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In Singular Points of Complex Hypersurfaces, Milnor shows that if $f: \mathbb{C}^{n+1} \rightarrow \mathbb{C}$, holomorphic, has an isolated singularity at the origin, then $f^{-1}(\epsilon) \cap B^{2n+2}_\delta$, which is a smooth manifold, has the homotopy type of a wedge of n-spheres for some sufficiently small $\delta$; the number of spheres is the Milnor number of the singularity and the f is smooth if and only if $f^{-1}(\epsilon) \cap B^{2n+2}_\delta$ is diffeomorphic $B^{2n}_\delta$.

I am wondering what the topology of more general affine varieties with isolated singularities is. More precisely, consider $f_i: \mathbb{C}^{n+1} \rightarrow \mathbb{C}, 1\le i \le k$, holomorphic for some $k < n$ such that the Jacobian matrix $\partial f_i /\partial x_j$ has full rank everywhere in a small neighborhood of the origin, except for the origin. Let $V = \{f_i = \epsilon \mbox{ all i} \} \cap B^{2n+2}_\delta$, which is a smooth manifold. What can be said about the homotopy type of $V$? Can someone provide examples of $V$ with interesting topology when $k > 1$?

I thought I proved that for $k > 1$, $V$ is always diffeomorphic to $B^{2n}$ but this seems extremely suspicious. Counterexamples would be much appreciated.

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  • $\begingroup$ Are you familiar with Golubitsky and Guillemin's "Stable Mappings and their Singularities"? It's more elementary than what you're describing here and is mostly concerned with the real case, but it might be a decent place to start. $\endgroup$ – anomaly Feb 13 '16 at 22:41
  • $\begingroup$ Also, although it's mostly a tangent, are you familiar with the Brieskorn spheres? $\endgroup$ – anomaly Feb 13 '16 at 22:48
  • $\begingroup$ @anomaly I'm not familiar with "Stable Mappings and their Singularities"; I'll take a look. Yes, Brieskorn spheres are the boundary of certain Milnor fibers of certain polynomials; this is the $k = 1$ case and I am interested in $k > 1$. $\endgroup$ – user39598 Feb 13 '16 at 23:46
  • $\begingroup$ Right; my thought was just that it might be a useful source of possible counterexamples (in diffeomorphism category, if not the homotopy one) for the $k > 1$ case. $\endgroup$ – anomaly Feb 14 '16 at 0:24
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Your hypothesis on the rank of the jacobian implies that $f^{-1}(0)$ is an isolated complete intersection singularity (ICIS) of dimension n-k. H. Hamm (Topology of isolated singularities of complex spaces. Proceedings of Liverpool Singularities Symposium, II (1969/1970), 213–217. Lecture Notes in Math., Vol. 209, Springer, Berlin, 1971) showed that the Milnor fibre of an ICIS (a nearby non-singular level set intersected with a suitably small ball) has the homotopy type of a wedge of spheres of middle dimension (i.e. n-k in this case). The number of spheres is called the Milnor number of the ICIS. See the book of Looijenga (Isolated Singular Points on Complete Intersections, London Math Soc Lecture Notes No. 77, 1984) for a modern treatment.

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