# Links of singular points of complex hypersurfaces

## 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$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}3$-dimensional sphere $S^3$$S^3$.

## 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 $V$$V$ is the algebraic set consisting of points $z=(z_1, \dots, z_{n+1})$$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 $V$$V$.

Near a regular point $z$$z$, the complex hypersurface $V$$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 $V$$V$ is more complicated. One way to look at the topology near $z$$z$, due to Brauner, is to look at the intersection of $V$$V$ with a $(2n+1)$$(2n+1)$-dimensioanl sphere of small radius $\epsilon$$\epsilon$ $S_{\epsilon}$$S_{\epsilon}$ centered at $z$$z$.

Theorem 2.1. The space $K=V\cap S_{\epsilon}$$K=V\cap S_{\epsilon}$ is $(n-2)$$(n-2)$-connected.

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

Theorem 2.2.(Fibration Theorem) For $\epsilon$$\epsilon$ sufficiently small, the space $S_{\epsilon}-K$$S_{\epsilon}-K$ is a smooth fiber bundle over $S^1$$S^1$, with projection map $\phi \colon S_{\epsilon}-K \to S^1$$\phi \colon S_{\epsilon}-K \to S^1$, $z \mapsto f(z)/|f(z)|$$z \mapsto f(z)/|f(z)|$. Each fiber $F_{\theta}$$F_{\theta}$ is parallelizable and has the homotopy type of a finite CW-complex of dimension $n$$n$.

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

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

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

Theorem 2.3. Each fiber $F_{\theta}$$F_{\theta}$ is a smooth parallelizable manifold, the closure $\overline{F_{\theta}}$$\overline{F_{\theta}}$ has boundary $K$$K$ and the homotopy type of a bouquet of $n$$n$-spheres $S^n\vee \cdots \vee S^n$$S^n\vee \cdots \vee S^n$.

## 3 Invariants

Seen from the above section, the link $K$$K$ of an isolated singular point $z$$z$ of a complex hypersurface $V$$V$ of complex dimension $n$$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)$$(2n-1)$-dimensional closed manifold $M/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_tJK8Zw$$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$$K$ is closely related to the singular point $z$$z$ of the complex hypersurfaces, some of the topological invariants of $K$$K$ are computable from the polynomial.

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

Theorem 3.1. The middle homology group $H_n(F_{\theta})$$H_n(F_{\theta})$ is a free abelian group of rank $\mu$$\mu$.

Furthermore, the homology groups of the link $K$$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)$$(\overline{F_{\theta}},K)$. The map $j_*$$j_*$ is the adjoint of the intersection pairing on $\overline{F_{\theta}}$$\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$$s$ is called the Milnor lattice of the singular point. Thus the homology groups of the link $K$$K$ is completely determined by the Milnor lattice of the singular point.

## 4 Topological spheres as links of singular points

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

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

Lemma 4.1. For $n \ne 2$$n \ne 2$ the manifolds $K$$K$ is a topological sphere is and only if the integer $\Delta(1)=\det(I-h_*)$$\Delta(1)=\det(I-h_*)$ equals to $\pm 1$$\pm 1$.

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

• if $n$$n$ is even, the diffeomorphism class of $K$$K$ is determined by the signature of the intersection pairing
$\displaystyle s \colon H_n(\overline{F_0}) \otimes H_n(\overline{F_0}) \to \mathbb Z$
• if $n$$n$ is odd, the diffeomorphism class of $K$$K$ is determined by the Kervaire invariant
$\displaystyle c(\overline{F_0}) \in \mathbb Z_2$

which was computed in [Levine1966]

$\displaystyle c(\overline{F_0})=0 \ \ \textrm{if} \ \ \Delta(-1) \equiv \pm 1 \pmod 8$
$\displaystyle c(\overline{F_0})=1 \ \ \textrm{if} \ \ \Delta(-1) \equiv \pm 3 \pmod 8$

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

\begin{thm}(Brieskorn-Pham) The group $H_n(F_{\theta})$$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$$\omega_j$ ranges over all $a_j$$a_j$-th root of unit other than $1$$1$. \end{thm}

The link $K$$K$ is called a Brieskorn variety.

For $a_1=\cdots=a_{n+1}=2$$a_1=\cdots=a_{n+1}=2$, it's seen from the defining equations that the link $K$$K$ is the sphere bundle of the tangent bundle of the $n$$n$-sphere, i.~e.~the Stiefel manifold $V_{n+1,2}(\mathbb R)$$V_{n+1,2}(\mathbb R)$.

The simplest nontrivial example is $a_1=\cdots =a_n=2$$a_1=\cdots =a_n=2$, $a_{n+1}=3$$a_{n+1}=3$. Then $\omega_1 = \cdots \omega_n=-1$$\omega_1 = \cdots \omega_n=-1$, $\omega_{n+1}=(-1\pm \sqrt{3})/2$$\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$$n=2k+1$ we have $\Delta(1)=1$$\Delta(1)=1$ so the link $K$$K$ is a topological sphere of dimension $4k+1$$4k+1$; $\Delta(-1)=3$$\Delta(-1)=3$, thus by [Levine1966] $K$$K$ has nontrivial Kervaire invariant. Especially for $k=2$$k=2$ $K^9$$K^9$ is the Kervaire sphere.

The above example is a special case of the $A_k$$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$$k$ being an integer $\ge 1$$\ge 1$. The Milnor lattice of an $A_k$$A_k$-singularity is represented by the Dynkin diagram of the simple Lie algebra $A_k$$A_k$. When $n=3$$n=3$, the diffeomorphism classification of the link $K$$K$ is obtained from its Milnor lattice and the classification of simply-connected 5-manifolds:

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

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

## 6 Further discussion

The link $K$$K$ of a singular point $z$$z$ is the intersection of the hypersurface $V$$V$ defined by $f$$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

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

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