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 $S^1$ 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 $2n$. Note that the closure of each fiber is a compact manifold with boundary. Here the boundary corresponds to the intersection of $V_f$ with the $(2n+1)$-sphere (of sufficiently small radius) and therefore it is a real manifold of dimension $(2n-1)$. 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 closed $(2n+2)$-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 $f(z,w)=z^2+w^3$ 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