Links of singular points of complex hypersurfaces
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\end{thm} | \end{thm} | ||
− | The homeomorphism type of $K$ is independent of the small | + | The homeomorphism type of $K$ is independent of the small parameter $\epsilon$, it is called the '''link''' of the singular point $z$. |
\begin{thm}(Fibration Theorem)\label{fibration} | \begin{thm}(Fibration Theorem)\label{fibration} | ||
Line 59: | Line 59: | ||
\begin{lemma} | \begin{lemma} | ||
− | For $n \ne 2$ the manifolds $K$ is a topological sphere is and only if the | + | For $n \ne 2$ the manifolds $K$ is a topological sphere is and only if the integer $\Delta(1)=\det(I-h_*)$ equals to $\pm 1$. |
\end{lemma} | \end{lemma} | ||
Line 77: | Line 77: | ||
The Brieskorn singularitites is a class of singular points which have been extensively studied. These are defined by the polynomials of the form | The Brieskorn singularitites is a class of singular points which have been extensively studied. These are defined by the polynomials of the form | ||
$$f(z_1, \dots, z_{n+1})=(z_1)^{a_1}+\dots +(z_{n+1})^{a_{n+1}}$$ | $$f(z_1, \dots, z_{n+1})=(z_1)^{a_1}+\dots +(z_{n+1})^{a_{n+1}}$$ | ||
− | where $a_1, \dots, a_{n+1}$ are | + | where $a_1, \dots, a_{n+1}$ are integers $\ge 2$. The origin is an isolated singular point of $f$. |
\begin{thm}(Brieskorn-Pham) | \begin{thm}(Brieskorn-Pham) | ||
Line 89: | Line 89: | ||
The link $K$ is called a [[Exotic spheres#Brieskorn varieties|Brieskorn variety]]. | The link $K$ is called a [[Exotic spheres#Brieskorn varieties|Brieskorn variety]]. | ||
− | For $a_1=\cdots=a_{n+1}=2$, it's seen from the defining equations that the | + | For $a_1=\cdots=a_{n+1}=2$, it's seen from the defining equations that the link $K$ is the sphere bundle of the tangent bundle of the $n$-sphere, i.~e.~the [[wikipedia:Stiefel manifold|Stiefel manifold]] $V_{n+1,2}(\mathbb R)$. |
The simplest nontrivial example is $a_1=\cdots =a_n=2$, $a_{n+1}=3$. Then $\omega_1 = \cdots \omega_n=-1$, $\omega_{n+1}=(-1\pm \sqrt{3})/2$. The characteristic polynomial is | The simplest nontrivial example is $a_1=\cdots =a_n=2$, $a_{n+1}=3$. Then $\omega_1 = \cdots \omega_n=-1$, $\omega_{n+1}=(-1\pm \sqrt{3})/2$. The characteristic polynomial is | ||
Line 99: | Line 99: | ||
The above example is a special case of the $A_k$-singularities, whose defining polynomial is | The above example is a special case of the $A_k$-singularities, whose defining polynomial is | ||
$$f(z_1, \dots, z_{n+1})=z_1^2+\cdots+z_n^2+z_{n+1}^{k+1}$$ | $$f(z_1, \dots, z_{n+1})=z_1^2+\cdots+z_n^2+z_{n+1}^{k+1}$$ | ||
− | $k$ being an | + | $k$ being an integer $\ge 1$. The Milnor lattice of an $A_k$-singularity is represented by the Dynkin diagram of the simple Lie algebra $A_k$. When $n=3$, the diffeomorphism classification of the link $K$ is obtained from its Milnor lattice and the classification of [[5-manifolds: 1-connected|simply-connected 5-manifolds]]: |
* $K$ is diffeomorphic to $S^2 \times S^3$ if $k$ is odd; | * $K$ is diffeomorphic to $S^2 \times S^3$ if $k$ is odd; | ||
* $K$ is diffeomorphic to $S^5$ if $k$ is even. | * $K$ is diffeomorphic to $S^5$ if $k$ is even. |
Revision as of 10:52, 21 June 2010
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
1 Introduction
The links of singular points of complex hypersurfaces provide a large class of examples of highly-connected odd dimensional manifolds. A standard reference is [Milnor1968]. See also [Hirzebruch&Mayer1968] and [Dimca1992].
