en>Yobot: /* Classification */WP:CHECKWIKI error fixes using AWB (8976)

2013-03-15T09:47:26Z

Classification: WP:CHECKWIKI error fixes using AWB (8976)

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{{refimprove|date=February 2009}}
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 [[fibration]]s, and the phrase '''Milnor fibration''' is more commonly encountered in the mathematical literature. The general definition is as follows.

Let <math>f(z_0,\dots,z_n)</math> be a non-constant [[polynomial function]] of <math>n+1</math> [[complex variables]] <math>z_0,\dots,z_n</math> such that <math>f(0,\dots,0)=0</math>, so that the set <math>V_f</math> of all complex <math>(n+1)</math>-vectors <math>(z_0,\dots,z_n)</math> with <math>f(z_0,\dots,z_n)=0</math> is a [[complex hypersurface]] of [[complex dimension]] <math>n</math> containing the origin of complex <math>(n+1)</math>-space. (For instance, if <math>n=1</math> then <math>V_f</math> is a [[complex plane curve]] containing <math>(0,0)</math>.) The [[argument]] of <math>f</math> is the function
<math>f/|f|</math> mapping the complement of <math>V_f</math> in complex <math>(n+1)</math>-space to the [[unit circle]] <math>S^1</math> in '''C'''. For any real radius <math>r > 0</math>, the restriction of the argument of <math>f</math> to the complement of <math>V_f</math> in the real <math>(2n+1)</math>-sphere with center at the origin and radius <math>r</math> is the ''Milnor map of <math>f</math> at radius <math>r</math>''.

Milnor's Fibration Theorem states that, for every <math>f</math> such that the origin is a [[Singular point of a curve|singular point]] of the hypersurface <math>V_f</math> (in particular, for every non-constant [[square-free polynomial]] <math>f</math> of two variables, the case of plane curves), then for <math>\epsilon</math> sufficiently small,

:<math>\dfrac{f}{|f|}: \left(S^{2n+1}_{\varepsilon} -V_f \right) \rightarrow S^1</math>

is a fibration. Each fiber is a non-compact [[differentiable manifold]] of real dimension <math>2n</math>. Note that the closure of each fiber is a compact [[manifold]] with boundary. Here the boundary corresponds to the intersection of <math>V_f</math> with the <math>(2n+1)</math>-sphere (of sufficiently small radius) and therefore it is a real manifold of dimension <math>(2n-1)</math>. Furthermore, this compact manifold with boundary, which is known as the ''Milnor fiber'' (of the isolated singular point of <math>V_f</math> at the origin), is diffeomorphic to the intersection of the closed <math>(2n+2)</math>-ball (bounded by the small <math>(2n+1)</math>-sphere) with the (non-singular) hypersurface <math>V_g</math> where <math>g=f-e</math> and <math>e</math> 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 <math>f(z,w)=z^2+w^3</math> at any radius is a fibration; this construction gives the [[trefoil knot]] its structure as a [[fibered knot]].

==References==
*{{citation
|last= Milnor
|first= John W.
|title= Singular points of complex hypersurfaces
|publisher= Annals of Mathematics Studies, No. 61. Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo
|year= 1968
|isbn= 0-691-08065-8}}

{{DEFAULTSORT:Milnor Map}}
[[Category:Knot theory]]
[[Category:Singularity theory]]

Minimal models - Revision history

en>Yobot: /* Classification */WP:CHECKWIKI error fixes using AWB (8976)