# 塞弗特－范坎彭定理

## 定理敍述

${\displaystyle X}$ 為拓撲空間，有兩個開且路徑連通的子空間${\displaystyle U_{1},U_{2}}$ 覆蓋${\displaystyle X}$ ，即${\displaystyle X=U_{1}\cup U_{2}}$ ，並且${\displaystyle U_{1}\cap U_{2}}$ 是非空且路徑連通。取${\displaystyle U_{1}\cap U_{2}}$ 中的一點${\displaystyle x_{0}}$ 為各空間的基本群的基點。那麼從${\displaystyle U_{1}\cap U_{2}}$ ${\displaystyle U_{1}}$ ${\displaystyle U_{2}}$ 包含映射導出相應基本群的群同態：（以下省略基本群中的基點。）

${\displaystyle \phi _{1}:\pi _{1}(U_{1}\cap U_{2})\to \pi _{1}(U_{1})}$
${\displaystyle \phi _{2}:\pi _{1}(U_{1}\cap U_{2})\to \pi _{1}(U_{2})}$

${\displaystyle \pi _{1}(X)=\pi _{1}(U_{1})*_{\pi _{1}(U_{1}\cap U_{2})}\pi _{1}(U_{2})}$

${\displaystyle \pi _{1}(U_{1})\leftarrow {\pi _{1}(U_{1}\cap U_{2})}\rightarrow \pi _{1}(U_{2})}$

• ${\displaystyle X}$ 為路徑連通拓撲空間，${\displaystyle x_{0}}$ ${\displaystyle X}$ 的一點，
• ${\displaystyle \{U_{\lambda }\}_{\lambda \in \Lambda }}$ 由路徑連通的開集組成，為${\displaystyle X}$ 的開覆蓋，
• 任何一個${\displaystyle U_{\lambda }}$ 都有點${\displaystyle x_{0}}$
• 對任何${\displaystyle \lambda ,\mu \in \Lambda }$ ，都有${\displaystyle \nu \in \Lambda }$ ，使得${\displaystyle U_{\lambda }\cap U_{\mu }=U_{\nu }}$

${\displaystyle U_{\lambda }\subset U_{\mu }}$ ，令

${\displaystyle \phi _{\lambda \mu }:\pi _{1}(U_{\lambda })\to \pi _{1}(U_{\mu })}$

${\displaystyle \psi _{\lambda }:\pi _{1}(U_{\lambda })\to \pi _{1}(X)}$

${\displaystyle H}$ 為群，對所有${\displaystyle \lambda \in \Lambda }$ 有群同態${\displaystyle \rho _{\lambda }:\pi _{1}(U_{\lambda })\to H}$ ，使得若${\displaystyle U_{\lambda }\subset U_{\mu }}$ ，則

${\displaystyle \rho _{\mu }\circ \phi _{\lambda \mu }=\rho _{\lambda }}$

${\displaystyle \rho _{\lambda }=\sigma \circ \psi _{\lambda }}$

## 參考

• Massey, William. A Basic Course in Algebraic Topology. Graduate Texts in Mathematics 127. Springer-Verlag. 1991.