Questions in category: 代数拓扑 (Algebraic Topology)

## 1. AHS (Alexander Horned Sphere)

Posted by haifeng on 2022-11-20 08:14:55 last update 2022-11-20 09:34:59 | Answers (0) | 收藏

AHS (Alexander Horned Sphere)

J. W. Alexander 构造了一个单连通曲面, 其(外面)所围区域非单连通. 文章很短, 见 [1].

References:

[1]  J. W. Alexander, Proceedings of the National Academy of Sciences of the United States of America, Jan. 15, 1924, Vol. 10, No. 1 (Jan. 15, 1924), pp. 8--10.

## 2. 三维欧氏空间中一个多面体假设有 $7n$ 个面, 证明: 存在 $n+1$ 个面, 这 $n+1$ 个面有相同的边数.

Posted by haifeng on 2022-06-10 15:17:06 last update 2022-06-10 15:50:59 | Answers (1) | 收藏

[Hint]  利用欧拉公式.

$V-E+F=2.$

Bilibili,  Up:  RealFiddie

## 3. 构建开球 $\mathrm{int}(D^n)$ 到 $\mathbb{R}^n$ 的同胚映射.

Posted by haifeng on 2022-01-24 15:47:00 last update 2022-01-25 21:26:35 | Answers (1) | 收藏

$B^n=\{x\in\mathbb{R}^n\mid |x| < 1\},$

[Hint] 利用 $y=\tan x$, $x\in(-\frac{\pi}{2},\frac{\pi}{2})$, 可以构建 $(-1,1)$ 到 $\mathbb{R}$ 的同胚映射:

$\begin{array}{rcl} \varphi:\ (-1,1)&\rightarrow&\mathbb{R}\\ x&\mapsto&\tan(\frac{\pi}{2}x) \end{array}$

## 4. CW 复形(CW complex)

Posted by haifeng on 2022-01-19 19:34:33 last update 2022-01-19 19:35:50 | Answers (0) | 收藏

CW 复形(CW complex)

$\mathbb{E}^n=(\mathbb{R}^n,d)$ 是 $n$ 维欧氏空间, 当然是度量空间, 从而也是拓扑空间. $\mathbb{E}^n$ 中的一个 $n$-维(闭)球是指

$D^n=\{x\in\mathbb{E}^n\mid |x|\leqslant 1\}$

References:

Soren Hansen,  CW complexes.

## 5. The Eckmann–Hilton argument

Posted by haifeng on 2017-03-04 15:16:43 last update 2017-03-04 15:16:43 | Answers (0) | 收藏

The Eckmann–Hilton argument

https://ncatlab.org/nlab/show/Eckmann-Hilton+argument

## 6. [吴文俊] $S^{4k}$ 不是近复流形.

Posted by haifeng on 2016-11-03 22:58:20 last update 2016-11-03 22:58:20 | Answers (0) | 收藏

[吴文俊] $S^{4k}$ 不是近复流形.

References:

S. S. Chern 93岁时的演讲

https://www.zhihu.com/question/52146571

## 7. 利用拓扑 K 理论证明: 若 $n > 3$, 则 $S^{2n}$ 上不存在近复结构.

Posted by haifeng on 2016-11-03 22:49:52 last update 2016-11-03 22:49:52 | Answers (0) | 收藏

## 8. 证明: $\mathbb{R}^n$ 与 $\mathbb{R}^m$ 同胚仅当 $n=m$.

Posted by haifeng on 2015-07-19 12:04:16 last update 2015-07-19 12:04:16 | Answers (1) | 收藏