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. These were introduced to study isolated singularities by constructing numerical invariants related to the topology of a smooth deformation of the singular space.

Definition

be a non-constant polynomial function complex variables where the vanishing locus of

is only at the origin, meaning the associated variety is not smooth) the Milnor fibrationpg 68 is defined as the map

1. ,

which is a locally trivial smooth fibration. Originally this was proven as a theorem by Milnor, but was later taken as the definition of a Milnor fibration. Note this is a well defined map since

1. ,

is the argument of a complex number.

Historical motivation

One of the original motivations for studying such maps was in the study of knots-ball around a singular point of a plane curve, which is isomorphic to a real 4-dimensional ball, and looking at the knot inside the boundary, which is a 1-manifold inside of a 3-sphere. Since this concept could be generalized to hypersurfaces with isolated singularities, Milnor introduced the subject and proved his theorem.

In algebraic geometry

Another closed related notion in algebraic geometry having a singularity at the origin, but now the polynomial

is considered. Then, the algebraic Milnor fiber.

Properties and Theorems

Parallelizability

One of the basic structure theorems about Milnor fibers is they are parallelizable manifoldspg 75.

Homotopy type

Milnor fibers are special because they have the homotopy type of a bouquet of spherespg 78. The number of these spheres is the Milnor number. In fact, the number of spheres can be computed using the formula

where the quotient ideal is the Jacobian ideal. These spheres deformed to the algebraic Milnor fiber are the Vanishing cycles of the fibrationpg 83. Computing the eigenvalues of their monodromy is computationally challenging and requires advanced techniques such as b-functionspg 23.

Milnor's fibration theorem

such that the origin is a singular point (in particular, for every non-constant square-free polynomial sufficiently small,

is a fibration. Each fiber is a non-compact differentiable manifold. Note that the closure of each fiber is a compact manifold. Furthermore, this compact manifold with boundary, which is known as the Milnor fiber 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.

Examples

at any radius is a fibration; this construction gives the trefoil knot its structure as a fibered knot.

See also

• Vanishing cycle

• Mixed Hodge structure

References