[Paper Review] Homotopy groups of complements and non-isolated singularities

Anatoly Libgober, Mihai Tibăr|ArXiv.org|Feb 6, 2002

Algebraic Geometry and Number Theory7 references4 citations

TL;DR

This paper establishes a direct link between the first non-trivial homotopy groups of complements to hypersurfaces with non-isolated singularities and the homology of Milnor fibers in both global (polynomial) and local (germ) settings. It shows that the homotopy group $π_{n-1}(B_H \setminus B_H \cap V)$ in a small ball around the origin is torsion and its order divides that of the corresponding group on the sphere, with the characteristic polynomial of monodromy matching this order in the local case.

ABSTRACT

We obtain sufficient conditions for the vanishing of higher homotopy groups of the complements to hypersurfaces in ${\mathbb C}^n$ in terms of the behavior at infinity and relate the monodromy of non isolated singularities to the position of singularities in generic plane sections.

Motivation & Objective

• To clarify the topological relationship between non-isolated singularities and the homotopy type of their complements.
• To extend results on higher homotopy groups of complements to polynomials with non-isolated singularities at infinity.
• To relate the first non-trivial homology group of a Milnor fiber to the first non-trivial homotopy group of the complement of the zero set in a generic linear section.
• To prove divisibility theorems for the orders of homotopy groups in both global and local settings.
• To establish a precise connection between the monodromy action on Milnor homology and the torsion order of the homotopy group in the local case.

Proposed method

• Analyzes the complement of a hypersurface $V = f^{-1}(0)$ in $\mathbb{C}^{n+1}$, focusing on the homotopy groups $\pi_i(\mathbb{C}^{n+1} \setminus V)$, particularly $\pi_{n-k}$ where $k = \dim \operatorname{Sing}(V)$.
• Uses a linear section $H$ of codimension $k$ to reduce the global problem to a local one, studying $\pi_i(H \setminus H \cap V)$.
• Applies the Zariski-Lefschetz theorem and the local cone structure to relate the homotopy type of $B_H \setminus B_H \cap V$ to that of $S_H \setminus S_H \cap V$, where $B_H$ is a small ball and $S_H$ is its boundary.
• Employs the Milnor fibration and the restriction of $f$ to a generic hyperplane to analyze the topology of the complement in a neighborhood of the singular locus.
• Uses the map $\phi = (h,g): \mathbb{C}^{n+1} \to \mathbb{C}^2$ and the polar curve $\Gamma(h,g)$ to construct homotopy equivalences and analyze cell attachments in $B_H \setminus B_H \cap V$.
• Applies the homotopy exact sequence and properties of $\pi_1$-modules to show that $\pi_{n-1}(S_H \setminus S_H \cap V)$ is a torsion module and that $\pi_{n-1}(B_H \setminus B_H \cap V) \otimes \mathbb{C}$ is a ${\mathbb{C}}[\mathbb{Z}]$-torsion module.

Experimental results

Research questions

• RQ1 How do the higher homotopy groups of the complement of a hypersurface with non-isolated singularities relate to the homology of its Milnor fiber?
• RQ2 What is the role of singularities at infinity in determining the homotopy type of the complement of a polynomial's fiber?
• RQ3 Can the order of the first non-trivial homotopy group of the complement be related to the monodromy action on Milnor homology?
• RQ4 How does the position and local type of non-isolated singularities affect the topology of the complement?
• RQ5 What conditions ensure the vanishing or non-vanishing of the first non-trivial homology group of a generic fiber in the presence of non-isolated singularities?

Key findings

• The first non-trivial homotopy group $\pi_{n-1}(S_H \setminus S_H \cap V)$ is a torsion $\pi_1$-module, where $S_H$ is the boundary of a small ball in a generic linear section $H$.
• The group $\pi_{n-1}(B_H \setminus B_H \cap V) \otimes \mathbb{C}$ is a ${\mathbb{C}}[\mathbb{Z}]$-torsion module, and its order divides that of $\pi_{n-1}(S_H \setminus S_H \cap V)$.
• The order of $\pi_{n-1}(B_H \setminus B_H \cap V) \otimes \mathbb{C}$ divides the product $\prod_i \Delta_i$, where $\Delta_i$ are certain invariants related to the singular locus.
• For a germ $g$ with $\dim \operatorname{Sing}(g) \leq 1$, the characteristic polynomial of the monodromy on $H_{n-1}(M_g, \mathbb{C})$ equals the order of $\pi_{n-1}(B_H \setminus B_H \cap V)$.
• The space $B_H \setminus B_H \cap V$ is homotopy equivalent to $\{|g| = \varepsilon\} \cap B_{H_t}$ with $n$-cells attached at the singular points of $g$ on $B_{H_t} \setminus B_{H_t} \cap V$.
• The map $\phi = (h,g)$ induces a homotopy equivalence between $\{|g| = \varepsilon\} \cap B_{H_0}$ and $\{|g| = \varepsilon\} \cap B_{H_t}$, enabling the construction of the homotopy type of the complement.