Links of singular points of complex hypersurfaces

From Manifold Atlas

(Difference between revisions)

Jump to: navigation, search

Latest revision as of 12:54, 14 January 2013

This page has not been refereed. The information given here might be incomplete or provisional.

[edit] 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 ${{Stub}} == Introduction == <wikitex>; The links of singular points of complex hypersurfaces provide a large class of examples of highly-connected odd dimensional manifolds. A standard reference is {{cite|Milnor1968}}. See also {{cite|Hirzebruch&Mayer1968}} and {{cite|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 $S^3$. </wikitex> == Construction and properties== <wikitex>; Let $f(z_1, \dots, z_{n+1})$ be a non-constant polynomial in $n+1$ complex variables. A complex hypersurface $V$ is the algebraic set consisting of points $z=(z_1, \dots, z_{n+1})$ such that $f(z)=0$. A regular point $z \in V$ is a point at which some partial derivative $\partial f /\partial z_j$ does not vanish; if at a point $z \in V$ all the partial derivatives $\partial f / \partial z_j$ vanish, $z$ is called a '''singular point''' of $V$. Near a regular point $z$, the complex hypersurface $V$ is a smooth manifold of real dimension n$; in a small neighborhood of a singular point $z$, the topology of the complex hypersurface $V$ is more complicated. One way to look at the topology near $z$, due to Brauner, is to look at the intersection of $V$ with a $(2n+1)$-dimensioanl sphere of small radius $\epsilon$ $S_{\epsilon}$ centered at $z$. \begin{thm} The space $K=V\cap S_{\epsilon}$ is $(n-2)$-connected. \end{thm} 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} For $\epsilon$ sufficiently small, the space $S_{\epsilon}-K$ is a smooth fiber bundle over $S^1$, with projection map $\phi \colon S_{\epsilon}-K \to S^1$, $z \mapsto f(z)/|f(z)|$. Each fiber $F_{\theta}$ is parallelizable and has the homotopy type of a finite CW-complex of dimension $n$. \end{thm} The fiber $F_{\theta}$ is usually called the '''Milnor fiber''' of the singular point $z$. A singular point $z$ is '''isolated''' if there is no other singular point in some small neighborhood of $z$. In this special situation, the above theorems are strengthened to the following \begin{thm} Each fiber $F_{\theta}$ is a smooth parallelizable manifold, the closure $\overline{F_{\theta}}$ has boundary $K$ and the homotopy type of a bouquet of $n$-spheres $S^n\vee \cdots \vee S^n$. \end{thm} </wikitex> == Invariants == <wikitex>; Seen from the above section, the link $K$ of an isolated singular point $z$ of a complex hypersurface $V$ of complex dimension $n$ is an $(n-2)$-connected $(2n-1)$-dimensional closed smooth manifold. In high dimensional topology, these are called [[highly-connected|highly connected manifolds]], since for a $(2n-1)$-dimensional closed manifold $M$ which is not a homotopy sphere, $(n-2)$ is the highest connectivity $M$ could have. Therefore to understand the classification and invariants of the links $K$ one needs to understand the classification and invariants of [[highly-connected]] odd dimensional manifolds. On the other hand, as the link $K$ is closely related to the singular point $z$ of the complex hypersurfaces, some of the topological invariants of $K$ are computable from the polynomial. Let $V$ be a complex hypersurface defined by $f(z_1, \dots, z_{n+1})$, $z^0$ be an isolated singular point of $V$. Let $g_j=\partial f /\partial z_j$ $j=1, \dots, n+1$. By putting all these $g_j$'s together we get the gradient field of $f$, which can be viewed as a map $g \colon \mathbb C^{n+1} \to \mathbb C^{n+1}$, $z \mapsto (g_1(z), \dots, g_{n+1}(z))$. If $z^0$ is an isolated singular point, then $z \mapsto g(z)/||g(z)||$ is a well-defined map from a small sphere $S_{\epsilon}$ centered at $z^0$ to the unit sphere $S^{2n+1}$ of $\mathbb C^{n+1}$. The mapping degree $\mu$ is called the multiplicity of the isolated singular point $z^0$. ($\mu$ is also called the Milnor number of $z^0$.) \begin{thm} The middle homology group $H_n(F_{\theta})$ is a free abelian group of rank $\mu$. \end{thm} Furthermore, the homology groups of the link $K$ are determined from the long homology exact sequence $$ \cdots \to H_n(\overline{F_{\theta}}) \stackrel{j_*}{\rightarrow} H_n(\overline{F_{\theta}}, K) \stackrel{\partial}{\rightarrow} \widetilde{H}_{n-1}(K) \to 0$$ of the pair $(\overline{F_{\theta}},K)$. The map $j_*$ is the adjoint of the intersection pairing on $\overline{F_{\theta}}$ $$s \colon H_n(\overline{F_{\theta}}) \otimes H_n(\overline{F_{\theta}}) \to \mathbb Z.$$ A symmetric or skew symmetric bilinear form on a free abelian group is usually referred to as a lattice. $s$ is called the '''Milnor lattice''' of the singular point. Thus the homology groups of the link $K$ is completely determined by the Milnor lattice of the singular point. </wikitex> == Topological spheres as links of singular points == <wikitex>; Especially, the link $K$ is an integral [[homology sphere]] if and only if the intersection form $s$ is unimodular, i.~e.~the matrix of $s$ has determinant $\pm1$. If $n\ge 3$, the [[Generalized Poincare Conjecture]] implies that $K$ is a topological sphere. By Theorem \ref{fibration}, there is a smooth fiber bundle over $S^1$ with fiber $F_{\theta}$. The natural action of a generator of $\pi_1(S^1)$ induces the characteristic homeomorphism $h$ of the fiber $F_0=\phi^{-1}$. Let $h_* \colon H_n(F_0) \to H_n(F_0)$ be the induced isomorphism on homology and $\Delta(t)=\det(tI-h_*)$ be the '''characteristic polynomial''' of the linear transformation $h_*$. It's a consequence of the [[Wang sequence]] associated with the fiber bundle over $S^1$ that \begin{lemma} 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} When $K$ is a topological sphere, as it is the boundary of an $(n-1)$-connected parallelisable n$-manifold $\overline{F_0}$, our knowledge of [[exotic spheres]] allows us to determine the diffeomorphism class of $K$ completely: * if $n$ is even, the diffeomorphism class of $K$ is determined by the [[Intersection form#Algebraic invariants|signature]] of the intersection pairing $$s \colon H_n(\overline{F_0}) \otimes H_n(\overline{F_0}) \to \mathbb Z$$ * if $n$ is odd, the diffeomorphism class of $K$ is determined by the [[wikipedia:Kervaire invariant|Kervaire invariant]] $$c(\overline{F_0}) \in \mathbb Z_2$$ which was computed in {{cite|Levine1966}} $$c(\overline{F_0})=0 \ \ \textrm{if} \ \ \Delta(-1) \equiv \pm 1 \pmod 8$$ $$c(\overline{F_0})=1 \ \ \textrm{if} \ \ \Delta(-1) \equiv \pm 3 \pmod 8$$ </wikitex> == Examples == <wikitex>; The Brieskorn singularitites is a class of singular points which have been extensively studied. These are defined by the polynomials of the form $$f(z_1, \dots, z_{n+1})=(z_1)^{a_1}+\dots +(z_{n+1})^{a_{n+1}}$$ where $a_1, \dots, a_{n+1}$ are integers $\ge 2$. The origin is an isolated singular point of $f$. \begin{thm}(Brieskorn-Pham) The group $H_n(F_{\theta})$ is free abelian of rank $$\mu = (a_1-1)(a_2-1)\cdots (a_{n+1}-1.)$$ The characteristic polynomial is $$\Delta(t)=\prod(t-\omega_1 \omega_2 \cdots \omega_{n+1}),$$ where each $\omega_j$ ranges over all $a_j$-th root of unit other than 3$ -dimensional sphere $S^3$ .

