When is a Brieskorn manifold diffeomorphic to an exotic sphere?

Question

This is a related question but what I'm asking here is more general. I find the wikipedia on Brieskorn manifolds a bit vague. The manifolds are defined to be the intersection of a "small sphere" centered at the origin with the variety $$x_1^{k_1}+...+x_n^{k_n}=0$$ After a bit of looking around on this website and overflow I found out that the $x_i$ are complex. My question is: what constraints on $k_i$ must there be and how "small" must the sphere with which we intersect the variety be for the Brieskorn manifold to be diffeomeophic to an exotic sphere?

Answer 1

The technical condition is that the sphere intersects transversally your variety, see the first pages of the book "Singular points of complex hypersurfaces" by J. Milnor. I think any sphere should do in your special case.

The $k_i$ should be all integers $\geq 2$, else you just get a usual sphere. Indeed locally you can assume your variety is given by say $z_1 = 0$ so the intersection $S \cap \{z_0 = 0\}$ is the sphere $|z_1|^2 + \dots + |z_n|^2 = 1$.

The simplest non trivial example is the cusp $y^2 = x^3$. Here, the Brieskorn sphere will be a circle, however embedded as a knot in a torus. More precisely let us look a the intersection of the curve with a sphere of radius $\varepsilon > 0$. We obtain the equation $|x|^2 + |y|^2 = \varepsilon, y^2 = x^3$. From this, it's easy to see that the modulus of $x$ and $y$ should be constant, say $\xi(\varepsilon)$ for $x$ and $\eta(\varepsilon) $ for $y$. This is exactly the equation of a torus. But in fact, the equation $y^2 = x^3$ tells us that $3\theta(y) = 2\theta(x)$. This means exactly that you get a torus knot of type $(3,2)$ inside a torus !

More generally, for small $\varepsilon$ let $X_{\varepsilon} = X \cap S_{\varepsilon}$ and $X_{\leq \varepsilon} = X \cap \overline B(0, \varepsilon)$. There is an homeomorphism of pair $(X_{\leq \varepsilon}, X_{\varepsilon} ) \cong (Cone (X_{\varepsilon}), X_{\varepsilon})$ ! This is a very stricking theorem. In particular the full topology of a neighorhood of a singular point is contained in the link $X_{\varepsilon}$. In the previous case it holds for any $\varepsilon > 0$ by construction, and I think the same should holds for the singularities you mentioned.