Milnor map
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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(z0,...,zn) be a non-constant polynomial function of n + 1 complex variables z0,...,zn such that f(0,...,0) = 0, so that the set Vf of all complex (n + 1)-vectors (z0,...,zn) with f(z0,...,zn) = 0 is a complex hypersurface of complex dimension n containing the origin of complex (n + 1)-space. (For instance, if n = 1 then Vf is a complex plane curve containing (0,0).) The argument of f is the function f / | f | mapping the complement of Vf 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 Vf 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 an isolated singular point of the hypersurface Vf (in particular, for every non-constant square-free polynomial f of two variables, the case of plane curves), the Milnor map of f at any sufficiently small radius is a fibration over S1. 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 Kf of Vf 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 Vf 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 Vg 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.
