Link of a singularity

Question

I would like to understand the topological type of a link of a singularity in a simple example. Consider for instance the cone ${xy-z^2=0}\subset\mathbb{C}^3$. If we set $x = x_1+ix_2, y = y_1+iy_2, z = z_1+iz_2$ then this corresponds to the $4$-dimensional real subvariety of $\mathbb{R}^6$ given as the complete intersection $\{x_1y_1-x_2y_2-z_1^2+z_2^2 = x_1y_2+x_2y_1-2z_1z_2 = 0\}$.

Therefore the link of the singularity is given by

$\{x_1y_1-x_2y_2-z_1^2+z_2^2 = 0, x_1y_2+x_2y_1-2z_1z_2 = 0, x_1^2+x_2^2+y_1^2+y_2^2+z_1^2+z_2^2 = 1\}$

What is the topological type of this link?

Can we determine the topological type of the link from these three equations?

Answer 1

More generally, consider the singularity given by $$x_1^2+\cdots+x_{n+1}^2=0$$ in $\mathbf{C}^{n+1}$. (Your case is $n=2$ after a change of variables.) Identifying $\mathbf{C}^{n+1}=\mathbf{R}^{n+1}\times\mathbf{R}^{n+1}$ we see at once that the link is the Stiefel manifold $\mathrm{V}_2(\mathbf{R}^{n+1})$ of pairs of orthonormal vectors in $\mathbf{R}^{n+1}$. (For $n=2$ this is $\mathrm{SO}(3)$, which is homeomorphic to $\mathbf{RP}^3$.) If you allow more complicated exponents in the defining equation of the singularity, the link can be very interesting. For example, the links of the singularities defined by $$x_1^2+x_2^2+x_3^2+x_4^3+x_5^{6k-1}=0 \ \ \ (1\leqslant k\leqslant 28)$$ give all 28 differentiable structures on $S^7$. (See for example E. Brieskorn's classic paper Beispiele zur Differentialtopologie von Singularitäten).

Answer 2

For singularities of the form $g(x,y)+z^n = 0$ there is a nice description: if you project onto the $xy$-plane (and you take a very small neighbourhood of the origin), you can view the link of the (surface) singularity as the cyclic $n$-fold cover of $S^3$ branched over the link of the (curve) singularity determined by $g$.

In the case at hand, $g(x,y) = xy$ and $n=2$, so its link is the Hopf link in $S^3$, and the double cover of $S^3$ branched over it is the lens space $L(2,1)$ (also known as $\mathbb{RP}^3$). More generally, for arbitrary $n$ (and the same $g$), you get the lens space $L(n,n-1)$ ($+n$-surgery on the unknot).

Answer 3

To add to the excellent answers already provided, here are some general facts in the case of rational surface singularities (1 and 2) and hypersurface singularities (3).

1. Many interesting singularities are obtained by quotienting $C^2$ by the action of a finite subgroup $G\subset U(2)$. Since $U(2)$ preserves the unit sphere, you can get the link just by quotienting $S^3$ by $G$. The example you gave is equivalent to the singularity you get by taking the quotient of $C^2$ by the subgroup $G=\{I,-I\}$, so that tells you the link is $RP^3$, as Marco pointed out. For cyclic quotient singularities, you'll get a lens space. There are more exotic examples, for example, for the spin double cover of the symmetry group of the icosahedron (under the covering map $SU(2)\to SO(3)$), the link of the quotient singularity will be the Poincaré homology sphere. If you want to understand these singularities as affine varieties, they're obtained by taking Spec of the ring of invariants for the finite group action on the ring of polynomial functions on $C^2$ (for $G=\{I,-I\}$the invariant functions are generated by $u=x^2,v=xy,w=y^2$ with the obvious relation $uw=v^2$, which is where your equation comes into it).

2. Take the minimal resolution of the singularity. Let's suppose the exceptional divisor is a tree T of spheres with negative self-intersection numbers. A neighbourhood of the exceptional divisor is then a plumbing (according to the tree) of disc bundles over spheres with Euler numbers given by these self-intersection numbers. This has a surgery description: you take a bunch (one for each vertex of T) of unknots in $S^3$ such that two of them link like a Hopf link if they are connected by an edge in T (and are pairwise unlinked otherwise). Give each unknot a framing of minus the self-intersection of the corresponding sphere. This is both a Kirby diagram of the minimal resolution and a Dehn surgery presentation of the link (the exceptional curves in the minimal resolution are given by slice discs for the unknots capped off by the cores of the 2-handles you attach).

3. If you have a complex hypersurface singularity (i.e. one defined by a single equation over C) then you can use the ideas of Milnor to get at the topology of the link (see his book on isolated singularities of complex hypersurfaces). This will give you an "open book decomposition" of the link, i.e. you end up finding a collection of knots (called the "binding") in the link of the singularity whose complement fibres over the circle (the fibres are real surfaces called the "pages"). In higher dimensions, the binding is a real codimension 2 contact submanifold of the link and the pages are real codimension 1 Stein domains.