1. 求 $\pi_1(S^n)$
Posted by haifeng on 2015-07-19 13:17:18 last update 2015-07-19 13:17:18 | Answers (0) | 收藏
$\pi_1(S^1)=\mathbb{Z}$,
$\pi_1(S^n)=0$, 对于 $n\geqslant 2$.
Posted by haifeng on 2015-07-19 13:17:18 last update 2015-07-19 13:17:18 | Answers (0) | 收藏
$\pi_1(S^1)=\mathbb{Z}$,
$\pi_1(S^n)=0$, 对于 $n\geqslant 2$.
Posted by haifeng on 2014-03-22 17:18:50 last update 2014-03-22 17:18:50 | Answers (0) | 收藏
求亏格为2的黎曼面的基本群
Posted by haifeng on 2013-03-01 15:25:39 last update 2013-03-01 15:30:24 | Answers (0) | 收藏
Def. 拓扑空间 $X$ 中的自由同伦回路所在的类 $\alpha$ 称为是“外围的“(peripheral), 如果对于 $X$ 中的任意紧子集 $R$, 都存在 $\alpha$ 类中的一个回路, 与 $R$ 不相交.
Reference:
[1] S. V. Buyalo, Euclidean planes in open 3-manifolds of nonpositive curvature, Algebra i Analiz 3 (1991), no.1, 102-117; English transl. in St.-Petersburg Math. J. 3 (1992).
[2] S. V. Buyalo, Three-manifolds with Cr-structure, Some questions of differential geometry in the large, Amer. Math. Soc. Transl. (2) Vol. 176, 1996
Posted by haifeng on 2012-07-20 15:22:53 last update 2012-07-20 15:22:53 | Answers (1) | 收藏
$\pi_1(\mathbb{R}P^n)=\mathbf{Z}_2$, $n\geqslant 2$.
Posted by haifeng on 2012-07-20 15:09:35 last update 2012-07-20 15:20:19 | Answers (1) | 收藏
在 $\mathbb{R}P^3$ 中定义关系:
\[P\sim Q\Leftrightarrow P=Q\ \text{或者}\ P=[x_1,x_2,x_3,x_4], Q=[-x_2,x_1,-x_4,x_3]\]
证明这是一个等价关系. 若设 $X=\mathbb{R}P^3/\sim$, 求 $\pi_1(X)$.