Sea
X
{\displaystyle X}
un espacio topológico ,
X
=
U
1
∪
U
2
{\displaystyle X=U_{1}\cup U_{2}}
, con
U
1
,
U
2
{\displaystyle U_{1},U_{2}}
subconjuntos abiertos y conexos por caminos , tales que
U
1
∩
U
2
≠
∅
{\displaystyle U_{1}\cap U_{2}\neq \varnothing }
también es conexo por caminos. Sea
x
0
∈
U
1
∩
U
2
{\displaystyle x_{0}\in U_{1}\cap U_{2}}
.
Supongamos que conocemos los grupos fundamentales
π
1
(
U
1
,
x
0
)
=
⟨
S
1
;
R
1
⟩
{\displaystyle \pi _{1}(U_{1},x_{0})=\langle S_{1};R_{1}\rangle \,}
,
π
1
(
U
2
,
x
0
)
=
⟨
S
2
;
R
2
⟩
{\displaystyle \pi _{1}(U_{2},x_{0})=\langle S_{2};R_{2}\rangle \,}
y
π
1
(
U
1
∩
U
2
,
x
0
)
=
⟨
S
;
R
⟩
{\displaystyle \pi _{1}(U_{1}\cap U_{2},x_{0})=\langle S;R\rangle }
.
Entonces,
π
1
(
X
,
x
0
)
=
⟨
S
1
∪
S
2
;
R
1
∪
R
2
∪
{
(
i
1
)
∗
(
s
)
(
(
i
2
)
∗
(
s
)
)
−
1
|
s
∈
S
}
⟩
{\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
i
1
:
U
1
∩
U
2
→
U
1
{\displaystyle i_{1}:U_{1}\cap U_{2}\rightarrow U_{1}}
y
i
2
:
U
1
∩
U
2
→
U
2
{\displaystyle i_{2}:U_{1}\cap U_{2}\rightarrow U_{2}}
son las inclusiones naturales,
entonces
(
i
1
)
∗
{\displaystyle (i_{1})_{*}}
y
(
i
2
)
∗
{\displaystyle (i_{2})_{*}}
son las aplicaciones inducidas tales que
(
i
1
)
∗
:
π
1
(
U
1
∩
U
2
,
x
0
)
→
π
1
(
U
1
,
x
0
)
{\displaystyle (i_{1})_{*}:\pi _{1}(U_{1}\cap U_{2},x_{0})\rightarrow \pi _{1}(U_{1},x_{0})}
que actúa
[
α
]
→
(
i
1
)
∗
(
[
α
]
)
:=
[
i
1
∘
α
]
{\displaystyle [\alpha ]\rightarrow (i_{1})_{*}([\alpha ]):=[i_{1}\circ \alpha ]}
,
y análogamente
(
i
2
)
∗
:
π
1
(
U
1
∩
U
2
,
x
0
)
→
π
1
(
U
2
,
x
0
)
{\displaystyle (i_{2})_{*}:\pi _{1}(U_{1}\cap U_{2},x_{0})\rightarrow \pi _{1}(U_{2},x_{0})}
que actúa
[
α
]
→
(
i
2
)
∗
(
[
α
]
)
:=
[
i
2
∘
α
]
{\displaystyle [\alpha ]\rightarrow (i_{2})_{*}([\alpha ]):=[i_{2}\circ \alpha ]}
.