Links of singular points of complex hypersurfaces

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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}}.
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 $3$-dimensional sphere $S^3$.
+
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 $3$-dimensional sphere $S^3$.
</wikitex>
</wikitex>
== Construction and examples ==
+
== Construction and properties==
<wikitex>;
<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 $V$.
+
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 $2n$; 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 $z$.
Near a regular point $z$, the complex hypersurface $V$ is a smooth manifold of real dimension $2n$; 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 $z$.
== Properties of links ==
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\begin{theorem}
+
\begin{thm}
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The space $K=V\cap S_{\epsilon}$ is $(n-2)$-connected.
The space $K=V\cap S_{\epsilon}$ is $(n-2)$-connected.
\end{thm}
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\end{theorem}
The homeomorphism type of $K$ is independent of the small paremeter $\epsilon$, it is called the link of the singular point $z$.
+
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)
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\begin{theorem}(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 $n$.
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}
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\end{theorem}
The fiber $F_{\theta}$ is usually called the Milnor fiber of the singular point $z$.
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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 $z$.
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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
In this special situation, the above theorems are strengthened to the following
\begin{thm}
+
\begin{theorem}
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 $S^n\vee \cdots \vee S^n$.
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}
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\end{theorem}
</wikitex>
</wikitex>
== Invariants ==
== Invariants ==
<wikitex>;
<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 a $(n-2)$-connected $(2n-1)$-dimensional closed smooth manifold. In high dimensional topology, these are called 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, for which see [[Category:Highly-connected manifolds|highly-connected manifolds]].
+
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.
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 $f(z_1, \dots, z_{n+1})$, $0$ 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 $g \colon \mathbb C^{n+1} \to \mathbb C^{n+1}$, $z \mapsto (g_1(z), \dots, g_{n+1}(z))$.
+
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{theorem}
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The middle homology group $H_n(F_{\theta})$ is a free abelian group of rank $\mu$.
+
\end{theorem}
+
+
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>
</wikitex>
== Classification ==
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== Topological spheres as links of singular points ==
<wikitex>;
<wikitex>;
...
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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}
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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 $2n$-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 [[Intersection form#Algebraic invariants|signature]] of the intersection pairing
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$$s \colon H_n(\overline{F_0}) \otimes H_n(\overline{F_0}) \to \mathbb Z$$
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* if $n$ is odd, the diffeomorphism class of $K$ is determined by the [[wikipedia:Kervaire invariant|Kervaire invariant]]
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$$c(\overline{F_0}) \in \mathbb Z_2$$
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which was computed in {{cite|Levine1966}}
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$$c(\overline{F_0})=0 \ \ \textrm{if} \ \ \Delta(-1) \equiv \pm 1 \pmod 8$$
+
$$c(\overline{F_0})=1 \ \ \textrm{if} \ \ \Delta(-1) \equiv \pm 3 \pmod 8$$
+
</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 $1$.
+
\end{thm}
+
+
The link $K$ is called a [[Exotic spheres#Brieskorn varieties|Brieskorn variety]].
+
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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
+
$$\Delta(t)=t^2-t+1 \ \ \textrm{for} \ \ n \ \ \textrm{odd}$$
+
$$\Delta(t)=t^2+t+1 \ \ \textrm{for} \ \ n \ \ \textrm{even}.$$
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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 {{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;
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* $K$ is diffeomorphic to $S^5$ if $k$ is even.
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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.
</wikitex>
</wikitex>
== Further discussion ==
== Further discussion ==
<wikitex>;
<wikitex>;
...
+
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}}).
</wikitex>
</wikitex>

Latest revision as of 11:54, 14 January 2013

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

Contents

[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 3-dimensional sphere S^3.

[edit] 2 Construction and properties

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 2n; 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.

Theorem 2.1. 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.

Theorem 2.2.(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

Theorem 2.3. 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 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 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.)

Theorem 3.1. 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

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.

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

Lemma 4.1. 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 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})=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}}

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.)

The characteristic polynomial is

\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{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}

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 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

\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

$ 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 $g \colon \mathbb C^{n+1} \to \mathbb C^{n+1}$, $z \mapsto (g_1(z), \dots, g_{n+1}(z))$. == Classification == ; ... == Further discussion == ; ... == References == {{#RefList:}} [[Category:Manifolds]]3-dimensional sphere S^3.

[edit] 2 Construction and properties

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 2n; 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.

Theorem 2.1. 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.

Theorem 2.2.(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

Theorem 2.3. 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 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 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.)

Theorem 3.1. 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

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.

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

Lemma 4.1. 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 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})=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}}

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.)

The characteristic polynomial is

\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{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}

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 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

\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

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