3: The monodromy of the cylinder bundle $Y$ around a twisted component $\Upsilon_j$: as the point $q \in \Upsilon_j$ goes around, in a transverse slice $\mathbb C^2 \subset S^5_{\epsilon}$ the two branches of the associated immersion -- i.e. the coordinates $x$ and $y$ -- interchange, hence the two boundary components of the cylinder $\{xy=\delta \}$ interchange as well; the meridian $m$ of the torus $\partial Y$ is a boundary component of an arbitrary cylinder fibre, while its longitude $c$ is the orbit of a generic point, rounding two times around $\Upsilon_j$ while $q$ goes around one time
• Figure 4: The decomposition of normal vectors of $\Upsilon$ and the map $\Xi$ drawn in $\mathbb R^5$
• Figure 5: Adding an extra twist in $v^+$

Theorems & Definitions (63)

• Proposition 2.1.1
• Remark 2.2.1
• Proposition 2.2.3
• proof
• Theorem 2.6.3: NP2, gtezis
• Corollary 2.6.5: NP2, gtezis
• Theorem 2.6.6: NP2, gtezis
• Proposition 3.2.1
• proof
• Remark 3.2.3