[edit] 2 Construction and properties

Let $f(z_1, \dots, z_{n+1})$ $f(z_1, \dots, z_{n+1})$ be a non-constant polynomial in $n+1$ $n+1$ complex variables. A complex hypersurface

Tex syntax error

$V$ is the algebraic set consisting of points $z=(z_1, \dots, z_{n+1})$ $z=(z_1, \dots, z_{n+1})$ such that $f(z)=0$ $f(z)=0$ . A regular point $z \in V$ $z \in V$ is a point at which some partial derivative $\partial f /\partial z_j$ $\partial f /\partial z_j$ does not vanish; if at a point $z \in V$ $z \in V$ all the partial derivatives $\partial f / \partial z_j$ $\partial f / \partial z_j$ vanish, $z$ $z$ is called a singular point of

Tex syntax error

$V$ . Near a regular point $z$ $z$ , the complex hypersurface

Tex syntax error

$V$ is a smooth manifold of real dimension $2n$ $2n$ ; in a small neighborhood of a singular point $z$ $z$ , the topology of the complex hypersurface

Tex syntax error

$V$ is more complicated. One way to look at the topology near $z$ $z$ , due to Brauner, is to look at the intersection of

Tex syntax error

$V$ with a $(2n+1)$ $(2n+1)$ -dimensioanl sphere of small radius $\epsilon$ $\epsilon$ $S_{\epsilon}$ $S_{\epsilon}$ centered at $z$ $z$ .

The space $K=V\cap S_{\epsilon}$ is $(n-2)$ -connected.

The homeomorphism type of $K$ is independent of the small parameter $\epsilon$ , it is called the link of the singular point $z$ .

(Fibration Theorem) For $\epsilon$ sufficiently small, the space $S_{\epsilon}-K$ is a smooth fiber bundle over $S^1$ , with projection map $\phi \colon S_{\epsilon}-K \to S^1$ , $z \mapsto f(z)/|f(z)|$ . Each fiber $F_{\theta}$ is parallelizable and has the homotopy type of a finite CW-complex of dimension $n$ .

The fiber $F_{\theta}$ is usually called the Milnor fiber of the singular point $z$ .

A singular point $z$ is isolated if there is no other singular point in some small neighborhood of $z$ .

In this special situation, the above theorems are strengthened to the following

Each fiber $F_{\theta}$ is a smooth parallelizable manifold, the closure $\overline{F_{\theta}}$ has boundary $K$ and the homotopy type of a bouquet of $n$ -spheres $S^n\vee \cdots \vee S^n$ .

[edit] 3 Invariants

Seen from the above section, the link $K$ $K$ of an isolated singular point $z$ $z$ of a complex hypersurface

Tex syntax error

$V$ of complex dimension $n$ $n$ is an $(n-2)$ $(n-2)$ -connected $(2n-1)$ $(2n-1)$ -dimensional closed smooth manifold. In high dimensional topology, these are called highly connected manifolds, since for a $(2n-1)$ $(2n-1)$ -dimensional closed manifold $M$ $M$ which is not a homotopy sphere, $(n-2)$ $(n-2)$ is the highest connectivity $M$ $M$ could have. Therefore to understand the classification and invariants of the links $K$ $K$ one needs to understand the classification and invariants of highly-connected odd dimensional manifolds.

On the other hand, as the link $K$ is closely related to the singular point $z$ of the complex hypersurfaces, some of the topological invariants of $K$ are computable from the polynomial.

Let

Tex syntax error

$V$ be a complex hypersurface defined by $f(z_1, \dots, z_{n+1})$ $f(z_1, \dots, z_{n+1})$ , $z^0$ $z^0$ be an isolated singular point of

