Milnor fibration theorem

Remarks

Milnor proved this in his 1968 book on singular points of complex hypersurfaces. The proof shows, with the curve selection lemma, that for small $\varepsilon$ the map $\phi$ has no critical points on $S_\varepsilon \setminus K_\varepsilon$. One then builds a vector field on $S_\varepsilon \setminus K_\varepsilon$ that $\phi$ sends to the unit tangent field of $S^1$ and that is controlled near $K_\varepsilon$. Its flow carries each fibre onto the others. An equivalent form, used often, is the fibration $f \colon B_\varepsilon \cap f^{-1}(\partial D_\delta) \to \partial D_\delta$, where $B_\varepsilon$ is the closed ball and $D_\delta$ a disc of radius $0 < \delta \ll \varepsilon$.

Example: for $f(z_0, z_1) = z_0 z_1$ on $\C^2$, the link is a pair of circles forming a Hopf link in $S^3$. The fibre in the tube form is $\set{z_0 z_1 = \delta} \cap B_\varepsilon$, an annulus, which has the homotopy type of a circle.

The radius must be small. On a large sphere, $f^{-1}(0)$ may meet the sphere non-transversally or other critical points of $f$ may intervene, and the conclusion can fail.

Going deeper

For a student

The Milnor fibration is the starting point of the topology of hypersurface singularities. Its fibre carries the Milnor number, and for an isolated singularity the fibre is a bouquet of spheres. Going once around the circle gives a diffeomorphism of the fibre, whose action on homology is the monodromy studied in the monodromy theorem. For plane curves, $n = 1$, the link is a link in $S^3$ and the Milnor fibre is a Seifert surface of it, which connects singularity theory with knot theory.