Milnor map

In mathematics, Milnor maps are named in honor of John Milnor, who introduced them to topology and algebraic geometry in his book Singular Points of Complex Hypersurfaces (Princeton University Press, 1968) and earlier lectures. The most studied Milnor maps are actually fibrations, and the phrase Milnor fibration is more commonly encountered in the mathematical literature. The general definition is as follows.

Let $f(z_0,\dots,z_n)$ be a non-constant polynomial function of n + 1 complex variables $z_0,\dots,z_n$ such that $f(0,\dots,0)=0$, so that the set V_f of all complex (n + 1)-vectors $(z_0,\dots,z_n)$ with $f(z_0,\dots,z_n)=0$ is a complex hypersurface of complex dimension n containing the origin of complex (n + 1)-space. (For instance, if n = 1 then V_f is a complex plane curve containing (0,0).) The argument of f is the function f / | f | mapping the complement of V_f in complex (n + 1)-space to the unit circle S¹ in C. For any real radius r > 0, the restriction of the argument of f to the complement of V_f in the real (2n + 1)-sphere with center at the origin and radius r is the Milnor map of f at radius r.

Milnor's Fibration Theorem states that, for every f such that the origin is a singular point of the hypersurface V_f (in particular, for every non-constant square-free polynomial f of two variables, the case of plane curves), then for $\epsilon$ sufficiently small,

$\dfrac{f}{|f|}: \left(S^{2n+1}_{\varepsilon} -V_f \right) \rightarrow S^1$

is a fibration. Each fiber is a non-compact differentiable manifold of real dimension 2n, and the closure of each fiber is a compact manifold with boundary bounded by the intersection K_f of V_f with the (2n + 1)-sphere of sufficiently small radius. Furthermore, this compact manifold with boundary, which is known as the Milnor fiber (of the isolated singular point of V_f at the origin), is diffeomorphic to the intersection of the (2n + 2)-ball (bounded by the small (2n + 1)-sphere) with the (non-singular) hypersurface V_g where g = f − e and e is any sufficiently small non-zero complex number. This small piece of hypersurface is also called a Milnor fiber.

Milnor maps at other radii are not always fibrations, but they still have many interesting properties. For most (but not all) polynomials, the Milnor map at infinity (that is, at any sufficiently large radius) is again a fibration.

The Milnor map of f(z,w) = z² + w³ at any radius is a fibration; this construction gives the trefoil knot its structure as a fibered knot.

References

• Milnor, John W. (1968), Singular points of complex hypersurfaces, Annals of Mathematics Studies, No. 61. Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, ISBN 0-691-08065-8