Link of a singularity

Remarks

For $\varepsilon$ small, $V \cap B_\varepsilon(p)$, where $B_\varepsilon(p)$ is the closed ball, is homeomorphic to the cone over $K_\varepsilon$ with vertex $p$; this is the conic structure theorem. If $V$ is smooth at $p$ of complex dimension $d$, the link is a sphere $S^{2d - 1}$. If $V = f^{-1}(0) \subset \C^{n+1}$ with an isolated singularity at $p$, the link is a smooth closed manifold of dimension $2n - 1$, and Milnor showed that it is $(n-2)$-connected.

Example: $V = \set{x^2 = y^3} \subset \C^2$ at $0$. The map $t \mapsto (t^3, t^2)$ is a bijection from $\C$ onto $V$. On the sphere $S_\varepsilon$, the points of $V$ are $(t^3, t^2)$ with $\abs{t}$ fixed, which trace the torus knot of type $(2, 3)$, the trefoil.

Non-example of the condition on $\varepsilon$: $V = \set{x y (y - 1) = 0} \subset \C^2$ at $0$. For $\varepsilon < 1$, $K_\varepsilon$ consists of the two circles cut out by $x = 0$ and $y = 0$, a Hopf link. For $\varepsilon > 1$, the line $y = 1$ also meets $S_\varepsilon$ in a third circle, so this sphere is too large to give the link.