In mathematics, Milnor maps are named in honor of John Milnor

John Milnor

John Willard Milnor is an American mathematician known for his work in differential topology, K-theory and dynamical systems. He won the Fields Medal in 1962, the Wolf Prize in 1989, and the Abel Prize in 2011. Milnor is a distinguished professor at Stony Brook University...

, who introduced them to topology

Topology

Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

 and algebraic geometry

Algebraic geometry

Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

 in his book Singular Points of Complex Hypersurfaces (Princeton University Press, 1968) and earlier lectures. The most studied Milnor maps are actually fibration

Fibration

In topology, a branch of mathematics, a fibration is a generalization of the notion of a fiber bundle. A fiber bundle makes precise the idea of one topological space being "parameterized" by another topological space . A fibration is like a fiber bundle, except that the fibers need not be the same...

s, and the phrase Milnor fibration is more commonly encountered in the mathematical literature. The general definition is as follows.

Let be a non-constant polynomial function of complex variables

Complex variables

In mathematics, the term complex variables may be taken to mean one of* Complex analysis* Several complex variables* Quaternion variable* Motor variable* Multicomplex numbers* Hypercomplex numbers* Tricomplex numbers...

  such that , so that the set of all complex -vectors with is a complex hypersurface of complex dimension

Complex dimension

In mathematics, complex dimension usually refers to the dimension of a complex manifold M, or complex algebraic variety V. If the complex dimension is d, the real dimension will be 2d...

  containing the origin of complex -space. (For instance, if then is a complex plane curve containing .) The argument

Argument

In philosophy and logic, an argument is an attempt to persuade someone of something, or give evidence or reasons for accepting a particular conclusion.Argument may also refer to:-Mathematics and computer science:...

 of is the function
mapping the complement of in complex -space to the unit circle

Unit circle

In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, "the" unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane...

  in C. For any real radius , the restriction of the argument of to the complement of in the real -sphere with center at the origin and radius is the Milnor map of at radius .

Milnor's Fibration Theorem states that, for every such that the origin is a singular point

Singular point of a curve

In geometry, a singular point on a curve is one where the curve is not given by a smooth embedding of a parameter. The precise definition of a singular point depends on the type of curve being studied.-Algebraic curves in the plane:...

 of the hypersurface (in particular, for every non-constant square-free polynomial

Square-free polynomial

In mathematics, a square-free polynomial is a polynomial with no square factors, i.e, f \in F[x] is square-free if and only if b^2 \nmid f for every b \in F[x] with non-zero degree. This definition implies that no factors of higher order can exist, either, for if b3 divided the polynomial, then b2...

  of two variables, the case of plane curves), then for sufficiently small,


is a fibration. Each fiber is a non-compact differentiable manifold

Differentiable manifold

A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...

 of real dimension , and the closure of each fiber is a compact manifold

Manifold

In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....

 with boundary bounded by the intersection of with the -sphere of sufficiently small radius. Furthermore, this compact manifold with boundary, which is known as the Milnor fiber (of the isolated singular point of at the origin), is diffeomorphic to the intersection of the -ball (bounded by the small -sphere) with the (non-singular) hypersurface where and 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 at any radius is a fibration; this construction gives the trefoil knot

Trefoil knot

In topology, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot. The trefoil can be obtained by joining together the two loose ends of a common overhand knot, resulting in a knotted loop...

 its structure as a fibered knot

Fibered knot

A knot or link Kin the 3-dimensional sphere S^3 is called fibered or fibred if there is a 1-parameter family F_t of Seifert surfaces for K, where the parameter t runs through the points of the unit circle S^1, such that if s is not equal to t...

.