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 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 be a non-constant polynomial function of complex variables where the vanishing locus ofis only at the origin, meaning the associated variety is not smooth at the origin. Then, for the Milnor fibrationpg 68 associated to is defined as the map
which is a locally trivial smooth fibration for sufficiently small. 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 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 -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 with having a singularity at the origin, but now the polynomialis considered. Then, the algebraic Milnor fiber is taken as one of the polynomials.
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 formulawhere the quotient ideal is the Jacobian ideal, defined by the partial derivatives. 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
Milnor's Fibration Theorem states that, for every such that the origin is a singular point of the hypersurface , then for sufficiently small,is a fibration. Each fiber is a non-compact differentiable manifold of real dimension. Note that the closure of each fiber is a compact manifold with boundary. Here the boundary corresponds to the intersection of with the -sphere and therefore it is a real manifold of dimension. Furthermore, this compact manifold with boundary, which is known as the Milnor fiber, is diffeomorphic to the intersection of the closed -ball with the 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 polynomials, the Milnor map at infinity is again a fibration.