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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 obtain 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.

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.

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 obtain in this way.

I want to know which element does $\Sigma_k$ 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 refers the proof to Brieskorn's paper.

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

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 obtain 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.