Tex syntax error

$V$ . Let $g_j=\partial f /\partial z_j$ $g_j=\partial f /\partial z_j$ $j=1, \dots, n+1$ $j=1, \dots, n+1$ . By putting all these $g_j$ $g_j$ 's together we get the gradient field of $f$ $f$ , which can be viewed as a map $g \colon \mathbb C^{n+1} \to \mathbb C^{n+1}$ $g \colon \mathbb C^{n+1} \to \mathbb C^{n+1}$ , $z \mapsto (g_1(z), \dots, g_{n+1}(z))$ $z \mapsto (g_1(z), \dots, g_{n+1}(z))$ . If $z^0$ $z^0$ is an isolated singular point, then $z \mapsto g(z)/||g(z)||$ $z \mapsto g(z)/||g(z)||$ is a well-defined map from a small sphere $S_{\epsilon}$ $S_{\epsilon}$ centered at $z^0$ $z^0$ to the unit sphere $S^{2n+1}$ $S^{2n+1}$ of $\mathbb C^{n+1}$ $\mathbb C^{n+1}$ . The mapping degree $\mu$ $\mu$ is called the multiplicity of the isolated singular point $z^0$ $z^0$ . ( $\mu$ $\mu$ is also called the Milnor number of $z^0$ $z^0$ .)

The middle homology group $H_n(F_{\theta})$ is a free abelian group of rank $\mu$ .

Furthermore, the homology groups of the link $K$ are determined from the long homology exact sequence

$\displaystyle \cdots \to H_n(\overline{F_{\theta}}) \stackrel{j_*}{\rightarrow} H_n(\overline{F_{\theta}}, K) \stackrel{\partial}{\rightarrow} \widetilde{H}_{n-1}(K) \to 0$ $\displaystyle \cdots \to H_n(\overline{F_{\theta}}) \stackrel{j_*}{\rightarrow} H_n(\overline{F_{\theta}}, K) \stackrel{\partial}{\rightarrow} \widetilde{H}_{n-1}(K) \to 0$

of the pair $(\overline{F_{\theta}},K)$ . The map $j_*$ is the adjoint of the intersection pairing on $\overline{F_{\theta}}$

$\displaystyle s \colon H_n(\overline{F_{\theta}}) \otimes H_n(\overline{F_{\theta}}) \to \mathbb Z.$ $\displaystyle s \colon H_n(\overline{F_{\theta}}) \otimes H_n(\overline{F_{\theta}}) \to \mathbb Z.$

A symmetric or skew symmetric bilinear form on a free abelian group is usually referred to as a lattice. $s$ is called the Milnor lattice of the singular point. Thus the homology groups of the link $K$ is completely determined by the Milnor lattice of the singular point.

[edit] 4 Topological spheres as links of singular points

Especially, the link $K$ is an integral homology sphere if and only if the intersection form $s$ is unimodular, i.~e.~the matrix of $s$ has determinant $\pm1$ . If $n\ge 3$ , the Generalized Poincare Conjecture implies that $K$ is a topological sphere.

By Theorem 2.2, there is a smooth fiber bundle over $S^1$ with fiber $F_{\theta}$ . The natural action of a generator of $\pi_1(S^1)$ induces the characteristic homeomorphism $h$ of the fiber $F_0=\phi^{-1}$ . Let $h_* \colon H_n(F_0) \to H_n(F_0)$ be the induced isomorphism on homology and $\Delta(t)=\det(tI-h_*)$ be the characteristic polynomial of the linear transformation $h_*$ . It's a consequence of the Wang sequence associated with the fiber bundle over $S^1$ that

For $n \ne 2$ the manifolds $K$ is a topological sphere is and only if the integer $\Delta(1)=\det(I-h_*)$ equals to $\pm 1$ .

When $K$ is a topological sphere, as it is the boundary of an $(n-1)$ -connected parallelisable $2n$ -manifold $\overline{F_0}$ , our knowledge of exotic spheres allows us to determine the diffeomorphism class of $K$ completely:

if $n$ is even, the diffeomorphism class of $K$ is determined by the signature of the intersection pairing

$\displaystyle s \colon H_n(\overline{F_0}) \otimes H_n(\overline{F_0}) \to \mathbb Z$ $\displaystyle s \colon H_n(\overline{F_0}) \otimes H_n(\overline{F_0}) \to \mathbb Z$

if $n$ is odd, the diffeomorphism class of $K$ is determined by the Kervaire invariant