These manifolds are the boundaries of highly-connected, stably parallelisable even dimensional manifolds and hence stably parallelisable themselves. In the case of singular points of complex curves, the link of such a singular point is a fibered link in the -dimensional sphere .
2 Construction and properties
Let be a non-constant polynomial in complex variables. A complex hypersurface is the algebraic set consisting of points such that . A regular point is a point at which some partial derivative does not vanish; if at a point all the partial derivatives vanish, is called a singular point of .
Near a regular point , the complex hypersurface is a smooth manifold of real dimension ; in a small neighborhood of a singular point , the topology of the complex hypersurface is more complicated. One way to look at the topology near , due to Brauner, is to look at the intersection of with a -dimensioanl sphere of small radius centered at .
\begin{thm} The space is -connected. \end{thm}
The homeomorphism type of is independent of the small parameter , it is called the link of the singular point .
\begin{thm}(Fibration Theorem) For sufficiently small, the space is a smooth fiber bundle over , with projection map , . Each fiber is parallelizable and has the homotopy type of a finite CW-complex of dimension . \end{thm}
The fiber is usually called the Milnor fiber of the singular point .
A singular point is isolated if there is no other singular point in some small neighborhood of .
In this special situation, the above theorems are strengthened to the following
\begin{thm} Each fiber is a smooth parallelizable manifold, the closure has boundary and the homotopy type of a bouquet of -spheres . \end{thm}
3 Invariants
Seen from the above section, the link of an isolated singular point of a complex hypersurface of complex dimension is an -connected -dimensional closed smooth manifold. In high dimensional topology, these are called highly connected manifolds, since for a -dimensional closed manifold which is not a homotopy sphere, is the highest connectivity could have. Therefore to understand the classification and invariants of the links one needs to understand the classification and invariants of highly-connected odd dimensional manifolds.
On the other hand, as the link is closely related to the singular point of the complex hypersurfaces, some of the topological invariants of are computable from the polynomial.
Let be a complex hypersurface defined by , be an isolated singular point of . Let . By putting all these 's together we get the gradient field of , which can be viewed as a map , . If is an isolated singular point, then is a well-defined map from a small sphere centered at to the unit sphere of . The mapping degree is called the multiplicity of the isolated singular point . ( is also called the Milnor number of .)
\begin{thm} The middle homology group is a free abelian group of rank . \end{thm}
Furthermore, the homology groups of the link are determined from the long homology exact sequence
of the pair . The map is the adjoint of the intersection pairing on
A symmetric or skew symmetric bilinear form on a free abelian group is usually referred to as a lattice. is called the Milnor lattice of the singular point. Thus the homology groups of the link is completely determined by the Milnor lattice of the singular point.
Topological spheres as links of singular points
Especially, the link is an integral homology sphere if and only if the intersection form is unimodular, i.~e.~the matrix of has determinant . If , the Generalized Poincare Conjecture implies that is a topological sphere.
By Theorem 2, there is a smooth fiber bundle over with fiber . The natural action of a generator of induces the characteristic homeomorphism of the fiber . Let be the induced isomorphism on homology and be the characteristic polynomial of the linear transformation . It's a consequence of the Wang sequence associated with the fiber bundle over that
Lemma 5.1. For the manifolds is a topological sphere is and only if the integer equals to .
When is a topological sphere, as it is the boundary of an -connected parallelisable -manifold , our knowledge of exotic spheres allows us to determine the diffeomorphism class of completely:
- if is even, the diffeomorphism class of is determined by the signature of the intersection pairing
- if is odd, the diffeomorphism class of is determined by the Kervaire invariant
which was computed in [Levine1966]
4 Examples
The Brieskorn singularitites is a class of singular points which have been extensively studied. These are defined by the polynomials of the form
where are integers . The origin is an isolated singular point of .
\begin{thm}(Brieskorn-Pham) The group is free abelian of rank
The characteristic polynomial is
where each ranges over all -th root of unit other than . \end{thm}
The link is called a Brieskorn variety.
For , it's seen from the defining equations that the link is the sphere bundle of the tangent bundle of the -sphere, i.~e.~the Stiefel manifold .
