How to calculate the exact differential structure of Brieskorn variety?

Question

As Kervaire and Milnor mentioned, an $n$-dim exotic sphere $\Sigma$ which bounds a parallelizable manifold $M$ is totally classified by the signature $\sigma(M)$ modulo the order of $bP_{n+1}$.

Let $n=2m$ be an even integer. Brieskorn had discovered that the singularity of complex hypersurfaces $V_k$, the zero set of $x_0^2+\cdots+x_{n-2}^2+x_{n-1}^3+x_n^{6k-1}=0$ has close relationship with exotic spheres. More precisely, if $\varepsilon>0$ is sufficiently small, let $S_\varepsilon$ be the $(2n+1)$-sphere with center $0$ and radius $\varepsilon$, then $\Sigma_k=S_\varepsilon\cap V_k$ is an exotic sphere, and actually every exotic sphere of dimension $(4m-1)$ which bounds a parallelizable manifold can be obtained in this way.

I want to know which element $\Sigma_k$ represents in $bP_{2n}$. In other words, I want to calculate the signature of the Milnor fibre. Since Brieskorn's original paper was written in German, I couldn't read it. Instead, I've read the papar 'Singularity and Exotic Sphere' written by Hirzebruch. In this paper, Hirzebruch gave the answer: actually $\Sigma_k$ represents $k$th multiple of the generator of $bP_{2n}$. However, he refered the proof to Brieskorn's paper.

Does anyone know a proof? Please tell me, thanks.

Answer 1

An english discussion of Brieskorn exotic spheres can be found in these slides of Ranicki, there is also a collection of links concerning exotic spheres here.

I can give you the gist of Brieskorn's argument, I hope this may help:

First, setting $a=(a_1,\dots,a_n)$, an explicit parallelizable manifold is given by $M_a=\Xi_a\cap D^{2n}$, $\Xi_a=\{(z_1,\dots,z_n)\in\mathbb{C}^n\mid \sum_{i=1}^n z_i^{a_i}=1\}$ and as usual $D^{2n}=\{(z_1,\dots,z_n)\in\mathbb{C}^n\mid \sum_{i=1}^n|z_i|^2\leq 1\}$. The intersection form for $\Xi_a$ is computed in F. Pham: Formules de Picard-Lefschetz généralisées et ramification des integrales. BSMF 93 (1965), 333-367. Then the steps of the proof are as follows:

1) there are automorphisms $\omega_k$ given by multiplying the $k$-th coordinate with $e^{2\pi i/a_k}$. these generate a finite automorphism group $\Omega_a\cong\prod_{i=1}^n\mathbb{Z}/(a_i)$, let $J_a=\mathbb{Z}[\Omega_a]$ be the group ring and $I_a$ the ideal generated by the elements $1+\omega_k+\cdots+ \omega_k^{a_k-1}$. Pham identified $H_{n-1}(\Xi_a,\mathbb{Z})\cong J_a/I_a$.

2) $H_{n-1}(\Xi_a,\mathbb{C})$ has a basis $v_j=\prod_{k=1}^n\sum_{r=0}^{a_k-1}e^{2\pi i j_k r/a_k}\omega_k^r$. In the basis $v_j+v_{a-j}$ and $i(v_j-v_{a-j})$ of $J_a/I_a\otimes\mathbb{R}$, the intersection form is diagonalized. The matrix of the intersection form can be found on p. 359 of Pham's paper, and implies $\langle v_j+v_{a-j},v_j+v_{a-j}\rangle=\langle i(v_j-v_{a-j}),i(v_j-v_{a-j})\rangle=2\langle v_j,v_{a-j}\rangle$ (and $0$ otherwise).

3) The final computation shows that $\langle v_j,v_{a-j}\rangle>0$ if and only if $0<\sum \frac{j_k}{a_k}<1\mod 2$, and $\langle v_j,v_{a-j}\rangle<0$ if and only if $-1<\sum\frac{j_k}{a_k}<0\mod 2$.

This implies the signature calculation, Satz 3 in Brieskorn's paper resp. the Theorem on p. 20 of Hirzebruch's paper "Singularities and exotic spheres". Actually, Brieskorn credits Hirzebruch for the result and proof.