[Paper Review] Fibered Multilinks and singularities $f \bar g$

Anne Pichon, José Seade|ArXiv.org|May 15, 2005

Algebraic Geometry and Number Theory18 references3 citations

TL;DR

This paper establishes that for holomorphic germs $f, g: (ℂ^n, 0) \to (ℂ, 0)$, the real analytic germ $f\overline{g}$ induces a Milnor fibration when $0$ is an isolated critical value, generalizing Milnor's fibration theorem to a broader class of real analytic singularities. The key result is that all fibred multilinks in plumbing 3-manifolds with positive multiplicities arise as links of such $f\overline{g}$ germs.

ABSTRACT

In this article we extend Milnor's fibration theorem for complex singularities to the case of singularities $f \bar g:(X,P) o (C,0))$ defined on a complex analytic singularity germ $(X,P)$, with $f, g$ holomorphic and $f \bar g$ having an isolated critical value at $0 \in C$. This can also be regarded as a result for meromorphic germs. Then we strenghten this fibration theorem when $X$ has complex dimension 2, obtaining a fibration theorem for multilinks that extends previous work by Pichon. We prove that the multilink $L_{f \bar g}$ in $L_X$ (the link of $X$), is fibred iff the map $f \bar g$ has an isolated critical value at $0 \in C$, and in this case the map $\frac{f \bar g}{|f \bar g|}$ defined on $L_X \setminus L_{f \bar g}$ is a multilink fibration.We also give a combinatorial criterium, easy to verify, to decide when is $L_{f \bar g}$ a fibred multilink. We finally prove a realization theorem for fibred multilinks.

Motivation & Objective

• To extend Milnor's fibration theorem to real analytic singularities of the form $f\overline{g}$, where $f$ and $g$ are holomorphic germs.
• To determine which fibred multilinks in 3-manifolds can be realized as links of real analytic germs $f\overline{g}$.
• To establish a realization theorem for fibred multilinks in plumbing 3-manifolds using $f\overline{g}$ germs on normal complex surface singularities.
• To explore the topological and geometric properties of the Milnor fibration associated with $f\overline{g}$, particularly in relation to open books and Seifert fibrations.
• To address the open problem of realizing non-taut or non-fibred multilinks via $f\overline{g}$ constructions under specific conditions.

Proposed method

• Use of the Milnor fibration theorem for complex singularities as a foundation, extending it to real analytic maps $f\overline{g}$ with isolated critical values.
• Application of open book decompositions and Seifert fibrations to analyze the fibration structure of $f\overline{g}$ on links in $S^3$ and plumbing 3-manifolds.
• Employment of plumbing graphs and intersection matrices to characterize the topology of links and verify fibredness via the condition $m_j^{(i)} \in \mathbb{Z}$.
• Leveraging results from G. Winters on degenerating families of complex surfaces to realize multilinks as links of holomorphic germs on normal surface singularities.
• Utilization of the map $f\overline{g}/|f\overline{g}|$ as a fibration over $S^1$ to realize $L_1 \cup (-L_2)$ as a fibred multilink.
• Proof of Theorem 7.2 by combining Theorem 7.1 (realization of fibred multilinks) with topological conditions on the plumbing manifold and multilink multiplicities.

Experimental results

Research questions

• RQ1 Which fibred multilinks in plumbing 3-manifolds arise as links of real analytic germs $f\overline{g}$ for holomorphic $f, g$?
• RQ2 Under what conditions is the map $f\overline{g}$ a Milnor fibration when $0$ is an isolated critical value?
• RQ3 Can all fibred multilinks with positive multiplicities be realized as links of $f\overline{g}$ germs on normal complex surface singularities?
• RQ4 What topological obstructions arise when $L_1$ and $L_2$ are not fibred, but $L_1 \cup (-L_2)$ is fibred?
• RQ5 Is the realization of $f\overline{g}$-links possible when the ambient singularity is not taut?

Key findings

• The Milnor fibration theorem extends to real analytic germs $f\overline{g}$ when $0$ is an isolated critical value, preserving the fibration structure over $S^1$.
• All fibred multilinks $L = n_1k_1 \cup \cdots \cup n_lk_l$ with positive multiplicities in a plumbing 3-manifold $M$ arise as links of $f\overline{g}$ germs on normal complex surface singularities.
• The fibration $f\overline{g}/|f\overline{g}|$ realizes $L_1 \cup (-L_2)$ as a fibred multilink if $L_1$, $L_2$, and $L_1 \cup (-L_2)$ are all fibred and $M$ is homeomorphic to a link of a taut surface singularity.
• The condition for a multilink to be fibred is equivalent to the integrality of the components $m_j^{(i)}$ of the solution to $M_\Gamma^t m = -b(L)$, where $M_\Gamma$ is the negative-definite intersection matrix.
• When $L_1$ and $L_2$ are not fibred but $L_1 \cup (-L_2)$ is, the realization of $f\overline{g}$-links remains open due to potential non-uniqueness of the analytic type of the surface singularity.
• The construction provides a realization of all knots and links in $S^3$ as algebraic links via $f\overline{g}$ germs, generalizing A’Campo’s and Akbulut-King’s results.