$\displaystyle c(\overline{F_0}) \in \mathbb Z_2$ $\displaystyle c(\overline{F_0}) \in \mathbb Z_2$

which was computed in [Levine1966]

$\displaystyle c(\overline{F_0})=0 \ \ \textrm{if} \ \ \Delta(-1) \equiv \pm 1 \pmod 8$ $\displaystyle c(\overline{F_0})=0 \ \ \textrm{if} \ \ \Delta(-1) \equiv \pm 1 \pmod 8$

$\displaystyle c(\overline{F_0})=1 \ \ \textrm{if} \ \ \Delta(-1) \equiv \pm 3 \pmod 8$ $\displaystyle c(\overline{F_0})=1 \ \ \textrm{if} \ \ \Delta(-1) \equiv \pm 3 \pmod 8$

[edit] 5 Examples

The Brieskorn singularitites is a class of singular points which have been extensively studied. These are defined by the polynomials of the form

$\displaystyle f(z_1, \dots, z_{n+1})=(z_1)^{a_1}+\dots +(z_{n+1})^{a_{n+1}}$ $\displaystyle f(z_1, \dots, z_{n+1})=(z_1)^{a_1}+\dots +(z_{n+1})^{a_{n+1}}$

where $a_1, \dots, a_{n+1}$ are integers $\ge 2$ . The origin is an isolated singular point of $f$ .

\begin{thm}(Brieskorn-Pham) The group $H_n(F_{\theta})$ is free abelian of rank

$\displaystyle \mu = (a_1-1)(a_2-1)\cdots (a_{n+1}-1.)$ $\displaystyle \mu = (a_1-1)(a_2-1)\cdots (a_{n+1}-1.)$

The characteristic polynomial is

$\displaystyle \Delta(t)=\prod(t-\omega_1 \omega_2 \cdots \omega_{n+1}),$ $\displaystyle \Delta(t)=\prod(t-\omega_1 \omega_2 \cdots \omega_{n+1}),$

where each $\omega_j$ ranges over all $a_j$ -th root of unit other than $1$ . \end{thm}

The link $K$ is called a Brieskorn variety.

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 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

$\displaystyle \Delta(t)=t^2-t+1 \ \ \textrm{for} \ \ n \ \ \textrm{odd}$ $\displaystyle \Delta(t)=t^2-t+1 \ \ \textrm{for} \ \ n \ \ \textrm{odd}$

$\displaystyle \Delta(t)=t^2+t+1 \ \ \textrm{for} \ \ n \ \ \textrm{even}.$ $\displaystyle \Delta(t)=t^2+t+1 \ \ \textrm{for} \ \ n \ \ \textrm{even}.$

For $n=2k+1$ we have $\Delta(1)=1$ so the link $K$ is a topological sphere of dimension $4k+1$ ; $\Delta(-1)=3$ , thus by [Levine1966] $K$ has nontrivial Kervaire invariant. Especially for $k=2$ $K^9$ is the Kervaire sphere.

The above example is a special case of the $A_k$ -singularities, whose defining polynomial is

$\displaystyle f(z_1, \dots, z_{n+1})=z_1^2+\cdots+z_n^2+z_{n+1}^{k+1}$ $\displaystyle f(z_1, \dots, z_{n+1})=z_1^2+\cdots+z_n^2+z_{n+1}^{k+1}$

$k$ 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 simply-connected 5-manifolds:

$K$ is diffeomorphic to $S^2 \times S^3$ if $k$ is odd;
$K$ is diffeomorphic to $S^5$ if $k$ is even.

In this dimension, the diffeomorphism classification of the link $K$ of other types of singular points can be obtained in the same way, once we know the Milnor lattice of the singular point.

