Probabalistic questions about singularities and exotic spheres

Question

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?

Answer 1

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 $0.286$.