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. These were introduced to study isolated singularities by constructing numerical invariants related to the topology of a smooth deformation of the singular space.
Definition.
Let formula_1 be a non-constant polynomial function of formula_2 complex variables formula_3 where the vanishing locus of
is only at the origin, meaning the associated variety formula_5 is not smooth at the origin. Then, for formula_6 (a sphere inside formula_7 of radius formula_8) the Milnor fibrationpg 68 associated to formula_9 is defined as the map
which is a locally trivial smooth fibration for sufficiently small formula_11. 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
where formula_13 is the argument of a complex number.
Historical motivation.
One of the original motivations for studying such maps was in the study of knots constructed by taking an formula_11-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 is the Milnor fiber of an isolated hypersurface singularity. This has a similar setup, where a polynomial formula_9 with formula_16 having a singularity at the origin, but now the polynomial
is considered. Then, the algebraic Milnor fiber is taken as one of the polynomials formula_18.
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, defined by the partial derivatives formula_20. These spheres deformed to the algebraic Milnor fiber are the Vanishing cycles of the fibrationpg 83. Unforunately, computing the eigenvalues of their monodromy is computationally challenging and requires advanced techniques such as b-functionspg 23.
Milnor's fibration theorem.
Milnor's Fibration Theorem states that, for every formula_9 such that the origin is a singular point of the hypersurface formula_22 (in particular, for every non-constant square-free polynomial formula_9 of two variables, the case of plane curves), then for formula_24 sufficiently small,
is a fibration. Each fiber is a non-compact differentiable manifold of real dimension formula_26. Note that the closure of each fiber is a compact manifold with boundary. Here the boundary corresponds to the intersection of formula_22 with the formula_28-sphere (of sufficiently small radius) and therefore it is a real manifold of dimension formula_29. Furthermore, this compact manifold with boundary, which is known as the "Milnor fiber" (of the isolated singular point of formula_22 at the origin), is diffeomorphic to the intersection of the closed formula_31-ball (bounded by the small formula_28-sphere) with the (non-singular) hypersurface formula_33 where formula_34 and formula_35 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.
The Milnor map of formula_36 at any radius is a fibration; this construction gives the trefoil knot its structure as a fibered knot.