[edit] 6 Further discussion

The link $K$ $K$ of a singular point $z$ $z$ is the intersection of the hypersurface

Tex syntax error

$V$ defined by $f$ $f$ and the sphere $S_{\epsilon}$ $S_{\epsilon}$ in the ambient space $\mathbb C^{n+1}$ $\mathbb C^{n+1}$ . Therefore it inherits certain symmetries from that of the singular point. As an example, consider an $A_k$ $A_k$ -singularity in $\mathbb C^4$ $\mathbb C^4$ . There is an orientation preserving involution

$\displaystyle \tau_k \colon \mathbb C^4 \to \mathbb C^4, \ \ (z_1, z_2, z_3, z_4) \mapsto (-z_1, -z_2, -z_3, z_4).$ $\displaystyle \tau_k \colon \mathbb C^4 \to \mathbb C^4, \ \ (z_1, z_2, z_3, z_4) \mapsto (-z_1, -z_2, -z_3, z_4).$

$\tau_k$ induces an orientation preserving free involution of $K\cong S^5$ or $S^2 \times S^3$ . For $k=0,2,4,6$ , $\tau_k$ 's provide all the 4 smooth free involutions on $S^5$ (see [Geiges&Thomas1998]).

[edit] 7 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 $5$ -manifolds with $\pi_1=Z_2$ , Ann. Inst. Fourier (Grenoble) 48 (1998), no.4, 1167–1188. MR1656012 (2000a:57069) Zbl 0912.57020
[Hirzebruch&Mayer1968] F. Hirzebruch and K. H. Mayer, $O(n)$ -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

$. \end{thm} 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 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 $$\Delta(t)=t^2-t+1 \ \ \textrm{for} \ \ n \ \ \textrm{odd}$$ $$\Delta(t)=t^2+t+1 \ \ \textrm{for} \ \ n \ \ \textrm{even}.$$ For $n=2k+1$ we have $\Delta(1)=1$ so the link $K$ is a topological sphere of dimension k+1$; $\Delta(-1)=3$, thus by {{cite|Levine1966}} $K$ has nontrivial [[wikipedia:Kervaire invariant|Kervaire invariant]]. Especially for $k=2$ $K^9$ is the [[Exotic spheres#Plumbing|Kervaire sphere]]. The above example is a special case of the $A_k$-singularities, whose defining polynomial is $$f(z_1, \dots, z_{n+1})=z_1^2+\cdots+z_n^2+z_{n+1}^{k+1}$$ $k$ 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 $S^5$ if $k$ is even. In this dimension, the diffeomorphism classification of the link $K$ of other types of singular points can be obtained in the same way, once we know the [[Links of singular points of complex hypersurfaces#Invariants|Milnor lattice]] of the singular point. == Further discussion == ; The link $K$ of a singular point $z$ is the intersection of the hypersurface $V$ defined by $f$ and the sphere $S_{\epsilon}$ in the ambient space $\mathbb C^{n+1}$. Therefore it inherits certain symmetries from that of the singular point. As an example, consider an $A_k$-singularity in $\mathbb C^4$. There is an orientation preserving involution $$\tau_k \colon \mathbb C^4 \to \mathbb C^4, \ \ (z_1, z_2, z_3, z_4) \mapsto (-z_1, -z_2, -z_3, z_4).$$ $\tau_k$ induces an orientation preserving free involution of $K\cong S^5$ or $S^2 \times S^3$. For $k=0,2,4,6$, $\tau_k$'s provide all the 4 smooth free involutions on $S^5$ (see {{cite|Geiges&Thomas1998}}). == References == {{#RefList:}} [[Category:Manifolds]]3-dimensional sphere $S^3$ $S^3$ .

[edit] 2 Construction and properties

Let $f(z_1, \dots, z_{n+1})$ $f(z_1, \dots, z_{n+1})$ be a non-constant polynomial in $n+1$ $n+1$ complex variables. A complex hypersurface

Tex syntax error

$V$ is the algebraic set consisting of points $z=(z_1, \dots, z_{n+1})$ $z=(z_1, \dots, z_{n+1})$ such that $f(z)=0$ $f(z)=0$ . A regular point $z \in V$ $z \in V$ is a point at which some partial derivative $\partial f /\partial z_j$ $\partial f /\partial z_j$ does not vanish; if at a point $z \in V$ $z \in V$ all the partial derivatives $\partial f / \partial z_j$ $\partial f / \partial z_j$ vanish, $z$ $z$ is called a singular point of

