[Paper Review] Milnor fibration and fibred links at infinity

Arnaud Bodin|ArXiv.org|Sep 19, 2003

Geometric and Algebraic Topology5 references3 citations

TL;DR

This paper establishes a characterization of fibred multilinks at infinity for polynomial maps $ f: \mathbb{C}^2 \to \mathbb{C} $, proving that the multilink $ K_0 = f^{-1}(0) \cap S^3_R $ is fibred if and only if all non-zero values are regular at infinity. Using resolution of singularities at infinity, the authors describe the Milnor fibration's fibre and monodromy in terms of combinatorial invariants from the resolution, providing a new proof of a result previously established under stronger hypotheses.

ABSTRACT

For a polynomial $f$ in two complex variables, we prove that the multi-link at infinity of the 0-fiber $f^{-1}(0)$ is a fibred multi-link if and only if all the values different from 0 are regular at infinity.

Motivation & Objective

• To characterize when the multilink $ K_0 = f^{-1}(0) \cap S^3_R $ is fibred at infinity for a polynomial $ f: \mathbb{C}^2 \to \mathbb{C} $.
• To provide a new proof of the fibred multilink characterization using resolution of singularities at infinity, avoiding the 'semitame' hypothesis used previously.
• To describe the fibre and monodromy of the Milnor fibration at infinity in terms of combinatorial invariants from the resolution process.
• To clarify the role of the value $ 0 $, showing it may be critical or regular at infinity without affecting the fibredness condition.

Proposed method

• Use of weak, partial, and total resolutions via blow-ups to resolve singularities at infinity and extend the map $ \tilde{f} $ to a morphism $ \phi $ on a resolved surface $ \Sigma_t $.
• Identification of critical values at infinity as those $ c \in \mathbb{C} \setminus \{0\} $ for which $ \phi_t(D) = c $ on components $ D $ of the critical divisor $ D_{\text{crit}} $, or as critical values of $ \phi_t|_{D_{\text{dic}}} $.
• Study of the map $ \theta = \phi / |\phi| $ restricted to the boundary sphere $ S \subset \Sigma_t $, which is homotopic to $ f / |f| $, to analyze the fibration structure.
• Application of Waldhausen decomposition to decompose $ S \setminus \phi^{-1}(0) $ into Seifert 3-manifolds associated to irreducible components of $ \pi_t^{-1}(L_\infty) $, proving fibration on each piece.
• Use of the minimal Waldhausen decomposition and virtual components to detect non-fibredness when $ c \neq 0 $ is a critical value at infinity.
• Proof by contradiction and case analysis, including the Hopf link case, to exclude the possibility of fibredness when $ c \neq 0 $ is critical at infinity.

Experimental results

Research questions

• RQ1 When is the multilink $ K_0 = f^{-1}(0) \cap S^3_R $ fibred at infinity for a polynomial $ f: \mathbb{C}^2 \to \mathbb{C} $?
• RQ2 What is the role of the value $ 0 $ in determining fibredness of $ K_0 $ at infinity, especially when $ 0 $ is a critical value at infinity?
• RQ3 How can the fibre and monodromy of the Milnor fibration at infinity be described using combinatorial invariants from resolution of singularities?
• RQ4 Under what conditions does the existence of a critical value $ c \neq 0 $ at infinity imply that $ K_0 $ is not fibred?
• RQ5 Can the fibredness of $ K_0 $ be characterized purely in terms of the critical values at infinity, independent of the regularity of $ 0 $?

Key findings

• The multilink $ K_0 = f^{-1}(0) \cap S^3_R $ is fibred if and only if all non-zero values are regular at infinity, regardless of whether $ 0 $ is regular or critical at infinity.
• The fibration structure of $ f/|f| $ on $ S^3_R \setminus f^{-1}(0) $ is homotopic to $ \phi / |\phi| $ on the resolved boundary $ S $, and this map is a fibration when all $ c \neq 0 $ are regular at infinity.
• The fibre and monodromy of the Milnor fibration at infinity are completely determined by the combinatorial data of the total resolution, including the degrees and configurations of dicritical and critical components.
• If $ c \neq 0 $ is a critical value at infinity, then $ K_0 $ is not a fibred multilink, as shown by the existence of a virtual component in the minimal Waldhausen decomposition that does not intersect a Seifert surface.
• In the special case where $ K_0 $ is the Hopf link, $ c \neq 0 $ must be a regular value at infinity, and the polynomial $ f $ is equivalent under automorphism to $ (xy + \lambda)^l $ or $ x^p y^q $, both of which have no critical values at infinity for $ c \neq 0 $.
• The proof establishes that non-fibredness arises from topological obstructions in the resolution: if $ \phi_p|_D $ has more than two zeroes for a dicritical component $ D $, then $ K_0 $ cannot be fibred, which is guaranteed when $ c \neq 0 $ is a critical value at infinity.