MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I was looking a bit at the connection between singularities of complex algebraic varieties and exotic spheres. I find it quite remarkable that you can obtain all 28 differentiable structures on the 7 sphere by intersecting a small sphere with some very innocent looking algebraic variety. It makes me wonder how common that is: if you take a "random" algebraic variety with a singularity at the origin in $\mathbb{C}^5$, and intersect it with a small sphere, what are the chances you get an exotic structure on the 7 sphere? Are some smooth structures more likely to appear than others? Has anyone actually computed probabilities?

share|cite|improve this question
You might wish to expand on the procedure(s) you have in mind for drawing at random (without the scare quotes) such an algebraic variety. – Did Feb 14 '12 at 10:10

Part of the problem is lack of precise probabilistic model for this question (i.e., what is a "random" singularity?). You also have to restrict to isolated hypersurface singularities in ${\mathbb C}^n$. The question starts to make sense if you further restrict to Brieskorn singularities of the form $$ z_0^{a_0}+...+ z_n^{a_n}=0 $$ where all $a_k\ge 2$ are integers. This situation was analyzed in great detail, for instance, in Milnor's book "Singular points of complex hypersurfaces," sections 8 and 9. (See also Hirzebruch's paper accessible here.) For isolated singularities, provided that $n\ge 3$, the link $K$ of the singularity is a topological sphere $\iff$ it is an (integer) homology sphere $\iff \Delta(1)=\pm 1$ (theorem 8.5 in Milnor's book). Furthermore, say, for odd $n$, one can decide if $K$ is the exotic or not by looking at $\Delta(-1)$ mod $8$. For Brieskorn singularities, $\Delta$ is quite computable, so you can get your hands dirty and start computing probabilities. It is also, probably, easier, to use for this computation the graph-theoretic criterion on page 18 of Hirzebruch's paper.

I looked at the case when $n=2$ and the relevant question is about the probability of $K$ to be an integer homology sphere, equivalently, the numbers $a_0, a_1, a_2$ being pairwise coprime. Such probabilities were computed here and the answer is approximately $0.286$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.