Tex syntax error

$V$ . Near a regular point $z$ $z$ , the complex hypersurface

Tex syntax error

$V$ is a smooth manifold of real dimension $2n$ $2n$ ; in a small neighborhood of a singular point $z$ $z$ , the topology of the complex hypersurface

Tex syntax error

$V$ is more complicated. One way to look at the topology near $z$ $z$ , due to Brauner, is to look at the intersection of

Tex syntax error

$V$ with a $(2n+1)$ $(2n+1)$ -dimensioanl sphere of small radius $\epsilon$ $\epsilon$ $S_{\epsilon}$ $S_{\epsilon}$ centered at $z$ $z$ .

The space $K=V\cap S_{\epsilon}$ is $(n-2)$ -connected.

The homeomorphism type of $K$ is independent of the small parameter $\epsilon$ , it is called the link of the singular point $z$ .

(Fibration Theorem) For $\epsilon$ sufficiently small, the space $S_{\epsilon}-K$ is a smooth fiber bundle over $S^1$ , with projection map $\phi \colon S_{\epsilon}-K \to S^1$ , $z \mapsto f(z)/|f(z)|$ . Each fiber $F_{\theta}$ is parallelizable and has the homotopy type of a finite CW-complex of dimension $n$ .

The fiber $F_{\theta}$ is usually called the Milnor fiber of the singular point $z$ .

A singular point $z$ is isolated if there is no other singular point in some small neighborhood of $z$ .

In this special situation, the above theorems are strengthened to the following

Each fiber $F_{\theta}$ is a smooth parallelizable manifold, the closure $\overline{F_{\theta}}$ has boundary $K$ and the homotopy type of a bouquet of $n$ -spheres $S^n\vee \cdots \vee S^n$ .

[edit] 3 Invariants

Seen from the above section, the link $K$ $K$ of an isolated singular point $z$ $z$ of a complex hypersurface

Tex syntax error

$V$ of complex dimension $n$ $n$ is an $(n-2)$ $(n-2)$ -connected $(2n-1)$ $(2n-1)$ -dimensional closed smooth manifold. In high dimensional topology, these are called highly connected manifolds, since for a $(2n-1)$ $(2n-1)$ -dimensional closed manifold $M$ $M$ which is not a homotopy sphere, $(n-2)$ $(n-2)$ is the highest connectivity $M$ $M$ could have. Therefore to understand the classification and invariants of the links $K$ $K$ one needs to understand the classification and invariants of highly-connected odd dimensional manifolds.

On the other hand, as the link $K$ is closely related to the singular point $z$ of the complex hypersurfaces, some of the topological invariants of $K$ are computable from the polynomial.

Let

Tex syntax error

$V$ be a complex hypersurface defined by $f(z_1, \dots, z_{n+1})$ $f(z_1, \dots, z_{n+1})$ , $z^0$ $z^0$ be an isolated singular point of

Tex syntax error

$V$ . Let $g_j=\partial f /\partial z_j$ $g_j=\partial f /\partial z_j$ $j=1, \dots, n+1$ $j=1, \dots, n+1$ . By putting all these $g_j$ $g_j$ 's together we get the gradient field of $f$ $f$ , which can be viewed as a map $g \colon \mathbb C^{n+1} \to \mathbb C^{n+1}$ $g \colon \mathbb C^{n+1} \to \mathbb C^{n+1}$ , $z \mapsto (g_1(z), \dots, g_{n+1}(z))$ $z \mapsto (g_1(z), \dots, g_{n+1}(z))$ . If $z^0$ $z^0$ is an isolated singular point, then $z \mapsto g(z)/||g(z)||$ $z \mapsto g(z)/||g(z)||$ is a well-defined map from a small sphere $S_{\epsilon}$ $S_{\epsilon}$ centered at $z^0$ $z^0$ to the unit sphere $S^{2n+1}$ $S^{2n+1}$ of $\mathbb C^{n+1}$ $\mathbb C^{n+1}$ . The mapping degree $\mu$ $\mu$ is called the multiplicity of the isolated singular point $z^0$ $z^0$ . ( $\mu$ $\mu$ is also called the Milnor number of $z^0$ $z^0$ .)