The simplest nontrivial example is , . Then , . The characteristic polynomial is
For we have so the link is a topological sphere of dimension ; , thus by [Levine1966] has nontrivial Kervaire invariant. Especially for is the Kervaire sphere.
The above example is a special case of the -singularities, whose defining polynomial is
being an integer . The Milnor lattice of an -singularity is represented by the Dynkin diagram of the simple Lie algebra . When , the diffeomorphism classification of the link is obtained from its Milnor lattice and the classification of simply-connected 5-manifolds:
- is diffeomorphic to if is odd;
- is diffeomorphic to if is even.
In this dimension, the diffeomorphism classification of the link of other types of singular points can be obtained in the same way, once we know the Milnor lattice of the singular point.
5 Further discussion
The link of a singular point is the intersection of the hypersurface defined by and the sphere in the ambient space . Therefore it inherits certain symmetries from that of the singular point. As an example, consider an -singularity in . There is an orientation preserving involution
induces an orientation preserving free involution of or . For , 's provide all the 4 smooth free involutions on (see [Geiges&Thomas1998]).
6 References
- [Dimca1992] A. Dimca, Singularities and topology of hypersurfaces, Springer-Verlag, New York, 1992. MR1194180 (94b:32058) Zbl 0753.57001
- [Geiges&Thomas1998] H. Geiges and C. B. Thomas, Contact topology and the structure of -manifolds with , Ann. Inst. Fourier (Grenoble) 48 (1998), no.4, 1167–1188. MR1656012 (2000a:57069) Zbl 0912.57020
- [Hirzebruch&Mayer1968] F. Hirzebruch and K. H. Mayer, -Mannigfaltigkeiten, exotische Sphären und Singularitäten, Springer-Verlag, Berlin, 1968. MR0229251 (37 #4825) Zbl 0172.25304
- [Levine1966] J. Levine, Polynomial invariants of knots of codimension two, Ann. of Math. (2) 84 (1966), 537–554. MR0200922 (34 #808) Zbl 0196.55905
- [Milnor1968] J. Milnor, Singular points of complex hypersurfaces, Princeton University Press, Princeton, N.J., 1968. MR0239612 (39 #969) Zbl 0224.57014
2 Construction and properties
Let be a non-constant polynomial in complex variables. A complex hypersurface is the algebraic set consisting of points such that . A regular point is a point at which some partial derivative does not vanish; if at a point all the partial derivatives vanish, is called a singular point of .
Near a regular point , the complex hypersurface is a smooth manifold of real dimension ; in a small neighborhood of a singular point , the topology of the complex hypersurface is more complicated. One way to look at the topology near , due to Brauner, is to look at the intersection of with a -dimensioanl sphere of small radius centered at .
\begin{thm} The space is -connected. \end{thm}
The homeomorphism type of is independent of the small parameter , it is called the link of the singular point .
\begin{thm}(Fibration Theorem) For sufficiently small, the space is a smooth fiber bundle over , with projection map , . Each fiber is parallelizable and has the homotopy type of a finite CW-complex of dimension . \end{thm}
The fiber is usually called the Milnor fiber of the singular point .
A singular point is isolated if there is no other singular point in some small neighborhood of .
In this special situation, the above theorems are strengthened to the following
\begin{thm} Each fiber is a smooth parallelizable manifold, the closure has boundary and the homotopy type of a bouquet of -spheres . \end{thm}
3 Invariants
Seen from the above section, the link of an isolated singular point of a complex hypersurface of complex dimension is an -connected -dimensional closed smooth manifold. In high dimensional topology, these are called highly connected manifolds, since for a -dimensional closed manifold which is not a homotopy sphere, is the highest connectivity could have. Therefore to understand the classification and invariants of the links one needs to understand the classification and invariants of highly-connected odd dimensional manifolds.
On the other hand, as the link is closely related to the singular point of the complex hypersurfaces, some of the topological invariants of are computable from the polynomial.
