Several complex variables and analytic spaces Theorem

Milnor fibration theorem

Builds on Fiber bundle Critical point Wedge sum Sphere

Statement

Let $f \colon \C^{n+1} \to \C$ be a nonconstant polynomial with $f(0) = 0$ , where $n \ge 0$ . For $\varepsilon > 0$ let $S_\varepsilon$ be the sphere of radius $\varepsilon$ about $0$ in $\C^{n+1}$ , and let $K = f^{-1}(0) \cap S_\varepsilon$ .

For all sufficiently small $\varepsilon > 0$ , the map $\phi = \frac{f}{\abs{f}} \colon S_\varepsilon \setminus K \to S^1$ is a smooth locally trivial fibration.
Suppose moreover that $0$ is an isolated critical point of $f$ , with Milnor number $\mu = \dim_\C \C\{z_0, \ldots, z_n\} \big/ \left( \frac{\partial f}{\partial z_0}, \ldots, \frac{\partial f}{\partial z_n} \right).$ Then each fibre of $\phi$ has the homotopy type of a wedge of $\mu$ spheres of dimension $n$ .

Remarks

The fibration is the Milnor fibration, its fibre is the Milnor fibre, and $K$ is the link of the singularity. In the isolated case $K$ is a smooth manifold of real dimension $2n - 1$ , and Milnor showed that it is $(n-2)$ -connected. Example: for $f = z_0^2 + z_1^3$ with $n = 1$ , the link $K$ in $S^3$ is the trefoil knot, and $\mu = 1 \cdot 2 = 2$ . The Milnor fibre is a punctured torus, which has the homotopy type of a wedge of two circles. For $f = z_0^2 + z_1^2$ the link is the Hopf link and the fibre is an annulus, homotopic to one circle, in agreement with $\mu = 1$ . The theorem holds equally for germs of holomorphic functions at $0$ . Milnor proved it in his book Singular points of complex hypersurfaces; the proof uses a curve selection lemma from real algebraic geometry.

Going deeper

For a student

The Milnor fibration is the basic topological structure attached to an isolated hypersurface singularity. The monodromy of the fibration acts on the middle homology of the fibre, a free abelian group of rank $\mu$ . Its eigenvalues and the Seifert form carry much of the topology of the singularity. For $n = 1$ the link is an algebraic knot or link in $S^3$ , and the fibre is a Seifert surface, so plane curve singularities give fibred links. The Milnor number appearing in the theorem is also the dimension of the local algebra used in finite determinacy and the classification of simple critical points.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Critical point concept
Fiber bundle concept
Sphere concept
Wedge sum concept
Milnor fibration theorem statement

Tag	`3CGMRT9W7J`
URI	`m2://3CGMRT9W7J`
Kind	statement, theorem
Writing	W0 stub
MSC	32S55, 58K10, 14B05
Status	This copy of the site has no build data. Run `m2 status 3CGMRT9W7J`.
About	Critical point
Uses	Fiber bundle, Critical point, Wedge sum, Sphere

Milnor fibration theorem

Statement

Remarks

Going deeper

For a student

Reading path

Further reading

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