The middle homology group $H_n(F_{\theta})$ is a free abelian group of rank $\mu$ .

Furthermore, the homology groups of the link $K$ are determined from the long homology exact sequence

$\displaystyle \cdots \to H_n(\overline{F_{\theta}}) \stackrel{j_*}{\rightarrow} H_n(\overline{F_{\theta}}, K) \stackrel{\partial}{\rightarrow} \widetilde{H}_{n-1}(K) \to 0$ $\displaystyle \cdots \to H_n(\overline{F_{\theta}}) \stackrel{j_*}{\rightarrow} H_n(\overline{F_{\theta}}, K) \stackrel{\partial}{\rightarrow} \widetilde{H}_{n-1}(K) \to 0$

of the pair $(\overline{F_{\theta}},K)$ . The map $j_*$ is the adjoint of the intersection pairing on $\overline{F_{\theta}}$

$\displaystyle s \colon H_n(\overline{F_{\theta}}) \otimes H_n(\overline{F_{\theta}}) \to \mathbb Z.$ $\displaystyle s \colon H_n(\overline{F_{\theta}}) \otimes H_n(\overline{F_{\theta}}) \to \mathbb Z.$

A symmetric or skew symmetric bilinear form on a free abelian group is usually referred to as a lattice. $s$ is called the Milnor lattice of the singular point. Thus the homology groups of the link $K$ is completely determined by the Milnor lattice of the singular point.

[edit] 4 Topological spheres as links of singular points

Especially, the link $K$ is an integral homology sphere if and only if the intersection form $s$ is unimodular, i.~e.~the matrix of $s$ has determinant $\pm1$ . If $n\ge 3$ , the Generalized Poincare Conjecture implies that $K$ is a topological sphere.

By Theorem 2.2, there is a smooth fiber bundle over $S^1$ with fiber $F_{\theta}$ . The natural action of a generator of $\pi_1(S^1)$ induces the characteristic homeomorphism $h$ of the fiber $F_0=\phi^{-1}$ . Let $h_* \colon H_n(F_0) \to H_n(F_0)$ be the induced isomorphism on homology and $\Delta(t)=\det(tI-h_*)$ be the characteristic polynomial of the linear transformation $h_*$ . It's a consequence of the Wang sequence associated with the fiber bundle over $S^1$ that

For $n \ne 2$ the manifolds $K$ is a topological sphere is and only if the integer $\Delta(1)=\det(I-h_*)$ equals to $\pm 1$ .

When $K$ is a topological sphere, as it is the boundary of an $(n-1)$ -connected parallelisable $2n$ -manifold $\overline{F_0}$ , our knowledge of exotic spheres allows us to determine the diffeomorphism class of $K$ completely:

if $n$ is even, the diffeomorphism class of $K$ is determined by the signature of the intersection pairing

$\displaystyle s \colon H_n(\overline{F_0}) \otimes H_n(\overline{F_0}) \to \mathbb Z$ $\displaystyle s \colon H_n(\overline{F_0}) \otimes H_n(\overline{F_0}) \to \mathbb Z$

if $n$ is odd, the diffeomorphism class of $K$ is determined by the Kervaire invariant

$\displaystyle c(\overline{F_0}) \in \mathbb Z_2$ $\displaystyle c(\overline{F_0}) \in \mathbb Z_2$

which was computed in [Levine1966]

$\displaystyle c(\overline{F_0})=0 \ \ \textrm{if} \ \ \Delta(-1) \equiv \pm 1 \pmod 8$ $\displaystyle c(\overline{F_0})=0 \ \ \textrm{if} \ \ \Delta(-1) \equiv \pm 1 \pmod 8$

$\displaystyle c(\overline{F_0})=1 \ \ \textrm{if} \ \ \Delta(-1) \equiv \pm 3 \pmod 8$ $\displaystyle c(\overline{F_0})=1 \ \ \textrm{if} \ \ \Delta(-1) \equiv \pm 3 \pmod 8$

[edit] 5 Examples