The boundary of the Milnor fibre and a linking invariant of finitely determined germs
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866909598132207616 |
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| author | Pintér, Gergő Terpai, Tamás |
| author_facet | Pintér, Gergő Terpai, Tamás |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_12672 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The boundary of the Milnor fibre and a linking invariant of finitely determined germs Pintér, Gergő Terpai, Tamás Geometric Topology Algebraic Topology 32S25, 32S50 (Primary) 32S55, 57R42 (Secondary) 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. |
| title | The boundary of the Milnor fibre and a linking invariant of finitely determined germs |
| topic | Geometric Topology Algebraic Topology 32S25, 32S50 (Primary) 32S55, 57R42 (Secondary) |
| url | https://arxiv.org/abs/2304.12672 |