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Brieskorn showed that $X_{k}=\{(z_0, \dots, z_k)\in \mathbb{C}^{k+1}| -z_{0}^{2}+\sum_{i=1}^{k} z_{i}^{2}=0\}$(k odd, k>2) is a topological manifold. Is it a smooth manifold?

In general, let $a_1, \dots, a_n \in \mathbb{N}$, and $M(a_1, \dots, a_n)=\{(z_1, \dots, z_n)\in \mathbb{C}^{n}| \sum_{i = 1}^{n} z_{i}^{a_i}=0\}$.

When is $M(a_1, \dots, a_n)$ a topological manifold?

When is $M(a_1,\dots, a_n)$ a smooth manifold?

When is $M(a_1, \dots, a_n)$ a topological manifold but it has no smooth structure?

When does $M(a_1, \dots, a_n)$ admit a smooth structure but the smooth structure is not unique?

Brieskorn showed that $X_{k}=\{(z_0, \dots, z_k)\in \mathbb{C}^{k+1}| -z_{0}^{3}+\sum_{i=1}^{k} z_{i}^{2}=0\}$(k odd, k>2) is a topological manifold. Is it a smooth manifold?

In general, let $a_1, \dots, a_n \in \mathbb{N}$, and $M(a_1, \dots, a_n)=\{(z_1, \dots, z_n)\in \mathbb{C}^{n}| \sum_{i = 1}^{n} z_{i}^{a_i}=0\}$.

When is $M(a_1, \dots, a_n)$ a topological manifold?

When is $M(a_1,\dots, a_n)$ a smooth manifold?

When is $M(a_1, \dots, a_n)$ a topological manifold but it has no smooth structure?

When does $M(a_1, \dots, a_n)$ admit a smooth structure but the smooth structure is not unique?

Brieskorn showed that $X_{k}=\{(z_0, \dots, z_k)\in \mathbb{C}^{k+1}| -z_{0}^{2}+\sum_{i=1}^{k} z_{i}^{2}=0\}$(k odd) is a topological manifold. Is it a smooth manifold?

In general, let $a_1, \dots, a_n \in \mathbb{N}$, and $M(a_1, \dots, a_n)=\{(z_1, \dots, z_n)\in \mathbb{C}^{n}| \sum_{i = 1}^{n} z_{i}^{a_i}=0\}$.

When is $M(a_1, \dots, a_n)$ a topological manifold?

When is $M(a_1,\dots, a_n)$ a smooth manifold?

When is $M(a_1, \dots, a_n)$ a topological manifold but it has no smooth structure?

When does $M(a_1, \dots, a_n)$ admit a smooth structure but the smooth structure is not unique?

Brieskorn showed that $X_{k}=\{(z_0, \dots, z_k)\in \mathbb{C}^{k+1}| -z_{0}^{2}+\sum_{i=1}^{k} z_{i}^{2}=0\}$(k odd, k>2) is a topological manifold. Is it a smooth manifold?

In general, let $a_1, \dots, a_n \in \mathbb{N}$, and $M(a_1, \dots, a_n)=\{(z_1, \dots, z_n)\in \mathbb{C}^{n}| \sum_{i = 1}^{n} z_{i}^{a_i}=0\}$.

When is $M(a_1, \dots, a_n)$ a topological manifold?

When is $M(a_1,\dots, a_n)$ a smooth manifold?

When is $M(a_1, \dots, a_n)$ a topological manifold but it has no smooth structure?

When does $M(a_1, \dots, a_n)$ admit a smooth structure but the smooth structure is not unique?

Brieskorn showed that $X_{k}=\{(z_0, \dots, z_k)\in \mathbb{C}^{k+1}| -z_{0}^{2}+\sum_{i=1}^{k} z_{i}^{2}=0\}$ is a topological manifold. Is it a smooth manifold?

In general, let $a_1, \dots, a_n \in \mathbb{N}$, and $M(a_1, \dots, a_n)=\{(z_1, \dots, z_n)\in \mathbb{C}^{n}| \sum_{i = 1}^{n} z_{i}^{a_i}=0\}$.

When is $M(a_1, \dots, a_n)$ a topological manifold?

When is $M(a_1,\dots, a_n)$ a smooth manifold?

When is $M(a_1, \dots, a_n)$ a topological manifold but it has no smooth structure?

When does $M(a_1, \dots, a_n)$ admit a smooth structure but the smooth structure is not unique?

Brieskorn showed that $X_{k}=\{(z_0, \dots, z_k)\in \mathbb{C}^{k+1}| -z_{0}^{2}+\sum_{i=1}^{k} z_{i}^{2}=0\}$(k odd) is a topological manifold. Is it a smooth manifold?

In general, let $a_1, \dots, a_n \in \mathbb{N}$, and $M(a_1, \dots, a_n)=\{(z_1, \dots, z_n)\in \mathbb{C}^{n}| \sum_{i = 1}^{n} z_{i}^{a_i}=0\}$.

When is $M(a_1, \dots, a_n)$ a topological manifold?

When is $M(a_1,\dots, a_n)$ a smooth manifold?

When is $M(a_1, \dots, a_n)$ a topological manifold but it has no smooth structure?

When does $M(a_1, \dots, a_n)$ admit a smooth structure but the smooth structure is not unique?