Teorema de Seifert-van Kampen

Sea ${\displaystyle X}$ un espacio topológico, ${\displaystyle X=U_{1}\cup U_{2}}$, con ${\displaystyle U_{1},U_{2}}$ subconjuntos abiertos y conexos por caminos, tales que ${\displaystyle U_{1}\cap U_{2}\neq \varnothing }$ también es conexo por caminos. Sea ${\displaystyle x_{0}\in U_{1}\cap U_{2}}$.

Supongamos que conocemos los grupos fundamentales

${\displaystyle \pi _{1}(U_{1},x_{0})=\langle S_{1};R_{1}\rangle \,}$,

${\displaystyle \pi _{1}(U_{2},x_{0})=\langle S_{2};R_{2}\rangle \,}$ y

${\displaystyle \pi _{1}(U_{1}\cap U_{2},x_{0})=\langle S;R\rangle }$.

Entonces, ${\displaystyle \pi _{1}(X,x_{0})=\langle S_{1}\cup S_{2};R_{1}\cup R_{2}\cup \{(i_{1})_{*}(s)((i_{2})_{*}(s))^{-1}|s\in S\}\rangle }$, donde,
si ${\displaystyle i_{1}:U_{1}\cap U_{2}\rightarrow U_{1}}$ y ${\displaystyle i_{2}:U_{1}\cap U_{2}\rightarrow U_{2}}$ son las inclusiones naturales,
entonces ${\displaystyle (i_{1})_{*}}$ y ${\displaystyle (i_{2})_{*}}$ son las aplicaciones inducidas tales que

${\displaystyle (i_{1})_{*}:\pi _{1}(U_{1}\cap U_{2},x_{0})\rightarrow \pi _{1}(U_{1},x_{0})}$ que actúa ${\displaystyle [\alpha ]\rightarrow (i_{1})_{*}([\alpha ]):=[i_{1}\circ \alpha ]}$,

y análogamente

${\displaystyle (i_{2})_{*}:\pi _{1}(U_{1}\cap U_{2},x_{0})\rightarrow \pi _{1}(U_{2},x_{0})}$ que actúa ${\displaystyle [\alpha ]\rightarrow (i_{2})_{*}([\alpha ]):=[i_{2}\circ \alpha ]}$.