For an isolated plane curve singularity, given by homogeneous equation $F=0 \subset \mathbb{C}^2$, one consider the curve $(F=0) \cap S^3 \subset S^3$, and we call it the link of singularity. some properties of the singularity are embedded in the corresponding link and one can start studying the singularity by studyng the invariants of the corresponding link, like Jones polynomial ...

Question: Is there any similar construction for more general singularities? Like when $X$ is an irreducible singular variety and the singular locus $X^{sing}$ is isomorphic to a smooth variety $V$?

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Yes, these sorts of things have been considered. For example, if $X$ is locally embedded into $\mathbb{C}^n$, the intersection of $X$ with a small sphere centred at an isolated singularity $p\in X$ is again called the link. Of course, there is a much longer story which someone might answer. – Donu Arapura Apr 4 '11 at 16:33
I think this method of studying algebraic varieties goes back at least as far as Wirtinger's work in 1895. – Ryan Budney Apr 4 '11 at 18:18

There are lots of results on the topology of the Milnor fibre. For example if $f : (\mathbb{C}^n,0) \to (\mathbb{C},0)$ is a holomorphic map germ with an isolated critical point at $0 \in \mathbb{C}^n$ then the Milnor fibre is homotopy equivalent to the bouquet of $\mu$-spheres, where $\mu$ denotes the Milnor number. The Milnor number is given by the absolute value of the Poincaré-Hopf index of the gradient vector field $\nabla f$ at $0 \in \mathbb{C}^n.$