Let be a complex hypersurface defined by , be an isolated singular point of . Let . By putting all these 's together we get the gradient field of , which can be viewed as a map , . If is an isolated singular point, then is a well-defined map from a small sphere centered at to the unit sphere of . The mapping degree is called the multiplicity of the isolated singular point . ( is also called the Milnor number of .)
\begin{thm} The middle homology group is a free abelian group of rank . \end{thm}
Furthermore, the homology groups of the link are determined from the long homology exact sequence
of the pair . The map is the adjoint of the intersection pairing on
A symmetric or skew symmetric bilinear form on a free abelian group is usually referred to as a lattice. is called the Milnor lattice of the singular point. Thus the homology groups of the link is completely determined by the Milnor lattice of the singular point.
Topological spheres as links of singular points
Especially, the link is an integral homology sphere if and only if the intersection form is unimodular, i.~e.~the matrix of has determinant . If , the Generalized Poincare Conjecture implies that is a topological sphere.
By Theorem 2, there is a smooth fiber bundle over with fiber . The natural action of a generator of induces the characteristic homeomorphism of the fiber . Let be the induced isomorphism on homology and be the characteristic polynomial of the linear transformation . It's a consequence of the Wang sequence associated with the fiber bundle over that
Lemma 5.1. For the manifolds is a topological sphere is and only if the integer equals to .
When is a topological sphere, as it is the boundary of an -connected parallelisable -manifold , our knowledge of exotic spheres allows us to determine the diffeomorphism class of completely:
- if is even, the diffeomorphism class of is determined by the signature of the intersection pairing
- if is odd, the diffeomorphism class of is determined by the Kervaire invariant
which was computed in [Levine1966]
4 Examples
The Brieskorn singularitites is a class of singular points which have been extensively studied. These are defined by the polynomials of the form
where are integers . The origin is an isolated singular point of .
\begin{thm}(Brieskorn-Pham) The group is free abelian of rank
The characteristic polynomial is
where each ranges over all -th root of unit other than . \end{thm}
The link is called a Brieskorn variety.
For , it's seen from the defining equations that the link is the sphere bundle of the tangent bundle of the -sphere, i.~e.~the Stiefel manifold .
The simplest nontrivial example is , . Then , . The characteristic polynomial is
For we have so the link is a topological sphere of dimension ; , thus by [Levine1966] has nontrivial Kervaire invariant. Especially for is the Kervaire sphere.
The above example is a special case of the -singularities, whose defining polynomial is
being an integer . The Milnor lattice of an -singularity is represented by the Dynkin diagram of the simple Lie algebra . When , the diffeomorphism classification of the link is obtained from its Milnor lattice and the classification of simply-connected 5-manifolds:
- is diffeomorphic to if is odd;
- is diffeomorphic to if is even.
In this dimension, the diffeomorphism classification of the link of other types of singular points can be obtained in the same way, once we know the Milnor lattice of the singular point.
5 Further discussion
The link of a singular point is the intersection of the hypersurface defined by and the sphere in the ambient space . Therefore it inherits certain symmetries from that of the singular point. As an example, consider an -singularity in . There is an orientation preserving involution
induces an orientation preserving free involution of or . For , 's provide all the 4 smooth free involutions on (see [Geiges&Thomas1998]).
6 References
- [Dimca1992] A. Dimca, Singularities and topology of hypersurfaces, Springer-Verlag, New York, 1992. MR1194180 (94b:32058) Zbl 0753.57001
- [Geiges&Thomas1998] H. Geiges and C. B. Thomas, Contact topology and the structure of -manifolds with , Ann. Inst. Fourier (Grenoble) 48 (1998), no.4, 1167–1188. MR1656012 (2000a:57069) Zbl 0912.57020
- [Hirzebruch&Mayer1968] F. Hirzebruch and K. H. Mayer, -Mannigfaltigkeiten, exotische Sphären und Singularitäten, Springer-Verlag, Berlin, 1968. MR0229251 (37 #4825) Zbl 0172.25304
- [Levine1966] J. Levine, Polynomial invariants of knots of codimension two, Ann. of Math. (2) 84 (1966), 537–554. MR0200922 (34 #808) Zbl 0196.55905
- [Milnor1968] J. Milnor, Singular points of complex hypersurfaces, Princeton University Press, Princeton, N.J., 1968. MR0239612 (39 #969) Zbl 0224.57014